Rick Block (talk | contribs) →Mediation: my position |
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I have read the article and understand its subject matter and all it details. As I begin delving through the talk page archives, I'll open the discussion with a call for opening statements. If you feel any archived passages are significant in summarizing the situation, it would help to include links, but please conclude your first post with a '''Summary of Position''' (''your'' opinion as it relates to the matter). And remember...''concise'' ;-)<br />--[[User:K10wnsta|K10wnsta]] ([[User talk:K10wnsta|talk]]) 05:28, 29 December 2009 (UTC) |
I have read the article and understand its subject matter and all it details. As I begin delving through the talk page archives, I'll open the discussion with a call for opening statements. If you feel any archived passages are significant in summarizing the situation, it would help to include links, but please conclude your first post with a '''Summary of Position''' (''your'' opinion as it relates to the matter). And remember...''concise'' ;-)<br />--[[User:K10wnsta|K10wnsta]] ([[User talk:K10wnsta|talk]]) 05:28, 29 December 2009 (UTC) |
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===Nijdam's position=== |
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I want the article clearly mention the remark made by some sources that the so called "simple solution" is not complete. It doesn't need initially mentioning the technical term "conditional probability". To make my point clear: the following resoning: |
I want the article clearly mention the remark made by some sources that the so called "simple solution" is not complete. It doesn't need initially mentioning the technical term "conditional probability". To make my point clear: the following resoning: |
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:''The player, having chosen a door, has a 1/3 chance of having the car behind the chosen door and a 2/3 chance that it's behind one of the other doors. Hence when the host opens a door to reveal a goat, the probability of a car behind the remaining door must be 2/3.'' |
:''The player, having chosen a door, has a 1/3 chance of having the car behind the chosen door and a 2/3 chance that it's behind one of the other doors. Hence when the host opens a door to reveal a goat, the probability of a car behind the remaining door must be 2/3.'' |
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Something alike holds for the so called "combined doors solution" and most of the other simple ways of understanding. That's all.[[User:Nijdam|Nijdam]] ([[User talk:Nijdam|talk]]) 08:36, 29 December 2009 (UTC) |
Something alike holds for the so called "combined doors solution" and most of the other simple ways of understanding. That's all.[[User:Nijdam|Nijdam]] ([[User talk:Nijdam|talk]]) 08:36, 29 December 2009 (UTC) |
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===Martin Hogbin's position=== |
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The MHP is essentially a ''simple mathematical puzzle'' that most people get wrong. At least the first part of the article should concentrate on giving a simple, clear, and ''convincing'' solution that does not involve conditional probability. All diagrams and explanations in this section should not show or discuss the possible difference that the door opened by the host might make, although I would be happy to include, 'this action does not give the player any new information about what is behind the door she has chosen' as in Nijdam's second statement above. The first section should give aids to understanding and discuss why many people get the solution wrong, without the use of conditional probability. The first section should be supported by sources which do not mention conditional probability |
The MHP is essentially a ''simple mathematical puzzle'' that most people get wrong. At least the first part of the article should concentrate on giving a simple, clear, and ''convincing'' solution that does not involve conditional probability. All diagrams and explanations in this section should not show or discuss the possible difference that the door opened by the host might make, although I would be happy to include, 'this action does not give the player any new information about what is behind the door she has chosen' as in Nijdam's second statement above. The first section should give aids to understanding and discuss why many people get the solution wrong, without the use of conditional probability. The first section should be supported by sources which do not mention conditional probability |
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The simple solution section should be followed by an explanation of why some formulations of the problem require the use of conditional probability, with reference to the paper by Morgan et al. and other sources. It should also include the various variations of the basic problem and other, more complex, issues. [[User:Martin Hogbin|Martin Hogbin]] ([[User talk:Martin Hogbin|talk]]) 10:19, 29 December 2009 (UTC) |
The simple solution section should be followed by an explanation of why some formulations of the problem require the use of conditional probability, with reference to the paper by Morgan et al. and other sources. It should also include the various variations of the basic problem and other, more complex, issues. [[User:Martin Hogbin|Martin Hogbin]] ([[User talk:Martin Hogbin|talk]]) 10:19, 29 December 2009 (UTC) |
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===Glkanter's position=== |
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I want the article to clearly mention that the remarks made by some sources, that the so called "simple solution" is not complete, is not shared by all sources. It need not mention "conditional probability" beyond saying that due to the symmetry forced by being a game show, the simple solution is equivalent to the symmetric 'conditional solution'. |
I want the article to clearly mention that the remarks made by some sources, that the so called "simple solution" is not complete, is not shared by all sources. It need not mention "conditional probability" beyond saying that due to the symmetry forced by being a game show, the simple solution is equivalent to the symmetric 'conditional solution'. |
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And the 'Variants - Slightly Modified Problems' section needs work. The MHP is from the contestant's state of knowledge (SoK). The versions in this section are not. This needs to be normalized for the reader in a few possible ways: An explicit statement that the contestant ''is'' aware of these new conditions (in which case these are no longer game show problems), or the explicit statement these problems are ''not'' from the contestant's SoK, and a comparison of the MHP from a non-contestant's SoK. [[User:Glkanter|Glkanter]] ([[User talk:Glkanter|talk]]) 13:14, 29 December 2009 (UTC) |
And the 'Variants - Slightly Modified Problems' section needs work. The MHP is from the contestant's state of knowledge (SoK). The versions in this section are not. This needs to be normalized for the reader in a few possible ways: An explicit statement that the contestant ''is'' aware of these new conditions (in which case these are no longer game show problems), or the explicit statement these problems are ''not'' from the contestant's SoK, and a comparison of the MHP from a non-contestant's SoK. [[User:Glkanter|Glkanter]] ([[User talk:Glkanter|talk]]) 13:14, 29 December 2009 (UTC) |
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===Rick Block's position=== |
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First, I think the basic issue is an NPOV issue. The primary question is whether the article currently expresses a "pro-Morgan" POV, i.e. takes the POV of the Morgan et al. source that "unconditional" solutions are unresponsive to the question and are therefore "false" solutions - and, if so, what should the remedy be. |
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There are a variety of sub-issues we need to discuss but I think the main event is how the solution section is presented. I strongly object to splitting the solution section into separate sections (this was done some time ago, well after the last FARC), which inherently favors whatever solution is presented in the first such section. I mildly object to including the "combining doors" explanation in the solution section rather than in a subsequent "aid to understanding" section. |
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What I would like is for the article to represent in an NPOV fashion both a well-sourced "unconditional" simple solution (e.g. vos Savant's or Selvin's) and a well-sourced conditional solution of the symmetric case (e.g. Chun's, or Morgan et al.'s, or Gillman's, or Grinstead and Snell's) in a single "Solution" section, more or less like the suggestion above (see [[#Proposed unified solution section]] - somewhat modified just now). This follows the guidelines at [[Wikipedia:Make technical articles accessible]], specifically most accessible parts up front, add a concrete example, add a picture, and do not "dumb-down". |
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Once we address this basic issue I think the other issues will be easier. -- [[user:Rick Block|Rick Block]] <small>([[user talk:Rick Block|talk]])</small> 19:43, 29 December 2009 (UTC) |
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One more simulation question
If one was to write a program to simulate this problem would the same rules apply or would the ratio of win/losses be 50-50 because the programing would not be choosing to switch or not like a human would but simply randomly choosing one of the remaining two doors after one of the initial three non-win doors was dropped?
Granted I'm not a computer programmer but it seems to me the program would run something like this:
- Choose a random number between 1~3, set that number as WIN.
- Again choose a random number between 1~3 set that number as CHOICE.
- Discard one non-WIN non-CHOICE number (randomly chosen if there are two possibilities).
- Randomly choose one of the remaining two numbers (this represents staying or switching).
- Log and graph results.
Is this an accurate simulation?
If this has been addressed somewhere referenceable can someone point me to it? Thanks! Colincbn (talk) 12:31, 12 November 2009 (UTC)
- If a person or a program randomly chooses whether to switch, they'll win 50% of the time - but this is not what the problem is asking. In either case (person or program) you should keep track of how many times you switch and win, how many times you switch and lose, how many times you don't switch and win, and how many times you don't switch and lose. The ratio of (switch and win) to (switch and lose) is the chance of winning by switching and will be about 2:1 (2/3). The ratio of (don't switch and win) to (don't switch and lose) is the chance of winning by not switching and will be about 1:2 (1/3). These two ratios must add up to 1. If you randomly pick whether to switch or not the ratio of switch (win + lose) to don't switch (win + lose) will be about 1:1, and the ratio of (switch and win) to (don't switch and win) will be about 2:1. This doesn't mean the chance of winning by switching is 50%, but that the chance of winning by making a random choice between switching and not is 50%. Regardless of how you choose whether to switch or not, if you switch your chance of winning is 2/3 and if you don't your chance of winning is 1/3.
- Specific numbers might help. If you do this 600 times, and always switch you'll win about 400 times and lose about 200 times. Similarly, if you never switch you'll win about 200 times and lose about 400 times. However if you randomly pick whether to switch, you'll switch about 300 times and not switch about 300 times. Of the times you switch, you'll win about 200 and lose about 100. Of the times you don't switch, you'll win about 100 and lose about 200. Overall, because you're randomly picking whether to switch or not, you'll win about 300 times.
- There is source code for some simulations (in various languages) at http://en.wikibooks.org/wiki/Algorithm_Implementation/Simulation/Monty_Hall_problem. -- Rick Block (talk) 20:19, 12 November 2009 (UTC)
- BTW - the discussion above contrasts the probability of winning if you choose to switch randomly vs. the probability of winning by switching. There's another layer to the problem as usually stated which is that you're asked to imagine having chosen door 1 and the host having opened door 3. Opinions differ about whether this is significant, but most simulations, and many solutions to the problem, don't address this particular situation (or any other specific combination of player pick and door the host opens) but rather what might be called the overall probability of winning by switching assuming (usually without an explicit justification) that this overall probability must be the same as the conditional probability of winning by switching given the player picks door 1 and the host opens door 3. If you're actually interested in the conditional probability of winning by switching given that you've chosen some particular door (like door 1) and the host has then opened some other particular door (like door 3), the simulation should be more like the one suggested above by Nijdam where you only record cases where the player initially chooses door 1 and the host opens door 3.
- Again, specific numbers might help.
- If you run 600 simulations (initially picking door 1 each time) we'd expect the car to be behind each door about 200 times. So if you switch every time, you'll win about 400 times meaning the probability of winning by switching is 2/3. However, how many times will the host open door 3, and how many of those times will you win if you switch? The host has to open door 2 if the car is behind door 3. This should happen about 200 times. And the host has to open door 3 if the car is behind door 2 - which also should happen about 200 times. If the car is behind door 1 the host can open either door. If he picks which door to open randomly (in this case) he'll open each of doors 2 and 3 about 100 times. In total, the host opens door 3 about 300 times - all 200 times the car is behind door 2 but only 100 of the times the car is behind door 1. So if you switch given you've chosen door 1 and the host has then opened door 3 you'll win about 200 out of 300 times (i.e. 2/3 of the time).
- Keeping track of it this way also lets us figure out the probability of winning by switching if the host doesn't pick which door to open randomly (in the case the car is behind the door the player picks). For example, if the player picks door 1 and the host opens door 3 with probability p when the car is behind door 1 (the player may or may not know what p is), in the 200 times the car is behind door 1 the host opens door 3 200p times. These are all cases where switching loses, so switching wins about 200 times out of a total of 200+200p times. Expressed as a fraction this is 200/(200+200p) which simplifies to 1/(1+p). Since p can theoretically be anything between 0 and 1, this means the probability of winning by switching given the player picks door 1 and the host opens door 3 could be anything between 1/2 and 1, depending on p. Again, the player may not know what p is. Many folks (on this page) insist the player CANNOT know what p is and therefore p MUST be taken to be 1/2, which leads to the 2/3 answer. Note that if all you're interested in is the overall chance of winning by switching, p doesn't matter at all. Staying with the numbers we've been using all along, if you always switch you'll win about 400 out of 600 times no matter what p is. If you're interested in the chances given the player picks door 1 and the host opens door 3 (or given any other specific combination, where p refers to the probability that the host opens the door he's opened in the case the car is behind the player's chosen door), then p does matter. -- Rick Block (talk) 02:51, 13 November 2009 (UTC)
- Thanks for the link, that is exactly what I was looking for!
- As far as I understand the p problem (hehe, "p problem" hehe...) this only applies in real world situations where "Monty" has some kind of preference when choosing goats, and that in any "true" random pattern of choosing this never comes into play (I realize that there is a different problem with whether or not there is such a thing as "true" randomness but for our purposes computer generated "system clock randomness" is most likely sufficient). But where the conditional/unconditional argument comes into play is if someone spent everyday of their adult life watching Let's Make A Deal then got invited on the show; that person might have a better than 2/3 chance given that they have a better understanding of Monty's preference of goat choosing, is this an accurate simplification?
- On a side note I actually ran through this in a pub last night with my sister in-law. At first she was always staying but after I explained the math (using mainly the Cecil Adams approach) she started switching and winning much more (about twice as much as it were). We used a deck of cards (ie: red twos and the ace of spades), and my randomizer was shuffling under the table and me trying not to think about it if I had to choose between two "goats". It was an interesting way to spend twenty minutes while guzzling Guinness and I would recommend it to anyone. Colincbn (talk) 07:11, 13 November 2009 (UTC)
Even If it's not stated, the host MUST choose randomly between two goats contestant's assesment of the host's choice must be that it appears random to her.
- You do have to treat p as 1/2 unless you are told what its value is, and which doors it relates to. Many people, even those who should know better, forget that "probability" is not a property of the specific result, it is a property of the random process that led to the result. As Kreyszig said on page 714 of his famous book (mine is the 1972 edition), "For this reason we now postulate the existence of a number P(E) which is called the probability of the event E in that random experiment. Note that this number is not an absolute property if E but refers to a certain sample space S, that is, to a certain random experiment."
- Here's my favorite example of the difference: Suppose I draw a card at random. I look at it by myself and see that it is the Queen of Hearts. I tell Ann that it is red, Bob that it is a heart, Carl that it a face card (that means TJQKA), and Dee that its value is even. I ask each what the probability is, that it is the Queen of Hearts. Ann says 1/26, Bob says 1/13, Carl says 1/20, and Dee says 1/24. But I know that this "probability" is actually 1/1. Since all the answers are different, who is wrong?
- Answer: Nobody. The probability is not about the card, it is about the process. And each person sees a different process, one that leads to the specific piece of information I gave to them. Since each piece is different, the process each is evaluating is different. In the Monty Hall problem, the card represents one particular day's broadcast of this game show. The probability we are trying to evaluate is not a property of this broadcast, but of all possible broadcasts. Even ones where Monty might change how he picks.
- Those people who think you can use a p other than p=1/2 to get an answer are wrong, by their own arguments. Consider that Monty might vary how he chooses p. He might choose p=1/3 when he is wearing white underwear, and p=3/4 when he is wearing colored underwear. But only Monty knows what kind of underwwear he randomly selected that day, or what mix exists in his wardrobe. So even if you track many games, and think you know what p is, you will still be wrong every time you use that p. It is never the value Monty uses. The assumption that p can be something other than 1/2 in the solution to the problem, leads to the contradiction that you have to know every factor that goes into determining what p is before you can use it. JeffJor (talk) 17:41, 13 November 2009 (UTC)
- What those who say p is important (e.g. Morgan et al. 1991, and Gillman 1992) actually say is
- unless you're told the host picks randomly if given the chance then assuming p is 1/2 is an unwarranted assumption [note the randomization procedure Gardner included in his version of the Three Prisoners problem)
- in this case (not being told what p is) the probability given a specific initial player choice and door the host has opened does not have a single numerical answer and the best we can say is that it is in the range [1/2, 1]
- if you're saying the answer is 2/3, either you're not addressing the conditional probability in a specific case of initial player pick and door the host has opened or you've assumed p is 1/2
- What those who say p is important (e.g. Morgan et al. 1991, and Gillman 1992) actually say is
- I think you disagree with them, but do you at least agree that this is what these sources say? -- Rick Block (talk) 20:36, 13 November 2009 (UTC)
- Those who say p is important are addressing a different problem than the one this article is about. So what they say isn't appropriate here. But their problem is actually unsolvable by the methods they use, since nothing can be assumed about the value of p by those methods. Especially not a "uniform distribution" of p. Those references should be dropped from this article, regardless of what you think they say about p.
- There are references that say p=1/2 is the only valid option, even when addressing Morgan's problem. That includes Gardner's version of the Three Prisoners problem. Gardner's assumption that p=1/2 is not part of the problem statement, it is part of his solution. It comes after he says "Now assume, by renaming the prisoners if necessary, that A is the one questioning the warden, and B the one the warden names as executed" which actually necessitates that p=1-p since the solution's treatment toward B (assume a value of pB) and C (which has pC=1-pB) has to be the same. Or just look up the Principle of Indifference.
- If you think you need to solve "the case of not being told what p is, then you are addressing the probability of a particular instance of the random experiment (e.g., The show that aired on March 27, 1993), not the random experiment itself. You are essentially saying that there is one, and only one, correct answer to my card problem about the Queen of Hearts. Which person do you think gave the one correct answer? Why is that person's answer "better" than the others? Or, just see my reference about probability being about the samples space - the set of possibilities - not E itself.
- I am saying the one, and only, possible answer to the problem this article is about, is 2/3. Many more references support this solution than the fradulent Morgan, et al, treatment. The 2/3 solution applies if the player picks a specific door (which isn't part of the problem) and the host also picks a specific door (also not part of the problem), AS LONG AS THE PROBLEM STATEMENT DOES NOT SPECIFY ANY BIAS IN HOW THEY WERE CHOSEN. If you can find such information in the problem statement, the answer might not be 2/3. But you can't speculate about it - you need to be told a value for p to use it. See the Principle of Indifference, particularly the parts about names not being useable as identification. JeffJor (talk) 14:36, 14 November 2009 (UTC)
- The real problem, Jeff, is that Morgan et al managed to get their paper published in a respectable journal. You know their paper is wrong and I know it is wrong but unfortunately that counts for nothing here, we must rely on 'reliable' sources. Their most serious error was that pointed out by Professor Seymann in his comment, published in the same journal. This is that Morgan do not make clear exactly what question they are answering. In the circumstances this is inexcusable. Martin Hogbin (talk) 18:24, 14 November 2009 (UTC)
- My point was to try to shift this discussion from an argument about what you or I or anyone else thinks the Truth(tm) might be, to a discussion about what the sources say. The former topic is a fine topic for the /Arguments page, but really has no relevance to the content of this article.
- And that is fine. Morgan, et al, and Gillman, do not address the problem that this article is about. Regardless of anybody's POV, THOSE REFERENCES DO NOT BELONG HERE. You are wrong whan you say they are references about this problem. Now, secondary to that, it is POV (and quite provable, which is a property that can be used to ignore a reference), that their treatment is wrong even for the problem they address. That is not the reason to remove the references, although it is a good reason to ignore them.
- The Eisenhauer reference you give acknowledges that what Morgan and Gillman handle is a different problem. All it does is acknowledge that the contestant still does better by switching in that different problem; it does not attempt to say what the probability is. So it does, indeed, supercede both Morgans's and Gillman's conclusions about the problem the article addresses. JeffJor (talk) 15:20, 15 November 2009 (UTC)
- The claim that Morgan et al. 1991, and Gillman 1992 (and Grinstead and Snell 2006, and Falk 1992) are addressing a different problem than the one this article is about is directly contradicted by what these sources say. For example, they all reference the problem statement from Parade.
- Morgan misquoted the problem, and misinterpreted what they misquoted. And (to comment to what you said next) no, Gardner does not state p=1/2 explicitly. He states that you flip a coin, and in his solution implicitly through the Principle of Indifference assumes that the probability of the two outcomes are each 1/2. The same principle could be applied if he didn't say to flip a coin, so that part is irrelevant in the problem statement. It just makes it more apparent to the masses who do not realize the same Principle applies to "The warden chooses between B and C" and "The warden flips a coin to choose between B and C." Or that whan a problem says "flip a coin" it really means "Assume the Principle of Indifference applies." JeffJor (talk) 15:20, 15 November 2009 (UTC)
- Gardner's description of the Three Prisoners problem (as published in Scientific American, Oct 1959) includes the following:
- "Then tell me", said A, "the name of one of the others who will be executed. If B is to be pardoned, give me C's name. If C is to be pardoned, give me B's name. And if I'm to be pardoned, flip a coin to decide whether to name B or C."
- There's no appeal to the principle of indifference. His problem description unambiguously specifies p=1/2.
- I agree there are plenty of references that say the answer is 2/3. However, (as far as I know) nearly all of these are not in peer reviewed math journals. Without sources to back up it up, the opinion that the Morgan et al. treatment is "fradulent" is irrelevant to the content of this article. From wp:reliable sources: Academic and peer-reviewed publications are usually the most reliable sources when available. However, some scholarly material may be outdated, superseded by more recent research, in competition with alternate theories, or controversial within the relevant field. The Morgan et al. paper is specifically about this problem, and it appears in a peer reviewed math journal. Is it outdated or superseded by more recent research? On the contrary, it is directly supported by a subsequently published paper (also in a peer reviewed journal) (see #Refereed paper agreeing with Morgan et al., above). Is it controversial within the relevant field? Rosenhouse's recent book (The Monty Hall problem, Jason Rosenhouse) doesn't like the tone, but even he goes on to say They, at least, were mathematically correct in their substantive points. Is it in competition with alternate theories? There are plenty of popular sources that perpetuate vos Savant's solution, or other solutions directly criticized by Morgan et al. To consider this treatment to be in competition with alternate theories, I would expect there to be academic sources that reference Morgan et al. and directly dispute it. I think the academic response has been to clarify the problem description, consistent with the Krauss and Wang version presented in the article. In this version, the constraints on the host behavior explicitly force p to be 1/2 so that the answer is unambiguously 2/3 even if the problem is interpreted as Morgan et al. do.
- Anyone can personally have whatever POV he or she would like about this problem. On the other hand, the article needs to reflect what reliable sources say regardless of whether you or I or anyone else agrees with them. -- Rick Block (talk) 19:52, 14 November 2009 (UTC)
- Jeff - So, your claim is that Morgan et al., and Gillman (and presumably Grinstead and Snell, and Falk, and Eisenhauer) do not address the problem this article is about. Your opinion about this clearly reflects your POV, and is worth much less than a published opinion like (for example) Barbeau's - see either Barbeau 1993 or Barbeau 2000. Martin has previously suggested trying to split the article into sections about two different problems, what he considers the "notable" one (which I think is consistent with what you seem to consider to be the one, true, interpretation) and an explicitly conditional version. The issue with this approach is finding appropriate sources. Many sources treat the problem unconditionally even though the usual form of the problem statement is fairly clearly conditional. You're saying even though it's conditional the probability of the host's choice (in the case the host has a choice) must be treated as 1/2 because of the principle of indifference. Do you know of a source that explicitly (not implicitly) treats it this way? -- Rick Block (talk) 19:02, 15 November 2009 (UTC)
- Rick - have you noticed that this article explicitly says "If both remaining doors have goats behind them, he chooses one randomly?" All those references EXCEPT MORGAN (and I just found a copy of Gillman today) acknowledge that they are talking about a different problem when that allow p<>1/2, and solve for a probability based on the parameter p THAT THE CONTESTANT HAS ACCESS TO (example, from Gillman: "In the extreme case of q=1, the host's opening of door #3 'gives you no information [emphasis added]. It is not inforrmation ONLY IF YOU KNOW q=1.) They all - EXCEPT MORGAN - acknowledge that the probability they give is based on the contestant's knowledge of p. Those treatments are correct. Only Morgan takes the additional, and quite incorrect, step of saying you can solve for a probability when p is not known or implied, even parametrically. Essentially, Morgan is claiming lat ln(2) is an answer to the unconditional problem, and it is not. If you don't know pp, the probability is 2/3 if you switch AND NOTHING IS CONTRADICTED if somebody who does know p claims a different probability. The two peoepl are answering different problems.
- And no, regardless of how many people treat the original problem conditionally, it is not. Definitely not this article's problem, and not Parade's, either. The door numbers used were examples (for a similar case, look at Grinstead & Snell's problem #7 on page 13. Are you suggesting that Roulette only allows people to bet on a 17, or that the correct answer needs to include whether the roulette wheel is unbalanced toward a 17? Or does every book on probabiltiy need to be re-written to make their exampels into conditions?). Those people are just grabbing onto a more interesting problem, one that shows that "switch" is a good strategy even if you allow for biases. JeffJor (talk) 18:01, 16 November 2009 (UTC)
- On thing that is undoubtedly true about Morgan is that they do not make clear exactly what question they are attempting to answer. This is not just what I say on my Morgan criticism page but it is what Prof Seymann says in his comment in the same peer-reviewed journal that Morgan were published in. Martin Hogbin (talk) 23:11, 16 November 2009 (UTC)
- To start with, is there anyone who can tell me whether Morgan are answering the problem from:
- The point of view (state of knowledge) of the player.
- The point of view (state of knowledge) of the audience.
- Only on the information given in Whitaker's question.
- Only on the information given in Morgan's restatement of the question.
- Some other basis - please specify.
- This is not a minor detail but something that is universally accepted as an essential requirement in statistics - that the question being asked is clear. Martin Hogbin (talk) 11:32, 17 November 2009 (UTC)
- Reply added at the /Arguments page. JeffJor (talk) 19:17, 17 November 2009 (UTC)
- Martin - you know perfectly well what problem Morgan et al. are answering (we've been over this innumerable times before). To save any newcomer to this argument the trouble of looking back through this thread, Morgan et al. answer the question based on the information given in Whitaker's version of the question, as printed in Parade (which they trivially misquote) which is the same as the point of view of the player and the audience. They interpret the question to be about a game show (involving physically identifiable, distinguishable doors) and, in this game show, the probability of winning by switching given that the player has initially selected a door (an identified door) and then the host has opened another (identified) door. They work through the example situation suggested in the problem statement where the player has picked door 1 and the host has then opened door 3 as a straightforward application of conditional probability. Their analysis covers a variety of host protocols (what JeffJor is calling different problems), including the one vos Savant's published clarifications suggest she was attempting to solve where nothing is said about the host's choice if the player initially selects the car (which means the conditional probability equation contains an unspecified variable) as well as the Krauss & Wang version where the host is known to select equally randomly in this case. I'll reply to JeffJor's comments at the /Arguments page there. -- Rick Block (talk) 02:13, 19 November 2009 (UTC)
- Rick, you know perfectly well that Morgan did not use Whitiker's version of the problem. They edited it into a different form that more closely resembled the problem they misinterpreted it for - and were ambiguous about (more below) in their solution. In Whitiker's version, the contestant chose a door (not "door No. 1"), and "#1" was given as an example in an aside that in no way implied it was meant to be a condition of the problem. The host opens a door (not "door #3), and "#3" was given as an example. Do you know what "a" means in English? It is an indefinite article, "used before a singular noun not referring to any specific member of a class or group" according to my Random House dictionary. The literal meaning of Whitiker's problem is nonspecific doors. I know you don't agree, but those newcomers might. And that difference doesn't actually matter, and isn't what Martin was referring to. Newcomers should know that, too.
- If you would read Martin's comment, it is plainly obvious he wasn't talking about the difference between "the conditional problem" and "the unconditional problem" as you are. He is asking "From whose point of view is the probability calculated?" Yes, if the conditional problem is intended, both the audience and the contestant can include that information in their point of view. IT CHANGES NOTHING. It does not matter if the doors are phsycially identifiable (see the Principle of Indifference), because the contestant has no way to know how the host uses such distinctions on this particular day. ANY NAMES ATTACHED TO THE DOORS CANNOT MATTER. And if it could matter? Moragn's entire analysis is invalid because they must assign a similar set of parameters to the probabiltiy that the car is placed behind the specific door numbers. Neither Whitiker's problem, nor Morgan's, ever say the car is placed randomly. That fact is just as implicit (or not implied, as Morgan assumes) as any bias the host has about opening a specific (by name) door.
- There is no absolute-correct probability for the two remaining doors, in either the conditional or unconditional problem. Because all of the random elements of the problem have already been determined at the time the decision is to be made. The car 'is behind one of the two doors. Any probability that can be assigned is a only measure of the information you assume is avaliable to calculate it, not a measure of that door's worth. If the contestant has only the information available to Marilyn's Little Green Woman, that probability is 1/2 (yes, I know that is less information than the problem says the contestant has). If the contestant has only the information that the problem says she has, the probability is 2/3 EVEN IF THE CONDITIONAL PROBLEM IS CONSIDERED. And if you are going to assume the contestant has more information than the problem says - i.e., that she knows the host has a bias measured by a parameter q - you might as well assume the contestant also knows everything that the host knows. Because only the host really knows how his q was determined on this particular day. Even if he is consistent with past games, and we assume the contestant knows all that history (which the problem does not allow), there could be unseen factors that influence the actual q so that on that day, it is not what this omniscient contestant thinks it is. The only virtue in considering a possible q, is that whatever its value is, it is never detrimental to switch. JeffJor (talk) 13:16, 19 November 2009 (UTC)
- Yes, that is my main point. Rick, you say that the question is being answered from the point of view (state of knowledge) of the contestant (or the audience, which I agree is likely to be the same). What Morgan have done, in that case, is to arbitrarily dis-apply the principle of indifference to the host's choice of goat door. As Jeff says, the player has no more idea of the host's door opening policy than she does of the car's initial position, yet Morgan tacitly take the car's initial position to be random (which is quite reasonable as the player is presumed to have no information on its whereabouts) but they take the host's door choice to be non-random and described by the parameter q, even though the player has no information about the host's strategy. Thus, as I have said all along, Morgan add an artificial layer of complexity and obfuscation to what should be a simple (but very unintuitive) problem. Martin Hogbin (talk) 17:01, 19 November 2009 (UTC)
- I'll try one last time, but this is a tiresomely repetitive argument. I think everyone (more or less) agrees the problem should be answered from the state of knowledge of someone who knows what is given in the problem statement (which we could assume is the player's SoK). I also think everyone (more or less) agrees the principle of indifference applies to the player's initial selection since, even if the player suspects the initial distribution is not random, the player has no way to know what the distribution is and is therefore making a random initial choice. In her followup columns, vos Savant removes any doubt about this explicitly saying the car is meant to be randomly located. I think so far, everyone is on the same page.
- Now, the host comes into the picture.
- The Parade problem statement says only "the host, who knows what's behind the doors, opens another door" and shows a goat. To make a decision about whether she should switch, the player now needs a little more information about the host's intent not specified in the Parade problem statement. Does the host always open a door showing a goat and make the offer to switch, or is this perhaps a one-time attempted swindle being offered only because the player's initial choice hides the car or even is the host opening a random other door which (this time) happens to not reveal the car? In her followup columns, vos Savant makes it very clear (even though these conditions are not in the problem as originally stated in Parade) the problem is meant to be interpreted that the host always opens a door, never reveals the car, and always makes the offer to switch. Even to this point I think everyone (more or less) still agrees.
- So, back to the state of the knowledge of the player. The player knows
- The initial distribution is explicitly 1/3:1/3:1/3 (and, even if it isn't, can be assumed to be because of the principle of indifference)
- The host is compelled to open a door showing a goat and make the offer to swtich, and is not acting randomly because doing so would involve the chance of revealing the car
- Sounds like a simple conditional probability problem to me. If we take a specific case (like the one mentioned in the problem statement, where the player initially picks door 1 and the host opens door 3), the probability of winning by switching is
- P(car is behind door 2) / (P(car is behind door 1)*X + P(car is behind door 2))
- where X is the probability the host opens door 3 in the case the car is behind door 1. This equation is well within the SoK of the player to come up with (and, by renumbering the doors, applies to any specific player's case), and would now (I think) lead the player to wonder about X. What exactly does this X mean? Is there anything in the player's SoK (i.e. the problem statement) that constrains it? The player definitely knows the host is not acting completely randomly, but should she assume the host is acting randomly in this subcase? Note that it's not the player who's choosing between these doors, but the host. Does the principle of indifference apply here?
- This is the point where opinions differ. Morgan et al. have an opinion, that they published in a peer reviewed journal. Gillman published his opinion, which agrees with Morgan et al. Grinstead and Snell finesse the issue, by explicitly requiring the host to choose randomly in this case (while acknowledging the difference between the unconditional and conditional problems and the effect of X in the conditional case).
- We can, and seemingly may, argue about this forever, but the bottom line is that from the state of knowledge of a Wikipedia editor Morgan et al. is a wp:reliable source and any arguments about what it says that are not based on equally reliable sources are simply wp:original research that has absolutely no bearing on the article content (by Wikipedia policy). -- Rick Block (talk) 19:27, 21 November 2009 (UTC)
- Yes, Rick, it is quite tiresome. The reason is that you won't even consider the validity of arguments that disagree with the approach taken by Morgan, et al, or Gillman. The Principle of Indifference applies equally to the initial car placement and the host's choice of doors to open. It is inconsistent for "all to agree" it should be applied to one, and accept it for that choice, while insisting you can ignore it for the other. To paraphrase you, the player has no way to know what the distribution of the Host's bias is, and therefore it must be treated as a random choice. And before you try to deny this, apply the exact same argument you intend to use to the car placement.
- You used the argument that "vos Savant [in later columns] removes any doubt about [the randomness of the car placement by] explicitly saying the car is meant to be randomly located." She also makes it clear in later columns that the host's choice is random. You can't apply these arguments differently to the two sources of uncertainty. Again, apply the same arguments to both choices.
- And nobody is claiming the host's choice is random between the unchosen doors. It is well established that the host knows where the car is, and where the goats are. This provides a way, not based on names, for the host to distinguish all three doors in some cases. So the host's choice is random between the set of unchosen doors with goats, which are the ones the Principle of Indifference applies to from his SoK. A choice can still be considered random, by the Principle of Indifference, when n=1." Your point #2 above does not apply to any situation in the problem - if it did, the host would also open the contestant's chosen door 1/3 of the time. The implicit assumption vos Savant makes is that the host opens a random unchosen goat door.
- Morgan's analysis is not an "opinion." It is a solution to a problem that is never stated in any of the traditional forms. Never. It does not belong in the article. JeffJor (talk) 20:37, 21 November 2009 (UTC)
- I'm perfectly willing to consider the validity of arguments that disagree with the approach taken by Morgan et al., or Gillman. However, to be relevant to the article content it takes a published argument. I did a search on Google Scholar for "Monty Hall problem" and "principle of indifference". As far as I can tell, none of the hits say anything about applying this principle to the issue we're talking about. Including this argument in the article would require a published source - whether it's valid or not is a secondary issue and, unless there's a source making this argument, from Wikipedia's perspective it doesn't matter if it's a valid argument.
- Do I personally agree that it's a valid argument? If you're attempting to justify 2/3 as the answer, sure - but I definitely don't agree that this means the approach Morgan et al. take is invalid. What I actually believe is that the approach Morgan et al. take applies to a broader range of specific wordings of the problem, and is therefore more useful, than an approach targeted at one and only one interpretation of the problem statement. In particular, the Morgan et al. approach addresses (among others) what might be called the current "standard form" (e.g. the K&R version with explicit constraints on the host including random pick between two goats if it comes up), and the "Monty forgets" variant (that vos Savant has mentioned in the last year or so in her column), as well as the "host might have a preference" generalization of the standard form. As has been previously mentioned, they don't do it but their approach also easily extends to an analysis where the initial probabilities are not 1/3 for each door. I know opinions about this differ, but I think the explicit conditional approach is an extremely clear and utterly convincing way to present the solution (even the 2/3 solution!).
- Separately, your repeated assertion that Morgan's analysis does not belong in this article is (quite frankly) bizarre. It is I believe the first (and perhaps only) paper in a peer reviewed math journal specifically about the mathematical intricacies of the Monty Hall problem in its usual form. Google Scholar lists 60 citations. The article would not be complete without this reference. -- Rick Block (talk) 23:11, 21 November 2009 (UTC)
- Rick you seem to want to confuse the issue in question by bringing up things that we all agree about. We all agree that (in the standard rules) the host always offers the swap and we all agree that the host knows where the car is and never opens a door to reveal it. The point is that, where the host has a choice of which door to open in other words in the case that the player has initially chosen the car, the player has no information on which to decide how the host will make this choice and thus we must take it that, in this case, the choice is random. This is not what Morgan do, they take it that this choice is not random and assign it a host door choice parameter.
- I think that the article needs to concentrate much more on the simple problem. This is the notable problem. The simple fact is that the answer (chance of winning on swapping) is 2/3 but nearly everybody thinks otherwise. Th article should concentrate on explaining why this is so.
- Regarding reliable sources and scholarly discussion, there are many reliable sources on this subject and there is no reason that we must use Morgan as the guiding light in our article. Although the paper was published in a peer-reviewed journal it has been the subject of much criticism, starting with Seymann's comment in the same publication. I suggest that the article should be based more of the article on K & W, for example, who actually start to address the reason that so many people get the answer wrong. If we take it that The Monty Hall Problem is the fully defined problem as stated by K & W (as we seem to do in the article now) then the Morgan paper is irrelevant as it does not address this problem, where the host choice is defined to be random.
- Although the article should be based on reliable sources there is some latitude to decide which sources are most relevant to the problem being addressed. I have seen more of a consensus to use sources other than Morgan or those based on it. Martin Hogbin (talk) 12:16, 22 November 2009 (UTC)
- Martin - please read the "comprehensive" and "well-researched" criteria from WP:FACR. Omitting significant references because they don't address what you think is the "notable" problem is contrary to both of these. You and Jeff have both asserted your opinion that if the host preference is not constrained by the problem statement then it must be taken to be random, citing principle of indifference. Since there are reliable sources for this specific problem (including a peer reviewed paper co-authored by 4 members of the faculty of a university math and statistics department!) that don't do this, your opinion is apparently wrong. Moreover, your opinion doesn't matter here - what does matter is what reliable sources say. Find a reliable source that explicitly makes this argument about this problem, ideally directly referencing Morgan et al. or Gillman or some similar source, and then we'll talk. Until then, what you're saying is simply WP:OR.
- And, I think you're absolutely incorrect about what the consensus is. As far I as I remember, you, Jeff, and one or two others we haven't heard from in a while are the only ones in favor of dropping Morgan et al. and similar sources, while Nijdam, I, and at least as many others we haven't heard from in a while (I think it's actually far more than one or two) are not in favor of this. I think the discussion on this page is so tiresomely repetitive that many, many folks have simply checked out, so trying to gauge consensus based on current participation is not likely to be very accurate. -- Rick Block (talk) 20:09, 22 November 2009 (UTC)
Rick, I corrected your header for this section. The mistake you made in naming it points out the issues that you are misintpreting. Morgan does not take a broader view of the problem's various wordings, it is a narrower view in the sense of approaching a solution. It is narrower because the simpler wordings, that Morgan deny, allow some assumptions to be made. Assumptions that are traditioanlly made in word problems. I.e., if you can draw N balls out of a bag, the problem does not need to state that each has a probability 1/N. Morgan disallows such assumptions, but only where the "conditional" solution can still provide an answer to the question "should you switch?" They aren't calcualting a probability, they are showing that whatever the calculation could produce, switching is a good idea.
I agree that published work, and not our opinions, must be the basis for the article. But the theses of Morgan, et al, and Gillman, are not about the general problem this article is about. Morgan, et al, even say "In general, we cannot answer the question 'What is the probability of winning if I switch, given that I have been shown a goat behind door #3?' unless we ... know the host's strategy ..." So Morgan is not answering the "general" problem, they are answering one where p13 takes on a specific value that is unknown ot the contestant, p22=p33=0, and (the assumption they make but don't state) p11=p21=p31=0. So they explicitly state that their solution applies to just one variation of the unknown Host's Strategy. Essentially, they solved a variation of the general problem that is interesting because you can get an answer without making the usual assumptions for one part of that strategy, but do for others. And that is where they are inconsistent. The article is flawed because it suggests someone did calulate a probability.
If their approach belongs at all, it should be moved to below the discussion about alternate Host stratgeies. It applies - according to the authors - only if the contestant knows the host's stragtegy except for the exact value of p23. And they also say that if the contestant does not know the host's strategy, that the best appraoch gives "the popular answer" of 2/3. (Note: there is a typo in their article. They gave the wrong value for that part, 1/2 instead of 2/3. And I think that proves that any oversight of their article treated it more as a curiosity, than as a scholarly work to be checked for accuracy.)
About the Principle of Indifference. It is so basic a principle in probability, that it never is listed as the reason whan it is applied. Authors just go ahead and state that they are applying an even distribution to the indistinguishable items, and it is universally understood why. Morgan does it, for the car's palcement. The only reason I brought it up (by name) at all is because it says the door numbers don't distinguish the doors unless there is some other known reason to treat them differently based on the numbers. When there is not, the solution treats them equally even if they are not. And so gets a correct answer that reflects how any possible bias could favor either door.
Finally, and you seem to keep ignoring this, the Parade version of the problem, and the K & W one this article is based on, do not describe the so-called "conditional" problem. Morgan misinterpreted it, and I have shown you exactly how their problem statement is different. The vast majority of peopel who see teh word puzzle do not even consider the "conditional" problem. But even in Morgan, they admit the "conditional" answer only applies when the host's strategy is known to fit within certain boundaries, that include knowing p23. Several times, Gillman points out that not knowing it makes the answers to his Game I and Game II equivalent. This is why they deserve to be a sidebar, at best, in the article. It simply is not "about the mathematical intricacies of the Monty Hall problem in its usual form." Their problem is just an interesting fact about one possible variation of what everybody sees the the usual form of the puzzle to be. JeffJor (talk) 19:25, 23 November 2009 (UTC)
Do Morgan et al. address 'The Monty Hall Problem'
I have added a heading here for ease of editing.
Rick, I see nothing in WP:FACR that tells us that we must use an academic paper that does not address the problem as stated in the article. As Jeff has said above, the problem statement given in the article, namely that from K & W, clearly states that the host door choice is random (subject to the agreed rules) and the initial car placement is random. The Morgan paper clearly refers to the specific case where the initial car placement is random (even though this is not explicitly stated) but the host door choice is not random. Thus the paper by Morgan et al is not relevant to the article, except as a special, and rather unrealistic, variation of the basic problem. Martin Hogbin (talk) 22:42, 23 November 2009 (UTC)
- No, they don't. And just to make it clear, this article is about the variation where these are the defining points of the problem:
- The car is placed randomly.
- The contestant can choose any door.
- The host must open a door that is randomly selected from all unopened doors that hide goats (this could be a choice of one door only).
- This is the "K&W" problem. It is what the casual reader feels any version (unless explicitly contradicting some part of it, which Parade's version does not do) means. (Documentation: some of the scholarly references listed in the article are not as much about the Mathematics, as about the perception of the problem by readers. Falk and Fox & Levav, to name two.) It is teh problem Marilyn vos Savant (who is notorious for expressing problems in an amgiguous way, but assuming peoepl will only see one) assumed people would see, and so is what she answered. But the most general case, using Morgan's approach for what makes for a more "general" problem, has these defining ponts:
- The car can be placed with a bias.
- The contestant always chooses Door #1 (chosing other doors is a similar problem, but with different paramater values based on point #5 below).
- The host can open an unchosen door with the car. If he does, the game ends immediately.
- The host can open the contestant's door. If he does, the game ends immediately. (Note that revealing Door #3 has the car in the previous point is the same as revealing Door #1 has a goat here.)
- The host has a set of biases that depend on the car placement and the contestant's choice.
- The contesant knows this bias exists, but not what it is.
- In the case of interest, the host actually reveals a goat behind Door #3.
- Morgan and Gillman do not address this most general case, because the question "should the contestant switch" cannot be answered for it. Instead, they apply (whether they say so or not) the Principle of Indifference to the placement of the car, because to not apply it is "unlikely to correspond to a real playing of the game." I take that to mean either (A) that the car's placement cannot be observed to be biased in the past history of the game show, or (B) even if it could, the contestant's choice is uninformed about that history. Either criterion makes the Principle of Indifference apply.
- They ignore point #4 but not point #3 which is essentially the same thing. And finally, they ignore the Principle of Indifference for how the contestant evaluates the host's choice, even though that is just as "unlikely to correspond to a real playing of the game." The host would no more want to (A) exhibit a bias than the stagehands would in placing the car, and (B) the contestant is just as uninformed about whether he is biased. So Morgan's problem is:
- The car is placed randomly (Morgan assumes it, Gillman actually edited this into the problem he claimed "appeared in the Ask Marilyn column in Parade.)
- The contestant always chooses Door #1 (chosing other doors is a similar problem, but with different paramater values based on point #5 below).
- The host can open an unchosen door with the car. If he does, the game ends immediately.
- The host cannot open the contestant's door.
- The host has a set of biases that depend on the car placement and the contestant's choice.
- The contesant knows this bias exists, but not what it is.
- In the case of interest, the host actually reveals a goat behind Door #3.
- The interesty in this "conditional problem" is that it restricts (not eliminates) the host's ability to trick her into switching, thereby reducing the probability she will win the car. Specifically, he can trick her if his strategy included the option to end the game without offering the choice to switch. Otherwise, her probability after switching goes up to at least 1/2 (from the 1/3 it was before he applied his strategy), and possibly more. It goes to 2/3 if she does not have any idea how he is biased. It goes to somewhere in the range of 1/2 to 1 if she thinks she knows something of his bias, and is trying to out-guess him.
- These conclusions are not OR, they are contained in both Morgan's, and Gillman's, paper. They just are hard to read in the text because the authors were focused more on addressing the interesting issues, than on the problem from the contestant's point of view. The wiki article does a poor job (i.e., not at all currently) in separating out those issues. "The" answer to the probabiltiy after switching is 2/3 based on the K&W problem the article is about. JeffJor (talk) 17:08, 24 November 2009 (UTC)
- Jeff, I am puzzled by your point 3 above - The host can open an unchosen door with the car. If he does, the game ends immediately. This possibility is excluded in what they call the vos Savant scenario. Martin Hogbin (talk) 17:34, 24 November 2009 (UTC)
- Jeff is listing the range of problems the Morgan et al. paper addresses. Although it briefly mentions the possibility of the host opening a door and revealing the car, the majority of the paper addresses a problem (their interpretation of what vos Savant intended, based on her clarifications in Parade prior to the publication of their paper) which could perhaps be more simply expressed as having these defining characteristics:
- The car is placed randomly.
- The contestant has initially chosen a door (which we'll call door 1).
- The host has opened a different door (which we'll call door 3) revealing a goat and is not precluded from having a bias if given a choice between two "goat doors".
- The contestant, knowing which door was initially selected and now seeing two unopened doors and one open door showing a goat, can switch to the other unopened door.
- I believe this differs from the K&R version only in that the K&R version adds the explicit constraint that the host must choose randomly between two goats if this comes up.
- The main point of the Morgan et al. paper is that this is a conditional probability problem and should be approached as such. They use the unspecified host bias mostly to show that there is a difference between solving for the unconditional probability of winning by switching vs. the conditional probability of winning by switching given knowledge of which door was initially picked and which door the host opened. I believe most people understand the door 1/door 3 combination to be representative of the general solution, and (whether they realize it or not) erroneously evaluate the conditional probability in this case using the "equal probability" assumption (as described by Falk, and Fox and Levav). Whether the host has a bias or not, the probability of interest is the conditional probability. If you're an expert in probability theory, you can perhaps immediately see that unless the host has a bias the unconditional and conditional solutions must be the same - but (IMO) most people do not understand that these are different questions and consequently find an unconditional solution not very convincing since they're actually trying to solve the conditional problem.
- The root question the article needs to address is WHY the 1/3 probability of the car being behind door 1 remains the same AFTER the player sees the host open door 3. The first step to understanding this is to understand that the probability after the host opens door 3 is a conditional probability. The second step is either to understand that this conditional probability is the same regardless of which door the player initially picks and which door the host opens and that this means the conditional probability (whatever it is) must be the same as the unconditional probability (which is trivial to compute), or to understand how to actually compute the conditional probability.
- IMO, the article would be more clear if we explained the difference between the unconditional and conditional questions up front, and (perhaps) said the Monty Hall problem as generally interpreted intends for these two different questions to have the same answer. The problem is most published unconditional solutions make no mention of the unconditional and conditional questions, and only implicitly equate them (leaving open the question of what exact problem they are addressing).
- Jeff - do you agree given more or less any set of Monty Hall rules that asking what is the (unconditional) probability of winning by switching is a different question than asking what is the (conditional) probability of winning by switching given which door the player picks and which door the host opens? If so, which question do most people think the Monty Hall problem asks? -- Rick Block (talk) 20:36, 24 November 2009 (UTC)
- Martin, that point represents the parameter Morgan call p22. They never use it explicitly in their results, but they do use p23 and the identity 'p22+p23=1. So anytiem they discuss, p23, they are using that point.
- So, in Morgan's terminology, the scenarios are:
- Marilyn vos Savant's actually-intended scenario: p23=1 and p13=1/2. Marilyn clarified both parts of that.
- Their pedantic misinterpretation of Marilyn vos Savant's scenario: p23=1.
- Their general scenario: no restrictions of their stated paramaners.
- The actual general scenario, alluded to in their conclusion, incldues a non-zero p11.
- Rick, I don't know how many different ways I can try to say this, but it is not getting through to you. The "given knowledge of which door was initially picked and which door the host opened" is useless to the contestant unless she knows both the value of q and which door is favored. Morgan's conclusions DO NOT use any such knowledge as a given. They only use the fact that, whatever q is in what they call the "vos Savant scenario," that specific strategy can't make the probability less than 1/2 by switching. They only answer teh quesiton "shoudl she switch," not "what is the exact probability if she switches?"
- And the "door 1/door 3" combination IS REPRESENTATIVE OF the general solution, even if the the host is biased, because the contestant does not know which door is favored. Look at it this way: suppose the host says to the contestant "I am using Moragn's stratgey based on a q of..." and names a value. "But," he adds, "I won't tell you which of the two doors, #2 and #3, is favored this way." From the contestant's SoK, WHICH IS THE ONLY ONE RELEVANT TO THE PROBLEM, the exact probability of winning is 2/3. Fporm the host's SoK, it is either 1 or 0, since he knows where the car is. Nobody possesses the SoK where the probabiltiy is 1/(1+q). That wasn't Morgan's point.
- We all seem to have fairly entrenched positions here so let me suggest that we leave the issue of conditional/unconditional for the moment. I agree that even if the host goat door choice is taken to be random, the problem can be treated as one of conditional probability. Whether this is necessary or desirable is another matter, which I suggest we leave for the moment.
- This leaves the point that Jeff makes above, which is the same point that I made at the start of this conversation when I came here in response to the RfC. If we are answering the question from the state of knowledge of the player, and the player does not know the host door choice strategy (the value of the parameter q) then we must take host the choice of door (from those permitted by the rules) to be random. Rick, do you accept this? Martin Hogbin (talk) 23:11, 24 November 2009 (UTC)
- I'll accept that if your goal is to determine a single numerical answer for the probability it is most sensible to take the host's choice (from those otherwise permitted by the rules) to be random, and that most people who pose this problem intend this to be the case. On the other hand, must is too strong. It's perfectly legitimate to leave this as a variable and to express the probability as a function of this variable. It is, in fact, helpful since (as Jeff notes) it can be used to show a specific player's chances never decrease. Saying the answer is "2/3" and saying it is "1/(1+q) for some unknown value of q" do not contradict each other. Both answers are right.
- In Jeff's example above where the contestant knows q but not which door it pertains to, I'd say the exact probability is either 1/(1+q) or 1/(2-q). The total probability works out to 2/3 since the probability that it's 1/(1+q) is (1/3)(1+q) and the probability that it's 1/(2-q) is (1/3)(2-q), but saying you can simply ignore q in this case takes an argument of some form. If you're given conditions in the problem statement that you're ignoring, you really need to say why it's OK to ignore them. Ignoring them because you simply "know" they have no effect on the answer doesn't cut it.
- Similarly, expressing the probability in terms of something that it depends on but is not constrained by the problem statement is not wrong but (I'd say) is actually a better answer. From the formula you can derive what the numeric average should be for a random set of trials, but it also can lead you to other factors that you could observe that might be important. -- Rick Block (talk) 01:42, 25 November 2009 (UTC)
- No, Rick, "must" is not too strong. The ultimate question is "should the player switch," and that can only be evaluated in the player's SoK. And let me reiterate that there is no SoK where the probability is 1/(1+q), because the only SoK that includes any knowledge of a value for q - that is, the host's SoK - also includes the exact knowledge of where the car is. In that SoK, the probability is not a choice between 1/(1+q) and 1/(2-q), it is between 1/1 and 0/1.
- Let me try another example by tweaking the problem to get certain results that might re-focus you. Suppose before the initial choice, the host says "The car has 50% chance to be behind #1, and a 25% chance to be behind #2 or #3." The contestant, naturally enough, chooses #1. The host then says "I will now open one of the other doors that does not have a goat. If both have goats, I will choose with a 75%/25% bias - but I won't tell you which is favored. You can then either keep your initial choice, or switch to the other closed door." He then flips two coins behind his back, and opens #3. Should the contestant switch? If my math is right, by your approach the chances are either 40% that switching will win, or 66.7%. In one possibility, the chances of winning go down; in the other, they go up, but by more than the other case's chances went down. But still, the player should switch. The possible probabilities 40% and 66.7% are not what is important to the contestant when making this decision, only the combined probability of 53.3%. (And incidentally, I did not say whether the assignment of the bias by the host is itself biased - it doesn't matter.)
- The so-called "conditional solution" is only useful to answer the question "should the contestant switch" if every possible value that q could take (in my example, I made two possibilities) is weighted (as I did with the weights 25% and 75%), and the corresponding values for P(Ws|D3) are averaged. This is not possible to do, in general, as Morgan correctly points out. We can't know the weights. But since every P(Ws|D3) is at least 1/2 when the car is randomly placed, regardless of q, they can conclude that the weighted average is also at least 1/2, regardless of the weights. That means the player should switch regardless of any value q could take. Not because of a q, regardless of it. (And if you check my unbalanced scenario, it has the same property as long as neither door is associated with a q, even though some qs do make for a lower probability.)
- Rick, you asked, given a specific host strategy, (1) if I thought whether asking for the "conditional" answer was different that asking for the "unconditional" answer, and (2) which one most people see in the Monty Hall problem. The fact that you have to ask either question shows that you haven't understood anything I've said. First off, I had just explicitly said that K&W, Marilyn vos Savant, the cited researchers who did psychological studies, and most people all see the problem, all see the only the "unconditional" problem unless it is explicitly stated how the "conditional" scenario applies. It IS the Monty Hall Problem. Anything else is a variation.
- Second, there really is no difference between the two problems. The apparent difference is created by whatever boundaries are placed on what you want to call "one" specific host strategy. What you call "the unconditional solution" does allow for the host to have a bias, but it does not specify which door the host favors with that bias. So whatever q may be, the contestant sees two host strategies in your one: one with a split of q:(1-q) for the two unchosen doors, and one with a split of (1-q):q for the same doors. But the contestant sees them as equally likely, where you want to see only one as possible. Since these two strategies are indistinguishable to the contestant, the Principle of Indifference says they must have equal weights in the solution. So the contestant can replace those two host strategies with one, where she averages the two weights given to any specific door. The average of q and (1-q) is 1/2. So whether or not the host is biased, unless the contestant is told which door is favored, the contestant must treat the host as being unbiased.
- Morgan, et al, and Gillman, and any others who address what you call "the conditional problem," are all adding additional boundaries to the host's startegies. By doing so, they create a set of strategies that cannot be used directly when answering the question. Their entire thesis is that no matter which of the strategies that fit in their boundaries is actually the case, the answer is still "switching is best." And now my entire argument can be encapsulated in one statement, so I will set it apart:
- The formula that Morgan, et al, give for P(Ws|D3) is not an answer to the Monty Hall Problem. It is
- the complete set of possible answers based on certain assumptions. Any one of them could be
- the correct answer, keeping within their assumptions, if the contestant had more information.
- But every answer that is correct, within their assumptions, is in that set. And every answer is
- at least 1/2. So regardless of what additional information the contestant could gain, again keeping
- within their assumptions, it will always be advantageous to switch.
- They state, several times, that there is no need to assume the host is unbiased. They fail to recognize that, although that is one possible way to view the assumption being made, it isn't the actual assumption. The assumption, derived directly from the fact that the problem statement does not say which door could be favored, is that the contestant is unbiased in how she treats each door. Just like if you always choose "rock" in "rock, paper, scissors," but I don't know that and choose randomly, I have a 50% chance of winning. You are biased, by my assessment of your possible choices must be unbiased so I consider it a 1/3 chance that you will pick any of the three. The solution to the actual Monty Hall problem has to do the same. Even if the host is biased in opening doors, the contestant does not know how and so treats the host as unbiased. This isn't the same as saying he is unbiased.
- And they make similar assumptions, in order to get their set of possible soluitions. Just not about the host's choice. They assume the car is randomly placed, which vos Savant never said, and whcih is similarly unnecessary It must be treated as unbiased because the contestant will choose any one with equal likelihood.
JeffJor (talk) 16:38, 25 November 2009 (UTC)
- Rick, I agree with you that [my italics] "if your goal is to determine a single numerical answer for the probability it is most sensible to take the host's choice (from those otherwise permitted by the rules) to be random", but I am not sure what other goal one might have for a probability question. Of course it is possible to produce a general answer where we do not apply the principle of indifference to the choices about which we are given no information. If we represent all the unknown choices by parameters we simply find that the probability of winning by switching is from 0 to 1, depending on where the car was initially placed and which door the player chooses. Not a very interesting result. Martin Hogbin (talk) 17:04, 25 November 2009 (UTC)
- Jeff - I believe I understand what you're saying perfectly well. I am on the other hand less than certain that you're understanding what I'm saying. Your opinion (or Martin's, or mine) about what IS the Monty Hall problem, no matter how fervently you believe it, is (from Wikipedia's perspective) original research and is not relevant to the content of the article. What is relevant is what reliable sources say. We have (of course) already discussed this on this page, and it's not even archived yet (see above, #What do the sources actually say).
- Rick, I am not basing what I think the MHP "is" on OR. It is the expressed opinion of those who wrote it (MVS specifically), it is explicitly contained in the problem statement this article is based on, and it is what is used by many of the references. Do you disagree? With what? In all those versions, the car is placed randomly, the host's strategy is to always open a door, with a goat, and to pick randomly. They may not explain why they think "randomly" is included in there either once or twice; but they all do at least once, including Morgan and Gillman (who edited it into what he claimed was a quote). They never explain why. You may think my providing the reason for them is OR, but it is not. I'm just explaining what they assumed was so implicit they did not need to say it.
- The question is never "what is the probability of winning?", it is "should the contestant in the problem switch?" Morgan answers that question IN STRAWMAN FASHION under a set of assumptions that they (not quite clearly) state, which include "car is placed randomly" and "host's strategy is to always open a door, with a goat, and to pick with a parametric bias based on door numbers." All they concluded is that under those conditions, switching is either a neutral choice or it improves the odds. They did not address how the contestant should know what q, becasue their conclusion did not require knowing q. Or say that their answer is "correct" for the MHP in general. And look at the second paragraph of their conclusions. They imply they are solving "the unconditional problem," but if you compare it to Grinstead and Snell, you will see that they are not. They are taking the two possible "unconditional" solutions and combining them is a way that becomes transparent to the contestant not knowing which door is favored; which is to say, they are putting the conditional solution into the SoK of the contestant. The answer they give for that SoK is 2/3.
- And read that Grinstead and Snell. They state quite clearly that the probability for both the conditional and unconditional MHP, as stated, is 2/3. They just require different solution methods. To illustrate that, they apply a specific q=3/4 and so it is a different problem. It is a problem based on the MHP, but it not the MHP itself. The "before" and "after" bit you refer to is a red herring. The timing isn't what is important, it is the knowledge. If the contestant chooses to switch "before the host opens a door" = "without any knowledge of how the host may use door numbers to pick" then the probability is 2/3. That's what establishes the difference. Morgan implies this is the important difference when they talk about their anlysis only affecting "informed players." JeffJor (talk) 19:49, 25 November 2009 (UTC)
- Martin - Strawman arguments are not helpful. The sources that introduce q don't "represent all the unknown choices by parameters", but rather a specific interesting one that highlights the difference between variants of the problem where the player might as well choose to switch before the host opens a door from those where the player specifically chooses after. IMO (to be clear, this is my opinion - not something from a published source) the difficulty most people have with the MHP is precisely because the point of decision is after the host has opened a door meaning most people treat it as a conditional probability problem. I definitely agree with Morgan et al. that whether the host is explicitly constrained to pick randomly between two goats, or whether we treat this choice as random because we don't know q, or whether we know q and which door it applies to "it [the MHP] is still a conditional probability problem". -- Rick Block (talk) 18:16, 25 November 2009 (UTC)
- Rick, I thought we had agree to drop the conditional/unconditional argument for the moment. Yes. Morgan have, for reasons known only to themselves, chosen to address a specific interesting problem. But, as you have already agreed, there is only a difference between the case where the player chooses whether to swap or not before a door is opened and the case where they chose after it has been opened if the player knows the host door opening strategy. If the player does not know this, even though they may have seen which door was opened, '...it is most sensible to take the host's choice (from those otherwise permitted by the rules) to be random'. This gives exactly the same probability of winning by switching as in the case where the player chooses whether to switch or not before the door has been opened. Do you not agree?
- What you call my strawman argument was intended simply to show the natural result of dis-applying the principle of indifference consistently. Martin Hogbin (talk) 19:34, 25 November 2009 (UTC)
What the Morgan paper does do.
I have added this section to try to find find some common ground regarding the Morgan paper. Basically I can agree that the Morgan paper was a response to vos Savant and her solution. As I understand it, vos Savant, proposed the set of rules that were not stated in the Whitaker question, namely that the host always offers the choice and always opens a door to reveal a goat. These rules have become the standard rules for the problem and were not criticized by Morgan.
Vos Savant also states that she takes the initial car placement to be random, this assumption is tacitly accepted by Morgan. She does not state that she takes the players initial choice to be random, and from the player's perspective I guess it is not. As it happens, this turns out not to be important and Morgan do not criticize her for not stating her assumptions in this respect.
The only mistake that vos Savant makes is that she fails to state that she takes the host's legal door choice to be random, even though it is clear that she does in fact do this and later stated that this was what she assumed. What the Morgan paper does do is to point out this omission by vos Savant.
Thus we can agree that the Morgan paper is a valid criticism of the vos Savant solution.
From my personal point of view it is a grossly excessive response to a simple failure to point out one assumption and a response that attempts to make this simple omission into a major problem requiring an 'elegant solution'.
The Morgan paper is not, in my opinion, a well-written, detailed, and comprehensive analysis of the Monty Hall problem which makes clear the exact question that it is answering or all the assumptions that it is making in its calculations. Martin Hogbin (talk) 17:40, 25 November 2009 (UTC)
- The entire issue, as you propose it, boils down to whether the "Whitiker question" is about one specific instance of the game show, in which case every fact in it is to be taken literally and apply to that day only; or if it was supposed to be an illustrative example of the game in general, in which case such facts are circumstantial. They illustrtate a point of variability between different days if the game allows that variability, otherwise they describe the game itself. The problem is, that the question (as asked, not as misquoted by Morgan or Gillman) is unanswerable in the former case. We can't know if the host always offers a switch, among other things. We would only know that he did on that day.
- The only way to justify the assumptions needed to make it solvable, is to treat it as an illustrative example that defines the game itself. And you can't pick-and-choose which interpretation you want for different facets of the problem. There are always three doors, because nobody in the illustrative example choose a number of doors. A door is always opened, because nobody choose whether to do it in the example. A goat is alwasy revealed, because it didn't say Monty choose between a car and a goat. However, any specific circumstances about those choices vary with each show, and can only do so randomly. That includes car placement, contestant choices (it says "you choose a door," not "you always hoose Door #1"), and host choices. Most people who read the problem do not even consider that it is a specific instance. This is effectively admitted by Morgan, who say few even considered the "conditional" problem of Choose 1 Open 3. Then they fail to recognize that those whom they claim did, addressed the solution based on the host's choice being between indistinguishable doors. So theirs was not the same "conditional" problem Morgan addressed, where bias is important.
- Rick, I see little evidence that what you attribute to Marilyn vos Savant is correct. She explained no assumption in the original column, she just implied her assumptions by using them in her solution. Specifically, she never said the placement is random. In followups, she only explained one of the assumptions from that interpretation. She said "So let's look at it again, remembering that the original answer defines certain conditions, the most significant of which is that the host always opens a losing door on purpose. (There's no way he can always open a losing door by chance!) Anything else is a different question." She never even said "always opens a door." She never identified what the other conditions were, she just kept using "1/3" as the probability that the car was behind any door, and "1/2" for the probability when the host chooses between two goat doors. There was no discussion of why, contrary to what you think. But it is a clear implication of what she thought the other "certain conditions" were. All of the conditions were based on the illustrative example I described above.
- Thus, what we can see is that the Morgan critisism is about a question MVS never thought she asked. It is not valid. She did not "omit that the host's choice was random," she never considered it to be possible for the user to use such information even if it was true, which makes it random in the contestant's eyes. All the paper was, in my opinion, was an attempt to apply what tools could be applied to the problem in a real-worl setting. By scholars who were too familiar with applying their science to real-world cases, to see that they were requiring a specific game, and a specific host, in what can only be a hypothetical problem. The very suggestion that Bayesian Inference could be used to refine the host's strategy demonstrates how that could not separate the real-world from the hypothetical.
- But it does make its conclusions clearer than you allow for, once you read it with an educated mind. The primary conclusion is "The fact that P(Ws|D3)≥1/2, regardless of the host's strategy, is key to the solution." There is nothing unclear about that. They aren't saying what the probabiliy is, they are saying how its range affects the choice to switch. The fact that they never discuss the contestant using information about any of their parameters shows that they are keeping the two SoK's separate, even if they don't explain that to the reader. JeffJor (talk) 20:58, 25 November 2009 (UTC)
- Jeff - You do realize this section was started by Martin, not me (Rick) - right? I'll respond to both of you.
- Martin: the Morgan et al. criticism of MvS is not that she didn't specify that the host's choice between two goats is random, but that her solution is not the solution to the problem that is asked. We've been over this so many times I'm surprised you apparently continue to fail to understand this.
- I fail to understand this because there is no 'problem that is asked', but do not take my word for it, read the comment on the Morgan paper by Seymann. In order to get a clearly defined problem certain details must be added. Whitaker actually says, 'opens another door, say #3', Morgan take this to mean, 'opens door #3', which interpretation they support by misquoting the original question. Vos Savant takes the question to mean, 'opens another door (to reveal a goat)'. This is an equally valid interpretation of Whitaker's question, in fact it is much more likely to be what Whitaker actually wanted to know. Do you really believe that he was only interested in the case where the host opens door 3?
- Martin: the Morgan et al. criticism of MvS is not that she didn't specify that the host's choice between two goats is random, but that her solution is not the solution to the problem that is asked. We've been over this so many times I'm surprised you apparently continue to fail to understand this.
- Look at her enumeration of the cases that "exhaust all the possibilities" from her second column (reproduced here). In what she labels as games 1-3, the car location is placed behind door 1, door 2, and door 3 respectively. The listed result in these cases is the result of switching, assuming the player has initially picked door 1. Games 4-6 are the same but show the result of staying with the initial choice (of door 1). From the structure of this table we can clearly see she is assuming the doors are distinguishable (as implied by "Suppose you're on a game show"), she's assuming the car is uniformly placed initially, she's assuming the host must show a goat and make the offer to switch, and that she's following part, but not all, of the suggested case from the problem statement where the player picks door 1 and the host opens door 3. What this table literally addresses is:
- P(win by switching|player picked door 1)
- The probability is 2/3. She doesn't do it, but the same table can also be used to show the results given any other initial door choice, which is obviously also 2/3. No argument. But, this is not the probability that is asked about in the problem statement. What is asked about is
- P(win by switching|player picked door 1 and host opened door 3)
- This probability is not obvious from her table. In fact, if you casually try to infer it from her table you end up with 1/2 (the only excluded case is the one where the car is behind door 3). This is the error most people make when they initially think about this problem, i.e. they see three equally likely cases and one of them is made impossible by the problem statement leaving only two equally likely cases. And, to be clear, no one is claiming MvS is making this error, only that her approach is not addressing the problem that is asked. I suspect you've been following this page long enough to realize that there are people who "understand" the overall probability of winning by switching is 2/3 but who still think the conditional probability given the player has picked door 1 and the host has opened door 3 is 1/2.
- No not at all. I have never seen that particular misconception expressed here. Most people think the probability in both cases is 1/2, see K & W, for example. Martin Hogbin (talk) 18:39, 27 November 2009 (UTC)
- But you have missed my main point which is that the Morgan paper is criticism of vos Savant's explanation, it is not a well-written, detailed, and comprehensive analysis of the Monty Hall problem which makes clear the exact question that it is answering or all the assumptions that it is making in its calculations. It is thus not appropriate to base this article on the Morgan paper or its conclusions, despite its being published in a peer reviewed journal. Martin Hogbin (talk) 18:43, 27 November 2009 (UTC)
- Jeff: I agree with nearly everything you say here. I think the critical difference is whether you treat the doors as distinguishable or not. You treat them as indistinguishable. Morgan et al. don't. Assume for the moment the doors are distinguishable. Given this assumption, you must agree that
- P(win by switch|player picked door 1 and host opened door 2)
- P(win by switch|player picked door 1 and host opened door 3)
- P(win by switch|player picked door 2 and host opened door 1)
- P(win by switch|player picked door 2 and host opened door 3)
- P(win by switch|player picked door 3 and host opened door 1)
- P(win by switch|player picked door 3 and host opened door 2)
- are not all identically 2/3 unless q is 1/2 in each case, since if we're free to assign whatever q values we'd like in these cases we can make them each individually anything we want from 1/2 to 1. Getting back to where this thread started (two or three headings ago) I think what you're saying is that in a simulation of, say, 3000 iterations of the "problem" we can count ALL of them, if necessary renaming the door the player picks as door 1 and the door the host opens as door 3, and with this simulation the number of wins by switching will approach 2/3. I agree with this. However, if the doors ARE distinguishable, we can also keep track of the number of wins by switching separately, in each of the 6 combinations of initial player pick and host door, meaning we can ask about any of the following probabilities (which might be different):
- P(win by switching)
- P(win by switching|player picked door 1)
- P(win by switching|player picked door 1 and host opened door 3)
- Although it's a little harder to phrase unambiguously, I think the last of these probabilities is what most people understand as the question raised by the MHP, and is the one that most conflicts with people's intuition. -- Rick Block (talk) 18:05, 27 November 2009 (UTC)
- Rick - yes, I mistook who started it. I don't see that it matters much. Your statement, "The Morgan et al. criticism of MvS is not that she didn't specify that the host's choice between two goats is random" (emphasis added) is correct. Their criticism is that she didn't answer "the conditional problem," and they point out that it matters if the host's choice is not random. G&S are clearer when they say it only matters if the host's choice is not random. Morgan, et al, never say the contestant needs to know the host's bias to answer the question, they only say it isn't necessary to assume a uniform bias to answer the question. But no source suggests the contestant should consider that possibility, but the article suggests the contestant does.
- Morgan, et al, also says that MvS didn't consider a non-uniform placement of the cars. But they can't make the same conclusion here - that it isn't necessary to assume it is random - so they didn't placement bias at all. That is as much of an admission they give that it is a different problem. But it is such an admission, because they can't ignore it as they did if it is necessary. Had they provided their implied argument to justify ignoring car-placement bias, it would apply to host bias as well. I can't help it thay they omitted these arguments, but they did because even they consider host bias to be a different problem. And none of the sources you have quoted say it is necessary to consider in order to answer the MHP. They just say you don't need to assume it is uniform, either. But it is necessary to assume placement is.
- And in fact, I gave you the example (OR, I know, so I am not suggesting it go in the article - only that it be used to understand what Morgan is trying to say) where the car placement is non-random. If there is a 50% chance of being behind Door #1, and 25% for #2 and #3, then Morgan's approach does not work. The probability can be less than 1/2 that switching will increase the chances for a specific host stragegy favoring a specific door. But from the SoK of a contestant who does not know which door is favored and so must treat a bias either way with equal probility, it does always go up. Just not as much as with a random car placement. The point is that Morgan did not need to address why a contestant should assume a specific host bias, because it was not necessary. In their problem, it always went up.
- Yes, I will agree that those six probabilities you listed can be different, under one condition. That you agree that the contestant has no way to know which is different, and so must average the six probabilities before she can decide if switching is beneficial. That she cannot use any one of the those answers, which is EXCATLY' what Morgan means when they say the problem they are addessing can't be answered in general without knowing the host's strategy, but that she can use the non-informed combination of the three. And that if she does, the answer is 2/3m regardless of host strategy.
- The numbers MvS uses in her second column are only to distinguish the three doors "Chosen," "Opened," and "Unchosen, Unopened." She does it using the numbers in the example, rather than the more complicated names, because it is far more transparent to the reader. She assigns equal probability to them not because it is impossible for a bias to exist, but because the doors are truly interchangeable so they must appear that way to the contestant. If you read the text that goes with that table you refer to, she refers to the host openeinmg "a losing door," not "Door #3," and that she compares it to a shell game with indistinguishable shells. Then she uses face-down cards, which are also indistinguishable. She did not intend to ask, and in fact did not ask, "the conditional problem." Again, the numbers used in the original problem are examples only.
- And I disagree emphatically that "[P(win by switching|player picked door 1 and host opened door 3)] is what most people understand as the question raised by the MHP." Almost none do. Read the pages and pages of responses to that forum post you referenced. None see that problem, and I have only seen mathematicians who cannot see the forest for the trees do. When they do so, they never claim any typical readers to. So you need to stop inserting your own opinion, and OR, into the problem. JeffJor (talk) 18:55, 28 November 2009 (UTC)
- It's nice to see the conversation directed at the emphasis on Morgan, and not some of the old red herrings. Credible people were publishing solutions along the lines of the 'Combining Doors' solution before, during and after Morgan and the others. The idea that this problem can only be solved with 'conditional' or 'unconditional' probability formulas is bogus. Symbolic logic works perfectly well, and eliminates the 'hosts behaviour' canard. That the original door choice never changes from 1/3 based on Monty's actions is all that's required. Glkanter (talk) 00:43, 29 November 2009 (UTC)
- Jeff - again, I think we agree on many things. Morgan et al.'s criticism of MvS is that she didn't answer the conditional problem, and they say in this problem the host's protocol for choosing between two goats matters (to the extent that if it is not specified the we cannot answer what the probability is of winning by switching, only that it certainly doesn't go down).
- They don't exactly say why they assume uniform car placement, although they do say that non-uniform car placement could be considered. I'm not sure what your point is about this. The car placement can either be uniform or not. If you assume non-uniform the problem is really not very interesting (unless, perhaps, the player knows the distribution). Whether this distribution is uniform or not, it's still a conditional probability problem. If the distribution is as in your example Morgan's approach DOES work, they just didn't bother showing this generalization. Treating the doors as distinguishable, and the problem as a conditional probability problem, allows any variant to be easily analyzed.
- Regarding the six probabilities - I'll agree the contestant has no way to know which is different, but only if the doors are treated as indistinguishable. If "Door 1" means the door with the number 1 written on it (which I think it does - I understand you think it doesn't), then the contestant could know which is different. I think this boils down to what we take the introductory "Suppose you're on a game show" to mean. I (some WP:OR here) take it to mean the setting is a game show meaning real doors, on a stage, with identifying markings (numbers 1,2, and 3) so the player and host (and audience) can know which door is which. I think you're taking it to mean far less than this, perhaps only to help explain what we mean by "host" and "player" - but not "door". The assumption that the doors are distinguishable seems to me to be just as much implied as the roles of the host and player. They don't exactly say this either, but Morgan et al. are clearly treating the doors as distinguishable.
- The fact that MvS uses unconditional analogies, and addresses the MHP as an unconditional problem, is precisely the criticism Morgan et al. make. In their view, the problem is conditional (almost certainly because they view the doors to be distinguishable). It might have been nice if the problem used something that really is indistinguishable or explicitly said the doors are to be treated as indistinguishable, actually making it an urn problem, but that's not how the MHP is stated. On a game show, doors are distinguishable - the player knows which one she chose and can see which one the host opens. Modeling this as a math problem where the doors are indistinguishable addresses a different problem (which is what I understand you think IS the Monty Hall problem). This is exactly the point. Perhaps the MHP is BOTH of these problems. Certainly many solutions treat it as an urn problem, but this interpretation is what Morgan et al. criticize. I don't think I've seen anything published that explicitly makes the claim that in the MHP the doors should be treated as indistinguishable. From your POV they're simply solving the problem. What Morgan is saying is that they've oversimplified their model of the problem. -- Rick Block (talk) 16:19, 29 November 2009 (UTC)
- Rick, once again, nobody except the people who "solve" the "conditional" problem think that the MHP is about the "conditional problem." MvS did not. None of the lay people who read it thought it was. Very few of the mathematicians who read it though it was. Gardner did not. The literal interpretation of the MvS problem is not. The EXPLICIT statement of the MHP that defines what this article is about does not. I don't know how to more clear on this point, yet you refuse to acknowledge that it means the MHP is not "the conditional problem." Regardless of whether Morgan thinks so.
- Yes, the host's strategy in the "conditional problem" matters. But whether or not it matters is completely irrelevant, since the "conditional problem" is not the MHP. No matter how many times you ignore this, it still has to define how the article is written. The "conditional problem" is a different problem than the MHP, whether or not Morgan and Gillman think it is due to their misquoting it.
- And I can also repeat why the "car placement" distribution is important, but you will call it OR. It isn't. It is a trivial argument used universally in probability that is being applied inconsistently by Morgan as you read it. Where no explicit reason is given in the problem statement for why one option is more probable than the others, all equivalent options have to be treated equally in the solution of that problem. It is so trivial, that few people reference the reason when they use it. It is the Principle of Indifferece, it is well known, it is not OR, and it applies to both the host's choice and the car placement. Even if they are named, and so lok different. It is a principle that is NECESSARY for the solution to apply to both the host's strategy and the car placement. And that does not mean that anybody thinks thoses chocies are, in fact, uniform, which is where you keep making your mistake. It means that even if there is a bias, the contestant does not know what it is. So from the contestant's point of view, it LOOKS uniform. This is part of the trivial point of probability. It is part of the Principle of Indifference. It is not OR.
- When Morgan and Gillman take their TWO liberties, (1) addressing the conditional problem instead of the unconditional one that is intended and explict, and (2) letting the host have an undefined bias, they are addressing a different problem, and addressing it in a special way. They are saying that that one particular choice is unimportant EVEN IF IT IS CONSIDERED TO BE NONUNIFORM. The "interest" it has is not that it defines "the solution" as you seem to think, it is that it removes a question some might have about the correct solution. No refernce ever says it should be considered for the general MHP, they are just applying an interesting twist that says it is unimportant to that question. Morgan, et al, ignore the fact that there could be similar questions about car placement until the end, where they leave it as an exersize to the reader to apply it and find that there is no similar "interesting twist" there.
- About the "six probabilities." How do you suggest the contestant in this problem treat them differently? If you can suggest a way the contestant can do it, that is supported by the problem statement, I will withdraw everything I have said. But if you can't, you have to admit that the contestant has no way to do it and therefore can't. That means the contestant can't apply "the conditional solution," becasue teh contestant must treat the chocie as unbiased. And I will (again) reiterate that Morgan and Gillman never suggest a way, either. They don't say that the "conditional solution" is a solution to the MHP in general. They never say the doors can be treated as distinguishable. They only say that IF THEY ARE so treaed, it doesn't affect the answer to "should you switch?" You are confusing their hypothetical "if they are distinguishable" from a statement that "they are distinguishable." JeffJor (talk) 18:08, 29 November 2009 (UTC)
- Jeff - I'm sorry, but Morgan et al., Gillman, and Grinstead and Snell are very, very clear. All the following are quotes:
- Morgan et al.: Ms. vos Savant went on to defend her original claim with a false proof and also suggested a false simulation ...
- Morgan et al.: Solution F1: If, regardless of the host's action, the player's strategy is to never switch, she will obviously will the car 1/3 of the time. Hence, the probability that she wins if she does switch is 2/3. ... F1's beauty as a false solution is that it is a true statement! It just does not solve the problem at hand.
- Morgan et al.: Solution F2: The sample space is {AGG, GAG, GGA}, each point having probability 1/3, where the triple AGG, for instance, means the auto behind door 1, goat behind door 2, and goat behind door 3. The player choosing door 1 will win in two of these cases if she switches, hence the probability that she wins by switching is 2/3. ... That it [F2] is not a solution to the stated conditional problem is apparent in that the outcome GGA is not in the conditional sample space, since door 3 has been revealed as hiding a goat.
- Gillman: Marilyn's solution goes like this. The chance is 1/3 that the car is actually at #1, and in that case you lose when you switch. The chance is 2/3 that the car is either at #2 (in which case the host perforce opens #3) or at #3 (in which case he perforce opens #2)-and in these cases, the host's revelation of a goat shows you how to switch and win. This is an elegant proof, but it does not address the problem posed, in which the host has shown you a goat at #3.
- Grinstead and Snell: This very simple analysis [as a preselected strategy, staying wins with probability 1/3 while switching wins with probability 2/3], though correct, does not quite solve the problem that Craig posed. Craig asked for the conditional probability that you win if you switch, given that you have chosen door 1 and that Monty has chosen door 3. To solve this problem, we set up the problem before getting this information and then compute the conditional probability given this information.
- Your argument about what what people think the problem is would be much more convincing with references. I've been here before with Martin. The problem is, as far as I can tell, no one who treats the problem unconditionally bothers to make it clear exactly what they're talking about, but those who treat it conditionally do make it clear and go on to say that the unconditional interpretation is not what the problem says. So, who are we to believe - reliable sources that implicitly say X, or other reliable sources who explicitly say both not X and that the sources that say X are incorrect? As a matter of Wikipedia policy (WP:NPOV), the article needs to neutrally present both POVs here (which is what I think the current version of the article attempts to do). Whether you or I personally agree with any particular POV is irrelevant.
- Rather than continue this seemingly unproductive argument, can I ask what specific changes you think should be made to the article? It might be helpful to start a new section for this. -- Rick Block (talk) 01:11, 30 November 2009 (UTC)
- Rick, about your quotes: Morgan, et al, misquoted MvS. They thought she was trying to solve a problem where specific doors were included, and rewrote the problem so that it looked like she did. With her wording, to quote K&W, "semantically, Door 3 in the standard version is named merely as an example." That invalidates your quotes about it. Semantically, the actual MvS problem is not about Door #3. Semantically, the MHP is the so-called unconditional problem, not the conditional problem. Any reference that addresses the conditional problem has semantically changed the problem into a different one than was intended by MvS; whether they do it explicitly by changing the wording, like Morgan and Gillman did, or implicitly by allowing that the interpretation might be different, like G&S did.
- K&W also say that the solution to the conditional problem "focuses on the behavior of the host rather than on that of the contestant. Consequently, the change from the contestant’s perspective to Monty Hall’s perspective corresponds to a change from non-Bayesian to Bayesian thinking." Not one reference you have quoted ever justifies how a specific host perspective can be used BY THE CONTESTANT to answer the question "should I switch?" Or how a specific Bayesian prior could be chosen. They only establish the set of results that could exist, never a specific result that the contestant can use. They only establish that it is unnecessary to assume a specific host strategy w.r.t. opening a door, when the car is behind the chosen door, in order to answer that question. They all continue to make the standard assumptions for every other aspect - car placement, always opening a door, always revealing a goat - because those assumptions are necessary to answer the question. K&W tried to write those into the problem, to avoid having to assume them.
- The quote you gave from G&S is the one that is most (I would claim it is the only one that is) useful. You just skipped the parts that you don't want to include. "[The conditional problem] means only two paths [of the twelve in the unconditional problem) through the tree are possible (see Figure 4.4). For one of these paths, the car is behind door 1 and for the other it is behind door 2. The path with the car behind door 2 is twice as likely as the one with the car behind door 1. Thus the conditional probability is 2/3 that the car is behind door 2 and 1/3 that it is behind door 1, so if you switch you have a 2/3 chance of winning the car, as Marilyn claimed." In other words, under the standard assmptions, the two approaches to the solution give the same answer through different combinations of possibilities. G&S then go on to say "Now suppose instead that in the case that he has a choice, he chooses the door with the larger number with probability 3/4. In the 'switch' vs. 'stay' problem, the probability of winning with the 'switch' strategy is still 2/3. However, in the original problem, if the contestant switches, he wins with probability 4/7." All this does is demonstrate how the different solution methods give different answers under specific assumptions. It does not justify what specific assumptions should be made, and the difference in the answers is moot without such a justification.
- And while incorrect in several ways, Morgan takes the correct approach for how the conditional solution applies to the contestant when they say "The unconditional problem is of interest too, for it evaluates the proportion of winners out of all games with the player following the switch strategy. It is instructive to express this as a mixture of the two conditional cases: [derivation omitted] Pr(WS)=(p23+p32." This is incorrect because it is a still the conditional problem, it just uses four of G&S's twelve paths instead of two. Where they say "unconditional," they mean "not conditioned on knowing the host's strategy when Door 1 is chosen and has the car." They leave it in terms of two parameters that are a part of the host's strategy, but the usual assumption (explicit in the K&W statement) is that both are equal to one. In this conclusion, Morgan says the answer from the contestant's point of view is 2/3.
- I've given you references for people who treat it unconditionally. MvS. Gardner. Add Tierney. Mlodinow. I haven't read the others to find more such references, but I'm sure they are there. But all you need for proof is Morgan themselves, in their list of so-called false solutions. They all are employing the unconditional problem in one way or another. Your problem with seeing that, is that you only recognize one of the two ways of employing it. You can do it as G&S do, by making twelve paths through the set of possibilities; or you can do it by employing either two or four paths, and recognizing that the numbered doors are interchangable. THAT IS A CONCLUSION THAT CAN BE DRAWN DIRECTLY FROM THE UNCONDITIONAL INTERPRETATION, BUT ALLOWS USING DOOR NUMBERS IN THE SOLUTION. That is, assume the doors are lettered A, B, and C, and so are distinguishable in your approach (as opposed to your assumption, and it is an assumption, that the doors are numbered). Where the problem says "Door #1," that represents a uniform distribution of Doors A, B, and C. The twelve paths represented by G&S using A, B, and C reduce to four using 1, 2, and 3. Where the problem says "Door #3," that represents a uniform distribution of the two possible remaining doors (a different set in each case fo Door #1). That reduces four paths to two. The two-path solution does indeed represent the case where Door #1 is chosen, and Door #3 is opened. It is just form the contestant's SoK, which is the only SoK that is useful in answering the question.
- There is a good reason you have not seen more scholarly work on the unconditional problem. It is a simple problem that from a mathematical standpoint was completely understood and presented in citable works before MvS asked it. Nothing in any of the references you cite invalidates anything about it, and much supports it. The answer (for the probability) is 2/3, quite simply, with nothing to be learned about it.
- To "neutrally present" the conditional problem is what Martin and I are trying to accomplish. That isn't what is currently done. The current article's approach to it does not address the question in the MHP, "Should the contestant switch?" It addresses "given additional information, what can be said by a third party about the choice?" If you disagree, please show me a reference that specifically mentions how the contestant can use the conditional solution to decide. Not what some mathematician thinks the answer would be if additional information were available. This is about the third time I've asked, and you haven't done it. The conditional solution isn't an alternate POV, which Wikipedia policy says must be neutrally presented, it is a different problem that assumes there is a way to apply it that is unstated. A variant, to be sure, but a different problem. And I don't just mean conditional vs. unconditional. To use it requires additional information that is not in the problem statement, and so is not assumable.
- As we have said before, if that section is kept at all, it should be moved below the discussion of "Other Host behaviors" in the "Variants" section. It is not a "Probabilistic solution" to the general MHP, it is a solution to an alternate problem called "the conditional problem." It should say that some readers think specific doors are meant, but that the original fomulator did not (because she never said she did, and in every word she wrote she clearly thought the specific door numbers did not matter). As Seymann points out in the response to Morgan, she can't be held to rigorous standards in a sunday supplement. To prove that she emant the conditioanl problem, you need to provide positive proof that she intended those doors to be different. And there is none.
- The article should make clear that the Moragn's conclusions require two assumptions: (1) the conditional problem is considered, AND (2) the game-show strategy includes certain biases (but not others). Only then you can still conclude that switching is better. But this applies only to those specific biases, not to the general case. You can even cite Morgan for that last statement. The "Bayseian analysis" section can't be included as is without a discussion of the two different meanings of "Baysian." People too often forget that "Bayesian" can mean "Using conditional probabilities," or "Starting with a specific prior, using conditional probabilities to get answsers that apply in just that specific case." The latter meaning cannot be used in a hypothetical problem, since there is no basis for assuming a informed prior (a noninformative prior is not really assuming a prior, it is assuming that nothing can be distinguished even if there is nonuniformity). As such, it is really just the solution to the conditional problem, not a separate analysis that repeats the same conclusions. Regardless, it needs to be more closely associated with the conditioanl problem, since all it is, is that variant's solution.
- Finally, it is not a FAQ that people ask about the conditional problem. It is only mentioned by people who solve the conditional problem, or are reading an article about it so their quesiotn can be asnwered there. It needs to be de-emphasized anywhere it is mentioned. Oh, and most of this discussion needs to be archived. [Signature added later - I never get it right :)] JeffJor (talk) 19:11, 30 November 2009 (UTC)
- I can get on board with the changes to the article described here. Glkanter (talk) 16:27, 30 November 2009 (UTC)
- Claiming that Morgan is solving a "different" problem instead of the "real" one is at least as misguided as Morgan claiming, vos Savant was solving not the "real" problem. This is an ambiguous problem that can be modelled in several ways depending on which specifics you chose to replace the ambiguity and from which perspective you argue. Moreover it is not up to us to pick between vos Savant, Morgan or others, but we simply report their published approaches and criticism.--Kmhkmh (talk) 17:46, 5 December 2009 (UTC)
Is The Contestant Aware?
Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?
It seems to me that this is a (unstated) premise of the problem, as both vos Savant (Whitaker) and Krauss and Wang begin the problem statement with: 'Suppose you're on a game show'. I read this as clearly stating it is only the contestant's point of view we are concerned about. And, being a game show, the host is prohibited from divulging to the contestant either where the car is, or where the car is not.
Is there agreement on this, or is this in dispute? Glkanter (talk) 11:28, 29 November 2009 (UTC)
- As far as I know, this is the only known instance where a contestant at home was able to determine a game show's strategy. It was an aberration, an unexpected outcome, and steps were immediately taken to prevent it from happening again.
- In my humble opinion, a particular contestant gaining usable information from the hosts actions, which would make that contestant's SoK something different than the 'average' contestant's SoK of '2/3 likelihood of a car if I switch' is inconsistent with the published problem statement: 'Suppose you're on a game show'. The problem statement would become: 'Suppose you are not on a game show'. This is not merely a 'variant', or a new or changed premise. It is a completely different problem. And why this completely different problem should be referenced as often and prominently as it is in the Wikipedia article does not make sense to me.
- There are countless reliable published sources which use the combining doors solution to derive the '2/3 likelihood of a car if I switch'. The contestant uses this method to determine the 2/3, then properly says to himself, 'Monty's actions haven't given me any new information, so I'll go with the 2/3 for the average contestant'. Glkanter (talk) 14:49, 29 November 2009 (UTC)
- Is there a specific change you're suggesting? If not, I'd suggest moving this thread to the /Arguments page. -- Rick Block (talk) 01:15, 30 November 2009 (UTC)
- Rick, this is the central element in my criticism of the over-reliance on Morgan's paper in the article. In order to develop a consensus that includes you, I need to know your position on it, so I would greatly appreciate your response to the original question:
- "Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?."
- I see this as an 'editing' question more than a 'mathematics' question, so I'd prefer to leave it here. Thank you. Glkanter (talk) 13:51, 30 November 2009 (UTC)
- Rick, this is the central element in my criticism of the over-reliance on Morgan's paper in the article. In order to develop a consensus that includes you, I need to know your position on it, so I would greatly appreciate your response to the original question:
- Is there a published source that takes the stance that "Suppose you're on a game show" means what you're suggesting? If not, then what you're talking about is WP:OR which makes it a moot point as far as editing is concerned. I'm not saying it's a bad or invalid argument, just that if it's not published it's worthless for editing purposes. -- Rick Block (talk) 14:52, 30 November 2009 (UTC)
- Yes, I have previously linked to the Wikipedia articles showing the only 2 instances where individual contestants had a different State of Knowledge than would the average contestant. The 1950s Quiz Show Scandal and Whammy/Press Your Luck. Both were considered as extraordinary events.
- It wouldn't matter, even if it was OR. I'm not going to put this critical mistake of Morgan's in the article (unless other editors want to build a consensus for it). I'm only using it to decide, using facts rather than my personal opinion, how much emphasis Morgan should get in the article.
Rick, I have directly asked you this question many times, and have never seen a direct 'yes or no' answer from you. As this is a crucial element of the consensus that has been built, it is essential that we understand your reasons if you do not agree with the paragraph above:
- "It seems to me that this is a (unstated) premise of the problem, as both vos Savant (Whitaker) and Krauss and Wang begin the problem statement with: 'Suppose you're on a game show'. I read this as clearly stating it is only the contestant's point of view we are concerned about. And, being a game show, the host is prohibited from divulging to the contestant either where the car is, or where the car is not."
I look forward to your response. Glkanter (talk) 23:02, 1 December 2009 (UTC)
- Rick has responded under his comments section. For clarity and closure, I'll copy them into this discussion.
- "Glkanter asks why I haven't responded about his "Is The Contestant Aware?" question. Why should I? Glkanter has repeatedly demonstrated a complete lack of comprehension of nearly everything I've ever said. It's like trying to explain something to a cat. At some point you just have to give up. However, I'll give it another go. Meow, meeeow, meow, meowww. I'm not sure I have that quite right since I don't speak cat, but it's probably about as comprehensible to him as anything else I could say."
- Nijdam, your response to the question at the beginning of this section is also of great interest to the other editors. In the MHP that begins with "Suppose you are on a game show", is the contestant aware of Monty's door choice behaviour when there are 2 goats remaining? Thank you. Glkanter (talk) 15:14, 4 December 2009 (UTC)
- Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method? Yes, but that doesn't mean the contestant is prevented from wondering what affect this might have on her chances of winning by switching.
- Rick, how am I to parse 'yes, but' in the above paragraph? Is that "yes, I agree that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?" Or, "but..." Let the contestant wonder, it's a free country. But that doesn't affect the game play. Glkanter (talk) 18:51, 5 December 2009 (UTC)
- And, being a game show, the host is prohibited from divulging to the contestant either where the car is, or where the car is not. Is there agreement on this, or is this in dispute? The problem specifies that a game show is the setting. How much information the host divulges to the contestant depends on how the problem is specifically phrased, not on what "must" be true of game shows.
- These are both questions that are more appropriate for the /Arguments page than here, since as editors what we believe to be the TRUTH is ultimately of no importance. What is important is what reliable sources say. Our task as editors is to accurately represent what these sources say regardless of whether we individually agree. This one of the three fundamental policies of Wikipedia, see WP:NPOV. -- Rick Block (talk) 17:44, 5 December 2009 (UTC)
Variants - Other Host Behaviours
http://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_behaviors
The term 'Host Behaviour' is used earlier in the article to dis-prove the Popular solutions. And the concept of a host's bias, without the word 'behaviour' is used numerous times to dis-prove the Popular solution. In this section, the same issue is used for some unstated (fun?) purpose. That's probably confusing to readers.
This section is inconsistent with the Monty Hall Problem in that the results shown are not from the contestant's SoK. It seems to me that except for the 'forgetful host variant' which vos Savant described as 'random' (aka Deal or No Deal) the contestant must still rely on the 2/3 average. This table is likely to confuse people, rather than enlighten them.
Does the contestant get a do-over if Monty reveals the car in the 'forgetful host (random, aka Deal or No Deal) variant'? If so, then I guess the contestant could infer that Monty is acting randomly, and determine the odds are 50/50. **Wait a minute, if it's a do-over, and he shows a goat this time, wouldn't it be 2/3 if I switch?** Still no harm in switching, though.
The paragraph under the table relating to 'game theory' also uses the term 'host's behaviour' differently than the earlier occurrence in the article. And the conclusion may not be correct. With no knowledge of the hosts behaviour (motivations?), it's understood that it's best to switch with a 2/3 probability. Glkanter (talk) 17:25, 29 November 2009 (UTC)
- I'm not seeing the term used in any way other than to mean constraints on the host. In particular, it is not used at all in the section titled "Probabilistic Solution". The first three variants in the "Other host behaviors" section are all referenced, and are all notable. Sources should be added for the ones that are not sourced. In the paragraph below the table, "host behavior" also means constraints on the host's actions included in the problem statement. The "for example" sentence means if the problem does not constrain the host to always make the offer to switch, the host could be offering the switch only in the case the player has initially picked the car (per the NY Times interview with Monty Hall - see the Tierney 1991 reference). With no constraints on the host's behavior, it's understood that the probability of winning by switching can be anything. That is sort of the point of this section. -- Rick Block (talk) 05:18, 30 November 2009 (UTC)
- The most meaningful point is that the contestant, whose State of Knowledge the problem uses, never knows about these biases. When shown a goat, he always sees himself as the 'average contestant' with a 2/3 likelihood of winning by switching. As I mention elsewhere, once a host behaviour is introduced, the problem statement becomes the opposite of the Monty Hall problem statement mentioned twice in the article. I mentioned a few reasons I think a reader, who knows how game shows must work, would be confused by the irrelevant tables, narrative and the illogical change in whose State of Knowledge is being discussed. Glkanter (talk) 05:59, 30 November 2009 (UTC)
- The "host behaviors" we're talking about here are constraints that are either taken to be implied by the problem statement or explicitly given as part of the problem statement. The Parade version of the problem statement is widely regarded as ambiguous since these constraints are not explicitly listed, in particular whether the host must make the offer to switch and exactly how the host chooses what door to open. There are countless references that consider the impact of slightly different constraints on the host, that would presumably be the well known rules (i.e. explicit and within the SoK of the contestant). This is a subsection of a major section called "Variants". You're perhaps arguing to delete this section, but can you instead suggest a way to make this more clear? -- Rick Block (talk) 15:18, 30 November 2009 (UTC)
- However you describe these things, they all rely on the contestant having knowledge of the host's behaviour. At that point it's no longer the Monty Hall problem about a game show, which is the subject of the article. They're confusing, and do not further the reader's understanding of the Monty Hall problem paradox. Glkanter (talk) 15:33, 30 November 2009 (UTC)
- Again, what is being called the "host's behavior" are the rules under which the game show would be run, which would be known to the contestant. It could still be a game show if the host is not required to make the offer to switch (in fact, that might be an interpretation that would be more consistent with the actual Let's Make a Deal show on which the problem is obviously based). The fact that the Parade version doesn't fully specify the host's behavior is in all likelihood one of the reasons the "Monty Hall problem" is so very contentious. Different people read it as different problems, and then insist their interpretation is the "right" one. Discussing variants leads most people (perhaps not you) to a better understanding of the issues that are involved in the problem description. Many, many sources do this. If the article didn't do this, it would not be complete. -- Rick Block (talk) 19:42, 30 November 2009 (UTC)
- (1) And again, we must assume that any knowledge that is available to the contestant is also avaiable to the person who is trying to determine what the contestant should do - the puzzle solver. The fact that it is not mentioned, especially in a non-rigorous forum like a sunday supplement (see Seymann), means that it isn't necessary to answer the puzzle. For the elements of policy, like if a switch is always offered, that means what is implied in the one example is the full policy. MvS EXPLICITLY says this is her intent in one of the sources you like to cite when it suits your purpose, but apparently not when it contradicts it. For the elements of random choice, like car placement and host choice, it means it is uniform, something even your sources assume when to do otherwise would not make an "interesting" (see Morgan, and Gilman's intentional misquote of MvS) problem.
- (2) Read K&W, and don't give us your unsupported (and unsupportable) opinion. They conclude that the reason for contention has nothing to do with any ambiguity, it is an inability to assess how information based on part of the random nature affects the results. It isn't "different problems," it is not understanding what is essentially the Principle of restricted choice. The two "unconditional camps" - the 1/2'ers and the 2/3'ers are conceptualizing the same problem. And it isn;t the conditioanl problem.
- (3) The only thing that discussing variants helps is understanding the variants. Again, not one of your references applies their results to the contestant. Morgan even reduces it to what they call the unconditional problem (but really isn't, it just has the same answers under the assumptions they do make that are equivalent to what they try not to make) when they try to apply it to the contestant. And there is a good reason: they were addressing a variant, not the MHP itself, and they did it only because it was interesting. Not topical. JeffJor (talk) 22:27, 30 November 2009 (UTC)
Wikipedia Editing Works Best With A Consensus
I probably know the least about this topic. But it looks like there is a lot of agreement amongst active editors. Not unanimous, of course, but that's not a requirement, is it? Glkanter (talk) 21:07, 29 November 2009 (UTC)
- What specific changes are you thinking there's consensus for? -- Rick Block (talk) 01:23, 30 November 2009 (UTC)
- Hi Rick. Thank you for your response. As you see, I've started 4 new sections. I'm especially interested in your responses to the various questions I ask and thought-provoking statements I made, as you appear to not be part of the consensus. We may have differing opinions, but we can work together within the parameters of Wikipedia. I'm laying all my cards out. That's what collaboration's all about, right? So, what's the Wikipedia definition of a 'proper' consensus? Glkanter (talk) 01:38, 30 November 2009 (UTC)
- Again, consensus for what? Assuming you're talking about major changes to the article, I would think any reasonable definition of consensus would have to include at least most of the folks who commented at the last featured article review, archived here. There were a flurry of changes since then - running any major changes by the folks involved in those would be a good idea as well (e.g. Dicklyon, Nijdam, Henning Makholm, Glopk). -- Rick Block (talk) 04:54, 30 November 2009 (UTC)
- Yes, of course, major changes, de-emphasisng the reliance on Morgan. I think that's consistent with the published views of JeffJor, Martin and myself. With all the documented problems with Morgan, it's time to get to work. I mean, the filibustering has to end sometime, right? I'm sure all those other guys you mentioned are anxious to join the discussion. Because, as I read on the subject of Wikipedia Consensus:
- Not hypothetical
- While everyone on Wikipedia has the right to be heard, this does not mean that discussions remain open indefinitely until we hear from them. Nor does it mean that a consensus should be overridden by an appeal to "Wikipedians out there" who silently disagree. In essence, silence implies consent. If you believe that the current discussion does not represent real opinion, either prove it by referring to an existing discussion, or suggest starting a new discussion with a wider audience.
- You are misinterpreting what you've quoted from Wikipedia:What is consensus?. We're talking here about things that have already been discussed, not a new topic. We know a specific set of users who have previously expressed opinions on these topics. It's a simple matter to ask them. Do you want to, or should I? -- Rick Block (talk) 15:35, 30 November 2009 (UTC)
- Which buddy would that be? -- Rick Block (talk) 16:16, 30 November 2009 (UTC)
Morgan's Paper Doesn't Specifically Mention The Combined Doors Solution
Is it meaningful for our purposes then, to assume, infer, or offer opinions on why or why not?
Can we infer anything about Morgan's thoughts on The Combined Door Solution if it's not specifically mentioned?
I think any such discussion would be out of order. Glkanter (talk) 22:08, 29 November 2009 (UTC)
- Are you specifically talking about the sentence that says "Morgan et al. (1991) state that the popular solutions are incomplete, because they all make assumptions about the probabilities after the host has opened a door, without proof"? How about:
- Morgan et al. (1991) state that many popular solutions are incomplete, because they do not explicitly address the specific case of a player who has picked Door 1 and has then seen the host open Door 3.
- If this is the sentence you're talking about, would you find this wording more acceptable? -- Rick Block (talk) 15:53, 30 November 2009 (UTC)
- No, a bigger picture than that. The article is edited as if Morgan's claim trumps all others. I'm pointing out that Morgan didn't state that 'The Combined Doors solution' is false/incomplete/etc'. That's an interpretation of Morgan's paper that you've made, but it's not supported by anything that's been published. So, it's wrong for the article to say, or have a POV, that the Combining Doors solution is in any way inferior/incomplete/etc., because that has not been explicitly published. Glkanter (talk) 16:05, 30 November 2009 (UTC)
- Well, then, where in the article are you seeing this criticism of the Combining Doors solution? The only occurrence of the letters "combin" in the entire article is in the paragraph describing this solution. If you're not talking about the sentence I've suggested, what are you talking about? The fourth paragraph in "Sources of confusion"? We can work on that paragraph, too, but if you're not going to say specifically what you don't like it's rather hard to fix it. -- Rick Block (talk) 01:29, 1 December 2009 (UTC)
- BTW, if nobody objects in the next day or so, I'll make the change I suggested above. -- Rick Block (talk) 01:32, 1 December 2009 (UTC)
Excessive Reliance On Morgan
Each instance in the article of a host behaviour, or host bias, or host prejudice is indicative of a reliance on Morgan's paper. Regardless of what is being illustrated, this topic only exists among Morgan and a few others.
Since the problem statement of both vos Savant (Whitaker) and Krauss & Wang begin with: "Suppose you are on a game show", we know that this host behaviour will not be shared with the contestant, whose State of Knowledge is the only one asked for in The Monty Hall problem.
So, while Morgan is published, his argument is flawed. It reminds me of what Nijdam once said about Devlin, 'His name is Morgan, not God'. The moment the problem is restated to rely on a host behaviour, it's no longer the Monty Hall problem. The problem statement becomes: 'Suppose you are not on a game show'. Which is the exact opposite of how both Monty Hall problem statements in the article begin.
Yes, Morgan should be referenced, but with such an obviously erroneous argument, it hardly deserves the great emphasis it currently enjoys. Glkanter (talk) 05:50, 30 November 2009 (UTC)
- You are completely and utterly wrong about any mention of "host behavior" being indicative of a reliance on Morgan. The Parade description of the problem is nearly universally agreed to be under qualified. What most people intend by the problem is consistent with the Krauss and Wang version, but the difference between this version and the Parade version is precisely the topic of "host behavior". As Seymann says (the quote Martin keeps bringing up): "Without a clear understanding of the precise intent of the questioner, there can be no single correct solution to any problem." It is by specifying the "host behavior" that the precise problem can be understood. If aspects of the host's behavior are not specified you may be able to make reasonable assumptions about them, but your assumptions may not match what whoever is posing the question meant. -- Rick Block (talk) 16:14, 30 November 2009 (UTC)
- Rick, I am not sure what you mean when you say that the difference between the K & W version and the parade version is host behaviour. In the K & W statement quoted in this article the host is stated to choose randomly, just like the car is stated to be placed randomly. Martin Hogbin (talk) 10:02, 3 December 2009 (UTC)
- "Host behaviour" means several things. Just because you can model all the different aspects with probabilities, that does not make them all fall under one umbrella. When "most people" (and I will supply as much support for that as you just did, Rick: none) use "host behavior," they interpret it to mean "does he always open a door," "can he reveal the car," and even "can he open the chosen door" which Morgan and Gillman ignore. Not "is the host biased toward one functionally equivalent choice over another, based on door numbers?" These are diffferent issues, and only Morgan (plus those that follow using the conditional problem) consider that possibility. But it is not something that can be used to directly address the question "should the contestant switch?" It can only be used by a third party who knows every factor that determines what p13 is on that particular day, to address the question "given this additional information, can we second-guess the contestant?" The answer is "no, given all of these other assumptions about behaviour that are as unfounded as the one about the choice of doors." That answer is useless to the contestant herself unless the problem statement says she has knowledge of p13. And nobody - not Morgan, not Gilman, not G&S, not K&W - says she does, or contradicts what I am saying here. JeffJor (talk) 19:35, 30 November 2009 (UTC)
Changes suggested by JeffJor, Martin Hogbin, and Glkanter
If you're here because you've been invited to comment, there are ,two,. three (related) suggestions.
- #Glkanter's suggestion: Eliminate all 'host behaviour, etc' influenced discussion, save for the Wikipedia minimum necessary references to Morgan and his ilk, as the 'conditional' problem is the converse of "Suppose you are on a game show."
- If nobody minds, I'd like to revise my proposal to make it more reflective of the literature: 3 Sections to the article: The unconditional MHP, A brief discussion on why Morgan and the 'conditional variants' are not the MHP, and 'diversions' - which includes 'variants', etc.
- #JeffJor's suggestion: The so-called conditional problem needs to be a separate article, with "conditional" in its title.
- #Martin Hogbin's suggestion: This article should concentrate on the unconditional solution with the Morgan's conditional solution in a variations section.
Please indicate in subsections below whether you favor or oppose each of these suggested changes.
The intent is to try to determine whether there is community consensus for any of these changes. I would suggest one subsection per user who is commenting, and to avoid endless arguments, restricting your comments to your own section (this is modeled after the process used at Wikipedia:Arbitration Committee). I've precreated sections for everyone I've explicitly invited to comment. -- Rick Block (talk) 15:31, 2 December 2009 (UTC)
Discussion about setting up this section and inviting folks to comment
In this section please summarize the changes you're suggesting. I'll be asking the set of folks I mentioned to Glkanter above to come here and offer their opinions, so please keep it as brief as possible. Please let me know when you think this section is ready for others to comment on. -- Rick Block (talk) 01:07, 1 December 2009 (UTC)
- Rick, have you invited Boris Tsirel, William Connolley, or C S to contribute their opinions? What sort of time frame do you have in mind before 'In essence, silence implies consent' as per Wikipedia policy? Glkanter (talk) 15:47, 2 December 2009 (UTC)
- I haven't invited anyone yet. The list of folks is the set of users I added below, plus I'll post something at Wikipedia:WikiProject Mathematics. I can specifically invite Boris, William, and C S if you'd like. As far as the timeframe, I was thinking maybe something like a week or two. -- Rick Block (talk) 15:53, 2 December 2009 (UTC)
- I was going to, but had an edit conflict with Martin as he added his new section below. We need to straighten this out first. -- Rick Block (talk) 16:06, 2 December 2009 (UTC)
- I've notified all (including Wikipedia:WikiProject Mathematics) using template:please see referring them to this section. -- Rick Block (talk) 04:20, 3 December 2009 (UTC)
Glkanter's suggestion
- This is from the section above.
- Each instance in the article (and the various FAQs) of a host behaviour, or host bias, or host prejudice is indicative of a reliance on Morgan's paper. Regardless of what is being illustrated, this topic only exists among Morgan and a few others.
- Since the problem statement of both vos Savant (Whitaker) and Krauss & Wang begins with: "Suppose you are on a game show", we know that this host behaviour will not be shared with the contestant, whose State of Knowledge is the only one asked for in The Monty Hall problem.
- So, while Morgan is published, his argument is irretrievably flawed. The moment the problem is restated to rely on a host behaviour, it's no longer the Monty Hall problem. The problem statement becomes: 'Suppose you are not on a game show'. Which is the exact opposite of how both Monty Hall problem statements in the article begin: "Suppose you are on a game show". Morgan's criticism and his solutions are not relevant to the Monty Hall game show problem, which is the subject of this article.
Only because it's been published, Morgan should be referenced, but with such an obviously erroneous argument, it hardly deserves the great emphasis it currently enjoys. All other references to host behavior, etc., 'conditional vs unconditional', 'variants', and the Popular solutions being in any way inadequate should be removed from the article.
- A second section would explain why Morgan and 'conditional variants' are not the Monty Hall problem
- A final section on 'diversions' would include 'variants' and whatever else.
JeffJor's suggestion
- Rick, I've changed my mind on one thing. The so-called conditional problem needs to be a separate article, with "conditional" in its title. It can be linked to the MHP, but it is not the MHP. For justification, see (and cite in the article) [url=http://www.jstor.org/stable/187880] Maya Bar-Hillel's article "How to Solve Probability Teasers," Philosophy of Science, Vol. 56, No. 2 (Jun., 1989), pp. 348-358. That addresses several points critical to the problem, that are quite specific to all of the arguments we have had here, incuding documented evidence. Specifically: (1) It's just a puzzle. It isn't supposed to present a rigorously-defined mathematical problem, (2) The simple assumptions implied by the informal problem statement are intended, and almost universally accepted by anyone who isn't expecting such a rigorously-defined mathematical problem, (3) Even when presented with alternate wordings that explicitly include elements of host strategy, the general audience does not take that strategy into account in their solutions, and (4) the clear majority of respondents get the wrong answer (1/2) that is based on naive intuition rather than a formal solution.
- By separating the articles this way, Wikipedia can clearly present both problems in a fair and uncluttered manner, allowing any reader who wants to depend on the more formal approach to do so, and allowing those who do not see that formalism as necessary to limit themselves to the information that is of interest to them. JeffJor (talk) 15:03, 1 December 2009 (UTC)
Martin Hogbin's suggestion
We should take the current K & W statement as our starting definition of the MHP.
- I suggest that we give the Whitaker statement first then say that the K & W statement is how this is generally interpreted. The K & W paper itself supports this view.
The primary solution and explanation should not use conditional probability
- Although it can be argued that, even in the case where the host is defined to choose a legal door randomly, conditional probability should still be used because the action of choosing a particular door reduces the sample set and thus the opening of a specific door represents a conditioning of the sample set, it is clear that this is a trivial condition that it is not necessary to consider. This is quite evident either from the symmetry of the problem or from the fact that the revealing of random information tells us nothing. I am sure that we can find reliable sources to support this view.
The Morgan paper clearly does not answer the question as stated in the article and thus should not be regarded as our ultimate reliable source.
- The Morgan paper introduces a parameter q for something that is defined by the article problem statement to have only the value of 1/2. The Morgan paper thus answers a different problem (I suggest that we call it the Morgan scenario) from that posed in the article. In the Morgan scenario it is known that the host might have some preference for one of the legal goat doors.
The Morgan solution should be introduced in a later section of the article that deals with variations of the problem.
- There are many variations of this problem and the Morgan Scenario is just one of many.
Colincbn's comments
(referring to JeffJor's suggestion)
- Hear, hear!! Colincbn (talk) 15:09, 1 December 2009 (UTC)
(referring to Glkanter's suggestion)
I really don't know jack about probability and whatnot, but I still tend to agree with Glkanter's points. I came to this article through looking up various paradoxes and this was a really neat one that I got to try out in the real world (see simulation question above). As I understand it the "Monty Hall problem" states that the host chooses randomly, so any other discussion about host behavior should be limited to the "Variants" section under "Other host behaviors". Just my 2 cents, Colincbn (talk) 02:41, 1 December 2009 (UTC)
- Just to clarify, I think a mention of the Host behaviour/Conditional problem should be made in a subsection of this article, such as the Variants section, with a "main article" link (ie: {{main|MHP Conditional solution}} ) to a separate article that goes into Morgan's conditional problem in detail. I figure this will give the casual reader all the info he/she is looking for with an easy way to delve into the mathematics more deeply if they want. (also thanks to Rick for maintaining this section!) Colincbn (talk) 01:07, 3 December 2009 (UTC)
Martin Hogbin's comments
- I agree that this article should concentrate on the simple and notable interpretation of the MHP, namely the version in which a conditional solution is an unnecessary complication. Morgan's academic problem could be a section of this article or could form a new one. Martin Hogbin (talk) 22:02, 1 December 2009 (UTC)
Glkanter's comments
By my count, that's 4 in favor of the proposed changes, and 0 against. I've been championing these changes since October, 2008, Martin prior to that, and countless other editors for about 5 years. When can we declare an end to the pointless filibustering, acknowledge a consensus, and move on? Rick, will you be offering your comments? Have you contacted the others? Glkanter (talk) 22:29, 1 December 2009 (UTC)
About Martin Hogbin's suggestion - :I agree 100% with your proposed changes. I would like to add my 2 cents to the rationale, however. Morgan is criticizing and solving something other than the Monty Hall game show problem in the article. The introduction of the contestant being aware of any 'host behaviour' when selecting from 2 remaining goats changes the Problem Statement of both vos Savant/Whitaker and Karauss & Wang from "Suppose you are on a game show" to the converse, "Suppose you are not on a game show". Individual contestants on game shows are never provided more information than the 'average' contestant will have. There can be no 'condition'. It's illogical. Glkanter (talk) 15:33, 2 December 2009 (UTC)
JeffJor's comments
[Repeated in part from comments below]
The point of separating the articles is not to eliminate any POVs. It is to emphasize them. To not let one facet of the MHP (simple solution w nonintuitive result) become overpowered by the other (good teaching tool for conditional probabilites). If we don't physically separate them, we need to more clearly divide the article. The first part should be about the classic (unconditional) MHP, as stated by MvS (not K&W), and listing the set of assumptions she has said (and 99.9% of readers agree) are implied: interchangable doors, and any kind bias becomes irrelevant because of interchangeable doors. Then a section about game protocals (part of what some call host stratgies) such as always opening a door or revealing a goat, WITHOUT mention of bias or conditional problems. This mostly exists. Finally, you can cite Gillman (not Morgan) as a reference that introduces the possibility that the conditional problem is intended, but matters only if there is a bias. Use the K&W statement here, not Gillman's misquote. Gillman is better than Morgan because it is clearer, includes placement bias, and does not launch into possibilities that we are never told how to use. I think this is pretty consistent with Martin's suggestion. JeffJor (talk) 17:44, 4 December 2009 (UTC)
- Rick, no paper that uses q<>1/2 is addressing the K&W problem. They allow for it as a very specific variant of what they are addressing. But make no mistake: they are treating the problem statement we are supposed to be working with as the variant. That is wrong. There is nothing wrong with addressing their solutions as the variant, because it is a (more general) variation of what the article is supposed to be about. It isn't even a variant that is supposed to be used: no references use it, they just present it and say you don't need to use it. And I feel you have been just as much as stone wall on points relating to this as you accuse others of being. Meow. JeffJor (talk) 17:56, 4 December 2009 (UTC)
- Rick, you keep treating the Morgan POV as though it is involiate. It is not. Morgan misquotes the MvS problem statement, and so their claim that "the conditional problem is intended" cannot be taken as a reliable interpretation of the MHP. It is just a possible interpretation. Any reference that derives from Morgan is similarly suspect. Gillman misquotes, too, but in different ways. Bar-Hillel's survey proves that few (she found none) readers think of the conditional problem. More references exist that ignore it completely, than that address it. Krauss and Wang admit what the mis-quoters do not - I'll repeat it since when I said it before, it was apparently in cat language before - "Semantically, Door 3 in the standard version is named merely as an example." Grinstead and Snell separate the problem in the exact same two ways I suggested (and in fact, were a model for the suggestion). In short, it is a very minor POV that the conditional problem is meant, and it is based on citable misquotation and misinterpretation. JeffJor (talk) 18:24, 4 December 2009 (UTC)
Rick Block's comments
As a matter of fundamental Wikipedia policy, articles MUST be written from a neutral point of view. What the proponents of these changes are essentially suggesting is that this article take the POV that the interpretation of the problem described by a significant number of reliable sources (the Morgan et al. reference and others) is invalid. Even if this were a stance taken by reliable sources (which, as far as I know, is not the case), by relegating the "Morgan" interpretation to a "variant" subsection or splitting it into a POV fork this article would then be taking the "anti-Morgan" POV. I've made this point to these editors numerous times before, but yet they keep tendentiously arguing that the "Morgan" POV is wrong, or the Morgan et al. reference has errors, or (most recently) that the Morgan POV is NOT about the "real" Monty Hall problem (as if by convincing me that their POV is "correct" I would then agree with the changes they're suggesting).
I sincerely hope the "consensus" from this process is against making these changes, because even if there is a consensus for these changes they cannot be implemented - doing so would violate Wikipedia policy. -- Rick Block (talk) 04:01, 3 December 2009 (UTC)
You all realize Martin's proposal implies the article will not even mention conditional probability except in a "variant" section, don't you? How anyone can think this is not a blatant POV issue escapes me. -- Rick Block (talk) 20:19, 3 December 2009 (UTC)
- Yes Rick, all three proposals are consistent that way. It's based on this very recent and brief section of this talk page (following long and lengthy discussions on various 'talk' and 'argument' pages), 'Is The Contestant Aware?':
- I started this section at 11:28, 29 November 2009 (UTC). You responded with 2 vague, filibuster-style questions, and at 23:02, 1 December 2009 (UTC) I wrote this:
- "Rick, I have directly asked you this question many times, and have never seen a direct 'yes or no' answer from you. As this is a crucial element of the consensus that has been built, it is essential that we understand your reasons if you do not agree with the paragraph above:
- "Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?". [This question was ommitted when I asked Rick the 3rd time. I include it here for clarity]
- ""It seems to me that this is a (unstated) premise of the problem, as both vos Savant (Whitaker) and Krauss and Wang begin the problem statement with: 'Suppose you're on a game show'. I read this as clearly stating it is only the contestant's point of view we are concerned about. And, being a game show, the host is prohibited from divulging to the contestant either where the car is, or where the car is not.""
- To date, at 20:36, 3 December 2009 (UTC), you have still not responded directly to this question. Glkanter (talk) 20:36, 3 December 2009 (UTC)
I have to state the opposite view, which is that you have taken a ridiculously pro-Morgan POV. There are many reliable sources that relate to the MHP and not all of them have a host door choice parameter. Those that do generally quote Morgan as the source for this.
The article already takes a problem statement from a reliable source (K & W) and that same source confirms that this is how most people view the problem. In that statement, the host is defined to choose a legal goat door randomly. It is thus a simple matter of fact that the Morgan paper does not address that problem in so far as it allows a door choice parameter where none is permitted by the problem statement.
The Morgan paper clearly addresses a scenario where where the player is somehow aware of the host's policy for choosing a legal goat door. This rather bizarre scenario is not the one described by our problem statement and thus it should be viewed as a variant of the MHP as it is most commonly understood. Martin Hogbin (talk) 21:30, 3 December 2009 (UTC)
I thought I made it clear we were to use arbcom style rules here, which are that you only comment in your own section (it really does help keep the threads from getting absurdly long). However, since you've been rude enough to post here I'll respond to each of you, BUT please do not continue this as a thread here.
Glkanter asks why so dramatic? The argument has shifted from "present an unconditional analysis first (and don't criticize it)" to "exclude the conditional analysis completely (except as a variant)". This is a huge difference.
Glkanter asks why I haven't responded about his "Is The Contestant Aware?" question. Why should I? Glkanter has repeatedly demonstrated a complete lack of comprehension of nearly everything I've ever said. It's like trying to explain something to a cat. At some point you just have to give up. However, I'll give it another go. Meow, meeeow, meow, meowww. I'm not sure I have that quite right since I don't speak cat, but it's probably about as comprehensible to him as anything else I could say.
Martin (incorrectly) claims again that the Morgan et al. paper does not address the K&W version of the problem. Quote from the paper: "Incidentally, Pr(Ws | D3) = 2/3 iff p = q = 1/2". This is the solution to the K&R version of the problem statement. The Morgan et al. paper (and the Gillman paper and many, many others who approach the problem conditionally) absolutely address the K&R version. Because they also address other versions doesn't mean they don't address the K&R version.
- In the K & W statement q=1/2 by definition thus any problem in which q might not be equal to 1/2 must be a different problem. It is that simple. Martin Hogbin (talk) 19:04, 4 December 2009 (UTC)
Martin and Glkanter are both apparently completely incapable of understanding the main point of the Morgan et al. paper (and the Gillman paper, and what Grinstead and Snell have to say) which is that the MHP is fundamentally a conditional probability problem and that there's a difference between an unconditional and conditional solution. What these sources are saying is that a conditional solution clearly addresses the MHP (as they view the problem), but an unconditional solution doesn't unless it's accompanied by some argument for why it applies to the conditional case as well (and there are many valid arguments, but no argument at all which is what is generally provided with most unconditional solutions is not one of them). The fact that the problem can be (and typically is meant to be) defined in such a way that unconditional and conditional solutions have the same numeric answer in no way invalidates what these sources say. To have the article take the stance that the conditional solution is invalid (which would be truly absurd), or that the criticism these sources make of unconditional solutions is incorrect, or that a conditional solution applies only to a "variant" is making the article take a POV. This would be a direct violation of a FUNDAMENTAL Wikipedia policy. -- Rick Block (talk) 01:53, 4 December 2009 (UTC)
Antaeus Feldspar's comments
Glopk's comments
Delayed response (am not a very active editor at all these days), but here it is. Statement
I support Rick Block's statement as expressed above, and am in favor of keeping the article more or less in the state in which it passed the last FA review, with minor edits where needed. I am in strong disagreement with all three suggestions above (JeffJor, Glkanter, Martin Hogbin). In particular, I am in strong support of keeping the language that differentiates between the conditional (Bayesian) interpretation of the problem and the unconditional (elementary) one.
Motivation. The purpose of an encyclopedia is to present a "best" selection from the body of knowledge about each topic, being POV neutral as well as reader-neutral. --glopk (talk) 18:53, 29 December 2009 (UTC)
Father Goose's comments
Chardish's comments
Thanks for the invitation to comment. In my opinion, Martin Hogbin's suggestion seems the post prudent. The Monty Hall problem as popularly explained doesn't rely on conditional probability, and the Whitman explanation seems sufficient for anyone who is not a mathematician. Wikipedia is a general-purpose encyclopedia, and as such main articles should focus on explaining topics as they are popularly understood, with specific scientific analysis relegated to separate articles.
And, to be honest, the article as it stands is much harder to read and understand (as a layperson) than it was several years ago. NPOV isn't "pleasing everyone equally"; don't let efforts towards neutrality wind up hurting the article. - Chardish (talk) 02:53, 6 December 2009 (UTC)
Michael Hardy's comments
PMAnderson's comments
Melchoir's comments
Just from reading the present Wikipedia article, I agree with Martin Hogbin's suggestion, because I don't see why allowing the host to prefer one goat over the other is a more relevant generalization than allowing the host other behaviors. Melchoir (talk) 06:47, 3 December 2009 (UTC)
jbmurray's comments
Nijdam's comments
I fully support Rick's view. Nijdam (talk) 10:34, 3 December 2009 (UTC)
To make my position crystal clear: there is no such as an unconditional solution. There are different problems: an unconditional problem and a conditional one. The latter generally being called the MHP. Nijdam (talk) 22:24, 3 December 2009 (UTC)
- Please read my proposal. I do not claim that the MHP is an unconditional problem. What I say is that in the problem definition given in the article the host is taken to choose a legal goat door randomly. Morgan address the case where this choice is non-random,thus they do not address the problem as defined in this article. Martin Hogbin (talk) 22:39, 3 December 2009 (UTC)
- I'm in the audience looking at the stage. I see three doors and a player pointing to one of them. From the two remaining doors one is opened and shows a goat. That's what I call the MHP. (And I know of the random placement of the car and the random choice of the host.) Nijdam (talk) 22:50, 3 December 2009 (UTC)
- Quite, and that is not the problem that the Morgan paper addresses. The Morgan paper addresses the case where the host door choice is not random. Thus the Morgan paper addresses a variation on what we all agree is the MHP. Martin Hogbin (talk) 23:28, 3 December 2009 (UTC)
- The problem that you and K & W describe, which is the problem addressed by this article, is one in which q=1/2 by definition. Therefore, any problem in which there is a possibility that q might not equal 1/2 must be a different problem. Morgan clearly consider a problem in which it is possible for q to have a value other than 1/2. The problem they consider therefore must be different from that in which q is defined to be 1/2. Morgan do indeed address a (bizarrely) more general problem than the one we are considering but it is, for sure, a different problem. Martin Hogbin (talk) 19:35, 4 December 2009 (UTC)
Dicklyon's comments
I haven't been watching this article for a while; glad to see the K&W treatment up front; that looks like the most sensible article I've seen on it. As for the Morgan conditional approach, I think it's an unnecessary distraction, but it's out there in mainstream reliable sources about the topic, so we ought to cover it in the article. I think Martin Hogbin's proposal sounds best. Dicklyon (talk) 05:01, 3 December 2009 (UTC)
I agree with Rick Block that the other two proposals essentially violate WP:NPOV; but I disagree that moving the conditional stuff to a more minor position is a problem; his heavy promotion of the conditional approach violates WP:UNDUE in my opinion. Dicklyon (talk) 16:29, 3 December 2009 (UTC)
Henning Makholm's comments
I have long since given up on following these discussions, and am not even a very active editor these days. However, since somebody went to the length of creating a heading for me, here are my general recommendations -- for whatever they are worth:
- The article absolutely should discuss assumptions about the host's behavior. It is impossible to derive a valid answer without making some assumptions, and differences in which assumptions are implicit are one of the main reasons why smart people can disagree on the solution when the problem is stated sloppily. It would be a sorry encyclopedia that purported to treat the Monty Hall problem without explicitly pointing out this kind of confusion.
- The analysis that involves conditional probabilities and the one that considers whole-game expectations under different player strategies are both valid ways of approaching the problem, each with its own advantages and disadvantages. The article should present both, and must not suggest that one of them is inherently better or more correct than the other. (For this reason I would oppose splitting one of the analyses into a separate article, suggesting that it solves a fundamentally different problem, rather than being an alternative way of approaching the same problem).
- There has been far too much microlinguistic analysis about precise wordings of the problem in this source or that one, trying to argue that this analysis or that one is the one that most directly addresses the question being asked (implying that the other is a detour via a different but non-canonical presentation of the problem). Which analysis one chooses depends depends far more on which properties (besides being valid) one wants of it. For example, raw convincing power for a lay audience would favor the whole-game analysis, whereas a more in-depth discussion of the effect of different assumptions of the host's behavior is most easily done using conditional probabilities.
- Editors should keep in mind that Wikipedia is an encyclopedia, not a textbook, an question-and-answer database, or a Court of Public Opinion. The goal of an encyclopedia article is not to answer one particular question but to present a body of knowledge. Therefore the amount of energy spent on negotiating "the" question that this article should be about answering is fundamentally misspent. The body of knowledge the article ought to present encompasses several different but related questions (some of which are sometimes mistaken for each other), and several different way of approaching some of them. An approach that restricts ourselves to discussing just one of them would fail to cover the topic encyclopedically.
- I have no strong opinion about which analysis should be first in the article, as long as it is not being touted as inherently superior or inferior by virtue of its position. However, the general principle of progressing from the "quick and easily understood" to the "more complex but also more general and (possibly) enlightening" would seem to suggest starting with the whole-game analysis.
–Henning Makholm (talk) 07:13, 3 December 2009 (UTC)
- Most people, including Rick, think that the problems should be addresses from the player's point of view (state of knowledge). As has been pointed out by many people, it is extremely unlikely that the player would have any knowledge of the host's door opening policy, thus from the player's point of view the host policy must be taken as random (within the rules).
- I have no objection to the Morgan scenario (in which the payer is assumed to know the host's policy) as well as the more simple case being presented here provided that it is made clear exactly what case this applies to.
- What you call, 'microlinguistic analysis about precise wordings of the problem' was started by Morgan et al. who added a pointless layer of obfuscation to a simple puzzle that most people get wrong.
- The point is that the simple/symmetrical/non-conditional problem is the notable one and therefore it should come first. More complex versions should come later for the few that are interested in such complications. Martin Hogbin (talk) 22:36, 4 December 2009 (UTC)
Boris Tsirelson's comments
I summarize my position in two points:
- 1. The symmetric case is more important for an encyclopedia than the general case. (Likewise, a circle is more important for an encyclopedia than an arbitrary curve.)
- 2. The coexistence of the conditional and the unconditional can be more peaceful. (Not just "numeric coincidence" in the symmetric case; see #Not just words and #Formulas, not words.)
Boris Tsirelson (talk) 06:44, 9 December 2009 (UTC)
Being invited by Glkanter, I quote here some paragraphs of a discussion that happened on my talk page on February 2009. As far as I understand, my position is close to that of JeffJor. Boris Tsirelson (talk) 17:20, 2 December 2009 (UTC)
Why split? Because of different importance. The "conditional" article will be, say, of middle importance, while the "unconditional" article – of high importance. We surely have our point of view about importance (rather than content). Boris Tsirelson (talk) 05:54, 4 December 2009 (UTC)
The quotes follow.
Each time giving the course "Introduction to probability" for our first-year students (math+stat+cs) I spend 20-30 min on the Monty Hall paradox. I compare two cases: (a) the given case: the host knows what's behind the doors, and (b) the alternative case: he does not know, and it is his good luck that he opens a door which has a goat. Im addition I treat the case of 100 (rather than 3) doors (just like Monty Hall problem#Increasing the number of doors). And, I believe, students understand it.
I have no idea, why some people spend much more time on the Monty Hall paradox (and even publish papers). (Boris Tsirelson)
This simple little problem is deeper than it might appear, and likely well worth more than 20-30 mins of lecture time. Perhaps even worth revisiting once or twice during a term to explore its more subtle aspects. (Rick Block)
Deeper than it might appear? OK, why not; but still, for now I am not enthusiastic to deep into it. Tastes differ. I find it more instructive, to restrict myself to the simpler, symmetric case, and compare the two cases mentioned above.
If an article leaves many readers puzzled, why it is unnecessarily complicated, it is a drawback. (Boris Tsirelson)
If a problem that appears so simple to me, like the Monty Hall problem, is not sufficiently solved using my unconditional proof, in what circumstances is the unconditional proof appropriate? Thank you. (Glkanter)
The unconditional argument shows that "always switch" is better than "never switch". This is what it can do. Let me add: if you (that is, the player) are not informed about possible asymmetry then you cannot do better than these two strategies, either "always switch" or "never switch". (Boris Tsirelson)
- Well, I gotta ask. Do you still prefer JeffJor's proposal among the 3 proposals put forth? Glkanter (talk) 07:19, 9 December 2009 (UTC)
- Really, I have nothing to add to the two points that summarize my position (above). Any move toward them is good for me. My resolution power, and my acquaintance with the literature, are too low for choosing between different proposals; I leave this matter to more informed (and less lazy) editors. Boris Tsirelson (talk) 08:22, 9 December 2009 (UTC)
William Connolley's comments
C S's comments
kmhkmh's comments
I'll start with a clear statement and give some more detailed information afterwards:
- I strongly disagree with any of the 3 suggestions (JeffJor, Glkanter, Martin Hogbin) and aside from minor difference fully support Rick Block's approach
If one surveys the available literature literature/publications on the topic, you pretty much get an relatively obvious outline for the article: original problem (in vos savant's column), unconditional solution (basically vos savant and/or various math sources), conditional solution (Morgan and almost in any math source), various problem variation and caveats, history of the problem, application of the problem outside the math domain. Which is essentially for the most part, what we already had and what Rick managed to maintain. In that context I fully agree with Henning Makholm's comments above, who puts it fairly well. The article wouldn't have such problems if all participants would follow that rationale.
The fuzz over quality or minor mistakes in Morgan's paper is a somewhat ridiculous distraction, since Morgan's paper is not needed to argue the conditional solution or caveats to the unconditional solution at all. There is plenty of other math literature dealing with the problem in more or less the same manner.
My personal advice would be to pass the article for final thorough review and modification to the math or a science portal. During that review neither of the 4 disagreeing authors (JeffJor, Glkanter, Martin Hogbin, Rick Block) are allowed to participate/edit. After that review the article should be fully protected for good.
I've seen what happened to the German version, that had similar problems (without a Rick Block around to constantly remain some standard). So we had a lot of people with a somewhat fanatic approach constantly pushing for their favoured explanation and constantly ignoring wiki standards, common sense and more important the available literature on the subject. As result mathematicians and scientists basically dumped the article and gave up on improving it.An effect this article has partially seen as well.--Kmhkmh (talk) 16:45, 4 December 2009 (UTC)
- Kmhkmh, no one is proposing a reduction in the quality of this article but you miss some essential points out in your outline. We should have: 'original problem (in vos savant's column), unambiguous problem definition (K&W), solution to the unambiguous problem (which is trivially conditional but need not be treated so, basically vos savant and/or various math sources), the Morgan scenario (in which the player knows host door choice policy) the conditional solution (Morgan). Martin Hogbin (talk) 19:46, 4 December 2009 (UTC)
- I don't quite see how that is "missing" in my outline above nor do I see any particular reason to give (K&W) a preferred treatment, such an approach does not reflect the publications on the topic.--Kmhkmh (talk) 20:11, 4 December 2009 (UTC)
- I am not proposing that we give K&W and preference but must have a clear and unambiguous problem statement before we (or anyone else) can attempt to answer the problem. Morgan do not have such a statement in their paper so we must use one from another reliable source, in this case another published paper. Note that the lack of clear problem statement in the Morgan paper is not just my opinion, that same point is made clear in the comment by Prof Seymann published in the same journal immediately after the Morgan paper. Martin Hogbin (talk) 20:20, 4 December 2009 (UTC)
- I'm really not interested in repeating now here the discussion that you're pushing for almost over a year now and which frankly from my perspective is entirely pointless and misguided. The original problem in vos Savant's column was ambiguous and hence various articles on the topic and its variations provide their own specifications. As pointed out above already Morgan doesn't really matter in that regard. What the Wikipedia article has to do, is to describe the all various specification and not arbitrarily picking one like K&W as the "right" one. I'd recommend you to reread Henning Makholm's comments carefully. Or to put it rather bluntly - you asked for my comment here it is: Leave the article alone.--Kmhkmh (talk) 21:46, 4 December 2009 (UTC)
- Actually I did not ask for your comment here and I certainly did not ask for, and do not need, your permission to edit Wikipedia. Neither did I pick K&W as the 'right one' as you put it, somebody else put it in the article as a clear and unambiguous description of the problem. As it happens I agree with whoever did this as K & W is the only published paper to seriously address the question of how most people interpret the MHP. It is therefore an excellent place to start the article. Martin Hogbin (talk) 22:25, 4 December 2009 (UTC)
- I'm really not interested in repeating now here the discussion that you're pushing for almost over a year now and which frankly from my perspective is entirely pointless and misguided. The original problem in vos Savant's column was ambiguous and hence various articles on the topic and its variations provide their own specifications. As pointed out above already Morgan doesn't really matter in that regard. What the Wikipedia article has to do, is to describe the all various specification and not arbitrarily picking one like K&W as the "right" one. I'd recommend you to reread Henning Makholm's comments carefully. Or to put it rather bluntly - you asked for my comment here it is: Leave the article alone.--Kmhkmh (talk) 21:46, 4 December 2009 (UTC)
- I am not proposing that we give K&W and preference but must have a clear and unambiguous problem statement before we (or anyone else) can attempt to answer the problem. Morgan do not have such a statement in their paper so we must use one from another reliable source, in this case another published paper. Note that the lack of clear problem statement in the Morgan paper is not just my opinion, that same point is made clear in the comment by Prof Seymann published in the same journal immediately after the Morgan paper. Martin Hogbin (talk) 20:20, 4 December 2009 (UTC)
- I don't quite see how that is "missing" in my outline above nor do I see any particular reason to give (K&W) a preferred treatment, such an approach does not reflect the publications on the topic.--Kmhkmh (talk) 20:11, 4 December 2009 (UTC)
Gill110951's comments
No comment right now. But a lot of Christmas break reading to do here, to catch up. Happy Wikipedia Christmas, everyone! Gill110951 (talk) 13:27, 20 December 2009 (UTC)
Friday's comments
Summary of opinions
I have added names to the sections below based on comments above. If I have got it wrong please move yourself.
Please do not make comments in this section.
Editors are invited to sign against their names to confirm that they are in the right section. Martin Hogbin (talk) 11:29, 5 December 2009 (UTC)
- Martin, What exactly do you mean by "for change" and "against change"? Dicklyon and JeffJor's opinions (for example) seem very different to me. By categorizing them both as "for change" I think you may be misrepresenting the situation. It would be better to be more specific about what change you're talking about, i.e. Glkanter's suggestion (
remove any mention of conditional probability and any host behavior variants)Eliminate all 'host behaviour, etc' influenced discussion, save for the Wikipedia minimum necessary references to Morgan and his ilk, JeffJor's suggestion (separate articles - basically Glkanter's suggestion plus create a new article for the "conditional" treatment), your suggestion (I'm not exactly sure precisely how to summarize yours). In addition, rather than "against change" the other alternative should probably be described in terms of what it is for, which I think could be described as "present both unconditional and conditional solutions without taking a POV about the validity of either one". And, I'll note that for the article to say that Morgan et al. criticize the unconditional solutions is not the same as taking that POV. You do understand this difference, don't you? -- Rick Block (talk) 16:52, 5
- Although there may be some discussion over the details it is fairly clear that several people would like to see the simple/unconditional solution/problem given more prominence here. This is the change that I am referring to. 'Against change' is fairly self explanatory Martin Hogbin (talk) 16:58, 5 December 2009 (UTC)
For change
Colincbn
Martin Hogbin Martin Hogbin (talk) 11:29, 5 December 2009 (UTC)
Glkanter Glkanter (talk) 12:16, 5 December 2009 (UTC)
JeffJor
Melchoir
Dicklyon
Boris Tsirelson Boris Tsirelson (talk) 15:27, 5 December 2009 (UTC)
Gill110951 (talk) 13:28, 20 December 2009 (UTC)
Against change
Rick Block
Nijdam
kmhkmh
Glopk
Unable to classify
Please move your name to the correct section if appropriate. Martin Hogbin (talk) 11:24, 5 December 2009 (UTC)
Henning Makholm
Chardish (I object to summary classification of my comments. - Chardish (talk) 00:59, 11 December 2009 (UTC))
Deal or No Deal
Asked whether the contestant should switch, vos Savant correctly replied, "If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch" (vos Savant, 2006).
I read this, and then the following section on "increasing the number of doors" and had the instant comparisson in my mind to "Deal or No Deal". I came back here to see the debate over that game show, but to me it's quite obvious. If there are 22 cases, and you select one, your initial probability is 1/22 of having the mil.
The two situations are clearly fundamentally different as demonstrable:
- We've all see contestants painfully open a case that has the Mil$. This is because the contestant does NOT know the contents and is randomly selecting. The probabilities say that if there are 22 cases, this should happen 20/22 times with only 2/22 probability of getting down to your case and one other (the "switch" situation). This is why it is so uncommon for the Mil to still be in play at the switch point of that show [In reality it's even less common than two in every 22 contestants getting to the switch with a Mil$ due to the irrelevant fact that those who still have a Mil$ in play often take a deal rather than risk eliminating the Mil$ with three or four cases left]. But assuming the true math problem, If you happen to get down to the switch for a Mil$, your odds are equal. God knows which case you picked. You got lucky enough to leave the Mil in play. Now you're down to 50/50 odds.
- The Monty Hall situation would be if you select your case (1/22 odds) and then the HOST, KNOWING which case has the mil randomly selects 20 other cases to eliminate that DO NOT have the mil. It doesn't matter if he eliminates them one at a time or all at once because it's random and he can't eliminate the Mil$. He ends up with case 13 up there, and you have your case. What do you think the odds are that you picked right in the first place and he HAD to leave 13 because it has the Mil$, vs the odds that you picked right and he randomly left 13 up there? Well it's clearly 1/22 odds that you picked right. If you picked ANY of the 21 of the 22 cases that didn't have the Mil, he would have left the same case up there: The one with the mil. Only in that 1/22 case where you have the Mil does he need to be random. So the odds when the revealer KNOWS and AVOIDS revealing the million are the odds of your original choice being wrong.
The point I was going for here in this lengthly analysis is, perhaps the now-common format of Deal or No Deal would make for a good example of the contrast between Monty Hall Problem (revealer knows and avoids opening the prize door) and Deal or No Deal (revealer doesn't know and got lucky to leave two random cases). It's a very good everyday life scenario that many readers will understand and relate to in terms of getting the idea across. I guess the problem is it needs citation, but there must certainly be published works out there analysing the math of Deal or No Deal that can be cited... TheHYPO (talk) 06:29, 4 December 2009 (UTC)
- Yes, I also consider this aspect as the most instructive part of MHP. See above, in my comments: "I compare two cases: (a) the given case: the host knows what's behind the doors, and (b) the alternative case: he does not know, and it is his good luck that he opens a door which has a goat." Boris Tsirelson (talk) 06:41, 4 December 2009 (UTC)
- I agree, that is an interesting, and indeed surprising for most people, case to consider. Most people are amazed to discover that whether the host chooses a goat door by chance or because they know which door is which makes a difference. Of course, you need to decide what happens if the host chooses randomly and happens to choose the car.
- My explanation would be along the lines that when the host knows what is behind the doors you consider all games but when the host chooses randomly you consider a non-random sample of games.
- This is the kind of thing that this article should be concentrating on rather than the rather bizarre scenario where the player knows the hosts door opening strategy. Martin Hogbin (talk) 09:57, 4 December 2009 (UTC)
- Yes. And here is an analogy. The (elementary) geometry of a circle is a special case of the (differential) geometry of an arbitrary curve. However it would be terrible to treat the circle (in an encyclopedia) only inside an article on curves in general. The symmetric case is more notable, more important, this is the point. And let me repeat: we surely have our POV about importance (rather than content). Boris Tsirelson (talk) 12:19, 4 December 2009 (UTC)
- Exactly. The whole point about the symmetric MHP is that is is a very simple problem that nearly everybody gets wrong. The Morgan scenario (player knows the host door opening policy) obfuscates the central problem and leaves us with a more complicated and rather boring problem that most people are not interested in. Martin Hogbin (talk) 12:51, 4 December 2009 (UTC)
- Yes. And let me add: I have nothing against conditional probabilities. I only note that in the symmetric case there is no difference between conditional and unconditional probabilities. And it is not a coincidence. Not at all. An unconditional probability is an average of conditional probabilities (the total probability formula); these conditional probabilities are mutually equal (by symmetry); the equality follows. Boris Tsirelson (talk) 14:16, 4 December 2009 (UTC)
- (back from the weekend) Of course, different measures. Of course, for certain event. But Martin Hogbin understands what I mean, see below "Words, words, words". And again, I do not object to conditioning. I only say that the symmetric case is more important for an encyclopedia.
- Boris, when you teach this problem, do you or do you not introduce conditional probabilities? If not, how to you distinguish the "host forgets" case from the "host knows" case? Martin is still misinterpreting what Morgan et al. says, which is not that the player necessarily knows the host's door opening policy (in the "host knows" case), but that to solve for a specific conditional probability, e.g. P(car is behind Door 2|player picks Door 1 and host opens Door 3), you have to be given or make an assumption about how often the host opens Door 3 if the car is behind Door 1. Symmetry says this is 1/2 is a perfectly valid way to make this assumption. The player has no knowledge of a potential host preference between Door 2 or Door 3 so must treat these doors as indistinguishable is another perfectly valid way to make this assumption. If your solution says nothing about this, your 2/3 answer is not the answer to P(car is behind Door 2|player picks Door 1 and host opens Door 3) but something different, like perhaps P(win|player switches). Which of these conditional probabilities more accurately represents the conditions described by the MHP which specifically allows the player to choose whether to switch after seeing the host open a door?
- (back from the weekend) It seems, everything is already answered in the "Words, words, words" section below. Except for the "personal" question... Yes, I do introduce conditional probabilities. Indeed, I do not teach MHP; I just illustrate conditional probabilities by MHP. So what? First, Wikipedia is not a textbook. Second, I do not object to conditioning. I only say that the symmetric case is more important for an encyclopedia.Boris Tsirelson (talk) 15:18, 5 December 2009 (UTC)
- BTW - the paragraph just added to the article has no references. Citation standards for feature articles are quite demanding. Essentially every independent thought should be referenced (this ensures that what is said is verifiable). Unless a reference can be provided that uses the 100,000 door example to contrast MHP with Deal or No Deal, this paragraph should be deleted. -- Rick Block (talk) 15:20, 4 December 2009 (UTC)
- Rick, you said "Morgan et al. says ... that to solve for a specific conditional probability ... you have to be given or make an assumption about how often the host opens Door 3 if the car is behind Door 1." That isn't really what they say. They say that it is an interesting generalization of the MHP to consider one specific host strategy that avoids such an assumption. They also say that the fully gerneralized host strategy is not interesting in this way. They never say that the assumption q=1/2 is an invalid assumption, that it is not implied by the problem statement, or offer any way for a contestant to obtain information about q. They only say it isn't necessary to make it the assumption in order to answer the question. They do say that for other generalizations, such as placement bias, you are required to assume something to answer the question (because to not do so is "unlikely to correspond to a real playing" of the game). Their only point is to treat that one option as a vehicle for teaching conditional probability, not to say that any formula gives "the probability" for the MHP. So you are injecting your own POV into the article. The POV that a contestant can obtain this information and use it, even if how is unstated. That is an uncited, and uncitable, original thought that should not be included in the article by the very standards you insist others must adhere to. Why do you think you can avoid it?
- So it isn't Martin that is misinterpreting Morgan, it is you. Martin understands that not only are "symmetric assumptions" valid, but they are required for the very reason that no bias values are given. Martin undersatnd that mot published treatmens of the problem make that assumption, and yes I have told you some of them. He understands that what you call the "symmertic assumptions" can be implemented in two ways: either by assuming q=1/2, or by assuming the doors are symmetric and therefore not distinguished. And that the two are isomorphic problems that have different solution methods, but the same answer (of course). That the former symmetric assumption requires Morgan's analysis, but the latter can be solved with most of the methods Moragn calls "incomplete." And that the point of the MHP is to show that that simpler soluiton produces an unintuitive result, not to get an actual probability.
- How can you argue, that the doors cannot be distinguished? When obviously anybody can distinguish them (comletely independent on which solution approach you pursue)? Even more the original problem formulation itsself distinguishes them explicitly? You may argue a "symmetry assumption" for solving (or simplifying) the problem, but you cannot argue indistinguishable doors, unless you blindfold the candidate and rob him of any orientation.--Kmhkmh (talk) 12:48, 5 December 2009 (UTC)
- I don't argue that the doors cannot be distinguished somehow, each one from the others. (But please note, since you are trying to argue for a very strict reading of the Whitiker statement: it does not say the doors have numbers, or are distinguishable. That is an assumption you make. It does mention numbers, but it doesn't say whether the numbers are a manifestation you created in your mind, or are seen by everybody. And don't, as others have, tell me there were numbers on the Let's Make a Deal show - they never played a game remotely like this and it was also not mentioned in the Whitiker statement.) I argue that no factors relating to the uncertainty in the outcome can be applied to the doors, as they are distinguished that way, when solving the puzzle. Suppose I were to roll a pair of dice, one red and one green. If I ask you for the probability one die will roll higher than the other, would you give different answers based on color? How? JeffJor (talk) 13:55, 7 December 2009 (UTC)
- The doors are there to keep the contestant from seeing behind them. Which they accomplish 100% effectively. What if the contestant was sequestered away, and couldn't see the front of the doors or the relative locations of the doors? He relies on Monty's description, as verified by the studio audience. The contestant still has no more knowledge than that switching 2/3 of the time will win the car, before or after Monty has opened a goat door. How has this extra level of 'identicalness of doors', that you claim is required for the unconditional solution, helped or hurt the contestant in any way? How does it apply to the Combined Doors solution, in which Monty reveals 'which' door has a goat, but that has no effect on his offer to switch, or the response by the contestant?Glkanter (talk) 13:28, 5 December 2009 (UTC)
- All of what you've stated here does not affect the distinguishability of the doors, I was talking above. I was making no statement regading the solution and I'm definitely not going to rehash the seemingly endless (and imho rather boring) discussion on that. I just pointed out, that claiming the doors he doors are indistuingishable, is factually false as far as reality or the perception of the contestant concerned. The contestant always see a left, middle and right door, i.e. he can distinguish them. Whether it might make sense to declare the doors indistinguishable in a model to solve the problem is another matter I did not address.--Kmhkmh (talk) 13:50, 5 December 2009 (UTC)
- How can you argue, that the doors cannot be distinguished? When obviously anybody can distinguish them (comletely independent on which solution approach you pursue)? Even more the original problem formulation itsself distinguishes them explicitly? You may argue a "symmetry assumption" for solving (or simplifying) the problem, but you cannot argue indistinguishable doors, unless you blindfold the candidate and rob him of any orientation.--Kmhkmh (talk) 12:48, 5 December 2009 (UTC)
- So it isn't Martin that is misinterpreting Morgan, it is you. Martin understands that not only are "symmetric assumptions" valid, but they are required for the very reason that no bias values are given. Martin undersatnd that mot published treatmens of the problem make that assumption, and yes I have told you some of them. He understands that what you call the "symmertic assumptions" can be implemented in two ways: either by assuming q=1/2, or by assuming the doors are symmetric and therefore not distinguished. And that the two are isomorphic problems that have different solution methods, but the same answer (of course). That the former symmetric assumption requires Morgan's analysis, but the latter can be solved with most of the methods Moragn calls "incomplete." And that the point of the MHP is to show that that simpler soluiton produces an unintuitive result, not to get an actual probability.
- The point of separating the articles is not to eliminate any POVs. It is to emphasize them. To not let one facet of the MHP (simple solution w nonintuitive result) become overpowered by the other (good teaching tool for conditional probabilites). If we don't physically separate them, we need to more clearluy divide the article. The first half should be about the classic (unconditional) MHP, as stated by MvS (not K&W), and listing the set of assumptions she has said (and 99.9% of readers agree) are implied: interchangable doors, and any kind bias becoems irrelevant because of interchangeable doors. Then a section about game protocals (part of what you call host stratgies) such as always opening a door or revealing a goat, WITHOUT mention of bias or conditional problems. Finally, you can cite Gillman (not Morgan) as a reference that introduces the possibility that the conditional problem is intended, but matters only if there is a bias. Use the K&W sattemetn here, not Gillman's misquote. Gillman is better than Morgan because it is clearer, includes placement bias, and does not launch into possibilities that we are never told how to use. I think this is pretty consistent with Martin's suggestion. JeffJor (talk) 17:41, 4 December 2009 (UTC)
- Rick you keep saying that I am misinterpreting what Morgan says without proof or justification. Every mathematician here agrees that, if the player does not know the host door opening policy and we are to address the problem from the player's state of knowledge, we can only take it that the host is equally likely to open either goat door when he has a choice. No parameter q is required, it is fixed in value at 1/2.
- Whether the problem has to be considered from the players ("contestant") perspective or from an informed 3rd person's perspective ("problem solver") is entirely unclear and depends on the perspective you take. Meaning the original question could stand "How would you decide as a player?" or "what would you recomend the player?" Various publications on the topic have examined those different perspectives. Also if the player doesn't know the host's policy, he still has 2 options for his decision making. Simply pick the most reasonable assumption (fix q at 1/2) and base his conclusion on that. Or instead analyze a large variety of possible host policies to see whether all (or at least most) yield to the same conclusion anyway (q varies).--Kmhkmh (talk) 13:31, 5 December 2009 (UTC)
- Both problem statements begin: "Suppose you are on a game show..." I see no ambiguity in that we are taking only the contestant's perspective (State of Knowledge) into our analysis. Glkanter (talk) 13:37, 5 December 2009 (UTC)
- That is correct if you just look at those statements in a literal sense. However it is not so clear if look at some "interpretations" or variation of the problem in literature (see Mark Steinbach in German article as a source). But no matter how you regard that, it does not change the second part, where even from the player's perspective the bayesian analysis (variation of q) is still an option for his reasoning.--Kmhkmh (talk) 13:59, 5 December 2009 (UTC)
- Both problem statements begin: "Suppose you are on a game show..." I see no ambiguity in that we are taking only the contestant's perspective (State of Knowledge) into our analysis. Glkanter (talk) 13:37, 5 December 2009 (UTC)
- Whether the problem has to be considered from the players ("contestant") perspective or from an informed 3rd person's perspective ("problem solver") is entirely unclear and depends on the perspective you take. Meaning the original question could stand "How would you decide as a player?" or "what would you recomend the player?" Various publications on the topic have examined those different perspectives. Also if the player doesn't know the host's policy, he still has 2 options for his decision making. Simply pick the most reasonable assumption (fix q at 1/2) and base his conclusion on that. Or instead analyze a large variety of possible host policies to see whether all (or at least most) yield to the same conclusion anyway (q varies).--Kmhkmh (talk) 13:31, 5 December 2009 (UTC)
- Regarding Morgan's attempt to answer a more general problem, they make a very poor job of this. For the most part is is hard to determine exactly what problem they are trying to answer. But that is irrelevant here, we know the problem that we are trying to deal with and it is not the same one that Morgan address.Martin Hogbin (talk) 20:06, 4 December 2009 (UTC)
Words, words, words
Because in words one often doesn't distinquish between probability and conditional probability, a lot of the misunderstanding arises. As I tried before, I ask anyone of the discussiant to formulate the problem and the solution in rigorous mathematical notation. I'll do the kick-off:
- Three doors with numbers 1,2 and 3
- C is the random variable indicating the number of the door with the car
- X is the random variable indicating the number of the door chosen by the player
- C and X are independent
- H is the random variable indicating the number of the door opened by the host
As C and X are independent we may without loss of generality consider the case "X=1" and without explicit mentioning condition on this event. Probabilities are:
Did I leave something out? Is something wrong? Please comment! In my opinion the whole point of the MHP is the calculation of P(C=2|H=3). I'm looking forward to other opinions. Nijdam (talk) 12:40, 5 December 2009 (UTC)
- No, your calculation is fine and as I have made clear before I accept that the MHP problem (as defined by the K & W statement in which specific doors are mentioned) is a problem of conditional probability. The opening of a door by the host reduces the sample set (clearly the player can no longer change to that door the host has opened) thus the sample set has been conditioned and the problem is one of conditional probability.
But, even with this problem formulation we can ask the question as to whether we need to use conditional probability to answer the question. We can immediately note that there is an obvious symmetry in the question. Namely that:
and therefore that: P(C=2|H=3)=P(C=3|H=2) Thus, as Boris has ,said the conditional solution, whichever door the host chooses, must equal the unconditional solution. There is nothing wrong with this approach, many intractable mathematical problems have been solved by noticing some symmetry in the problem.
>>>Even so, as you say, we have to calculate the conditional probability. Nijdam (talk) 15:57, 5 December 2009 (UTC)
Alternatively we might say that, because the hosts choice is random (when he has a choice) it gives no information about the initial placement of the car and thus the original
must hold good after the host has opened a door.
>>>??It doesn't:
- You are right, of course. I should have just said that P(C=1)=1/3; Martin Hogbin (talk) 16:39, 5 December 2009 (UTC)
So to sum up, with the K & W formulation, the problem may be strictly conditional but it is clear that a simple unconditional solution will give the correct answer.
However there is more to the issue than this. The MHP is essentially a mathematical paradox and thus it is logical to formulate the problem so that the solution is as simple as possible. This is what was done before the Morgan paper was published. The door numbers were considered irrelevant. I have answered your question, now perhaps you will answer mine. Do you really believe that what Whitaker wanted to know was what the probability of winning was given only that specific doors had been chosen and opened? I would very much like to hear your answer to that question.
>>>Whitaker definitely wanted to know what for a chosen door and an opened one the (conditional) probability was for the car to be behind the remaining unopened door. Nijdam (talk) 15:57, 5 December 2009 (UTC)
I would say that the question that he actually wanted the answer to is: 'Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, and the host, who knows what's behind the doors, opens another door, which has a goat. He then says to you, "Do you want to pick the remaining door?" Is it generally to your advantage to switch your choice?'
He obviously wants to know what is the best strategy in playing the game: to switch or not. This a simple question with a simple answer that most people get wrong.
>>>Always given the situation the player is in. Nijdam (talk) 15:57, 5 December 2009 (UTC)
I can only repeat, yet again, the words of the wise Prof Seymann (with my emphasis) in the hopes that you will heed them: ' Without a clear understanding of the precise intent of the questioner, there can be no single correct solution to any problem. Thus, with respect to the three door problem, the answer is dependent on the assumptions one makes about the intent of the one who initially posed the question '. In my opinion Morgan et al. have completely misrepresented the original point of the question, resulting in all this conditional nonsense which only serves to obfuscate an interesting problem. Martin Hogbin (talk) 14:02, 5 December 2009 (UTC)
>>>There you have a point. The original problem may be interpreted in different ways. Also in the way of Morgan et al. BTW. But my main concern is that even in the precise definition (like the K&W formulation) many people, seemingly also mathematicians, probabalists and statisticians, give the wrong analysis. Nijdam (talk) 15:57, 5 December 2009 (UTC)
- I don't completely disagree with your description. However making claims about Whittaker's intentions is clearly WP:OR. Either his intentions are published somewhere, then they are known and we can incorporate them or they are not. In the latter case personal speculations of WP editors do not belong in the article. As far possible or real misinterpretation are concerned, WP has to describe or summarize how Whitaker was interpreted in various publication - period.--Kmhkmh (talk) 14:12, 5 December 2009 (UTC)
- I understand your point about sources; that is why I have not edited the article but continued to argue here. On the other hand may editors feel passionately that the current format of the article is wrong. There are many sources on this problem and it is, to some degree, up to us which ones we use and how we use them. See my suggestion below on mediation. Martin Hogbin (talk) 14:33, 5 December 2009 (UTC)
- I agree completely with the analysis done above by Nijdam and Martin Hogbin. Of course, the symmetry assumption should be formulated more accurately in the "official" text; not only C and X are independent, but also C is uniform on the 3-point set, and H is conditionally uniform on the rest (be it 1-point or 2-point set). Boris Tsirelson (talk) 15:04, 5 December 2009 (UTC)
- Actually Martin's description leaves me a bit confused since in my perception it is not quite in line with what he was pushing for earlier. However if we all essentially agree now, I don't quite get what the actual dissent is now. Are we just arguing about the order of various sections but agree about the general content (see also the outline under my comment)? I can live with more or less any order in the article as long as the (sourced) content stays (in whatever section). Or to be specific regarding the much debated Morgan, his approach (or that in similar publications) hast to be mentioned and his caveat to the vos savant's approach/simple solution as well, in which section of the article this happens however doesn't really matter.--Kmhkmh (talk) 15:22, 5 December 2009 (UTC)
- As for me, I only say that the symmetric case is more important for an encyclopedia than the general case. (Likewise, a circle is more important for an encyclopedia than an arbitrary curve.) Boris Tsirelson (talk) 15:32, 5 December 2009 (UTC)
- Actually Martin's description leaves me a bit confused since in my perception it is not quite in line with what he was pushing for earlier. However if we all essentially agree now, I don't quite get what the actual dissent is now. Are we just arguing about the order of various sections but agree about the general content (see also the outline under my comment)? I can live with more or less any order in the article as long as the (sourced) content stays (in whatever section). Or to be specific regarding the much debated Morgan, his approach (or that in similar publications) hast to be mentioned and his caveat to the vos savant's approach/simple solution as well, in which section of the article this happens however doesn't really matter.--Kmhkmh (talk) 15:22, 5 December 2009 (UTC)
- I agree completely with the analysis done above by Nijdam and Martin Hogbin. Of course, the symmetry assumption should be formulated more accurately in the "official" text; not only C and X are independent, but also C is uniform on the 3-point set, and H is conditionally uniform on the rest (be it 1-point or 2-point set). Boris Tsirelson (talk) 15:04, 5 December 2009 (UTC)
- Indeed; but I'm afraid, Rick Block will say that a featured article must be perfectly sourced... Boris Tsirelson (talk) 16:15, 5 December 2009 (UTC)
- I meant, please respond to my call. Formulate here your ideas about the symmetric case and its solution in the above terminology!! Nijdam (talk) 16:21, 5 December 2009 (UTC)
- OK, wait a bit. Boris Tsirelson (talk) 16:25, 5 December 2009 (UTC)
- I meant, please respond to my call. Formulate here your ideas about the symmetric case and its solution in the above terminology!! Nijdam (talk) 16:21, 5 December 2009 (UTC)
- Indeed; but I'm afraid, Rick Block will say that a featured article must be perfectly sourced... Boris Tsirelson (talk) 16:15, 5 December 2009 (UTC)
Not just words
Notation:
- Three doors with numbers 1,2 and 3
- C is the random variable indicating the number of the door with the car
- X is the random variable indicating the number of the door chosen by the player
- H is the random variable indicating the number of the door opened by the host
Assumptions:
- C is uniform on the set {1,2,3}
- C and X are independent
- H is conditionally uniform on the complement of the set {C,X} in the set {1,2,3} (be the complement a 1-point or 2-point set)
The "unconditional" solution:
A pure strategy of the player is a function that maps every possible pair (X,H) to either X or the "third" element of {1,2,3} (different from X and H).
A mixed strategy of the player is a function that maps every possible pair (X,H) to a probability measure on {1,2,3} that vanishes on {H}.
The winning probability is a function of a strategy. It is invariant under the rearrangement group of {1,2,3} (by the symmetry, of course). Therefore it is sufficient to consider only invariant mixed strategies. Such a strategy is nothing but a probability of switching. It is sufficient to consider only extremal values (0 and 1) of the probability. Thus, the question boils down to: to switch or not to switch. The simple unconditional calculation completes the analysis.
The "semi-conditional" solution:
The conditional winning probability without switching, P(X=C|X,H), is a function of X and H. It is invariant under the rearrangement group of {1,2,3} (by the symmetry, of course). Therefore it is the constant function. Taking into account that its expectation is equal to the unconditional probability (the total probability formula) we see that P(X=C|X,H) = P(X=C); that is, the conditional probability is equal to the unconditional probability. The simple unconditional calculation (the same as above) completes the analysis.
(You see, the conditional probability is used; but is reduced to the unconditional probability.)
There is also the "all-conditional" solution, not mentioning the unconditional probability; but probably I do not need to write it here.
What do you think about it? Boris Tsirelson (talk) 17:08, 5 December 2009 (UTC)
Maybe some words here sound a frightening math for some people; but they are just a formalization of the symmetry argument intuitively clear to most people. So much clear that they probably do not feel any need in formalization... Boris Tsirelson (talk) 17:27, 5 December 2009 (UTC)
- As I said; words, words, words and more words. Give me the correct formulation of this decision theoretical approach in formulas. BTW: I hope you don't tell this story to your first grade students in your introductory course. What I am actually interested in is what you tell in this course and then in the above formalism, with no more words than needed. Think you can manage? Nijdam (talk) 10:03, 6 December 2009 (UTC)
- Sorry, now I do not understand what do you mean by "words". Every mathematical paper consists of words (and formulas inside). The formulation above is correct. Yes, this is not what I teach. The reason is that I do not teach MHP; I teach probability, especially conditioning, and at that moment I use MHP as an instructive example. The goal of Wikipedia article is probably quite different. Boris Tsirelson (talk) 13:26, 6 December 2009 (UTC)
- I am not used to explain math in a sarcastic environment. If someone will ask me a reasonably specific question in a reasonably polite manner, I'll be reasonably helpful.
- I claim that the above texts are correct proofs (well, somewhat sketchy). I am not an anon; I am a professional mathematician.
- If you want to say that this is anyway too complicated for the article, just say so. In fact, I never proposed these solutions for the article. And I did not object against conditional probability. I only object against the general case treated as no less important (for Wikipedia, not for science) than the symmetric case. Boris Tsirelson (talk) 14:48, 6 December 2009 (UTC)
- I think the argument for this extensive treatment of the general case is, that is somewhat reflects the academic publications/treatment of that topic. I have hardly any objection against Martin's reaction to Nijdam's comment. Featuring the special case or unconditional solution and its benefits prominently in the first part of the article is perfectly fine. However if you review some of Martin's other or earlier comments and even more so some comments of JeffJor or Glkanter, then you can see they are pushing for things which partially factually false or unsourced and definitely not in line with bulk of reputable literature on the subject. Among these are glkanter "legal arguments", JeffJor's attempt to define the conditional approach as a "non MHP"-problem. There also seems to be an attempt to remove reputable sources from the article (via badmouthing) and ignoring that other sources more or less state same conclusions anyway. In particular there is one thing the WP-article cannot do, that is treating the MHP as a unconditional problem only, while the bulk of the literature treats it at least as a conditional problem as well. Another thing is, that the WP article cannot redefine the MHP to remove its ambiguity, since the original problem is ambiguous and much of the literature explicitly deals with the ambiguity. WP needs to report the definitions of others and not set its own.--Kmhkmh (talk) 15:38, 6 December 2009 (UTC)
- I agree. Boris Tsirelson (talk) 16:02, 6 December 2009 (UTC)
- I think the argument for this extensive treatment of the general case is, that is somewhat reflects the academic publications/treatment of that topic. I have hardly any objection against Martin's reaction to Nijdam's comment. Featuring the special case or unconditional solution and its benefits prominently in the first part of the article is perfectly fine. However if you review some of Martin's other or earlier comments and even more so some comments of JeffJor or Glkanter, then you can see they are pushing for things which partially factually false or unsourced and definitely not in line with bulk of reputable literature on the subject. Among these are glkanter "legal arguments", JeffJor's attempt to define the conditional approach as a "non MHP"-problem. There also seems to be an attempt to remove reputable sources from the article (via badmouthing) and ignoring that other sources more or less state same conclusions anyway. In particular there is one thing the WP-article cannot do, that is treating the MHP as a unconditional problem only, while the bulk of the literature treats it at least as a conditional problem as well. Another thing is, that the WP article cannot redefine the MHP to remove its ambiguity, since the original problem is ambiguous and much of the literature explicitly deals with the ambiguity. WP needs to report the definitions of others and not set its own.--Kmhkmh (talk) 15:38, 6 December 2009 (UTC)
- If you want to say that this is anyway too complicated for the article, just say so. In fact, I never proposed these solutions for the article. And I did not object against conditional probability. I only object against the general case treated as no less important (for Wikipedia, not for science) than the symmetric case. Boris Tsirelson (talk) 14:48, 6 December 2009 (UTC)
- I'm confused. If you agree then why are you listed here as supporting changing the article (to attempt to redefine the MHP to remove its ambiguity and to remove, badmouth, and ignore reputable sources). I suspect we don't have a common understanding of what changes are suggested. -- Rick Block (talk) 17:19, 6 December 2009 (UTC)
- Here is my position (once again): I only say that the symmetric case is more important for an encyclopedia than the general case. (Likewise, a circle is more important for an encyclopedia than an arbitrary curve.) Boris Tsirelson (talk) 18:43, 6 December 2009 (UTC)
- Then, I would suggest you change where your name appears in Martin's "summary" of opinions. I believe what Martin, and Glkanter, and JeffJor are actually arguing for is to banish any mention of conditional probability to a "variant" section. And they think you agree with this. -- Rick Block (talk) 20:20, 6 December 2009 (UTC)
- But I do support some change. First, prepare a list with a different formulation. Boris Tsirelson (talk) 20:42, 6 December 2009 (UTC)
- Rick, you are overstating my request. I do not want to banish any mention of conditional probability to a variant section but I do want to state that it is only really important if the host chooses non-randomly - the Morgan scenario. Martin Hogbin (talk) 21:55, 6 December 2009 (UTC)
Formulas, not words
Here is the "semi-conditional" solution rewritten for these that hate words and like formulas:
P(X=C|X=1,H=2) = P(X=C|X=1,H=3) = P(X=C|X=2,H=1) = P(X=C|X=2,H=3) = P(X=C|X=3,H=1) = P(X=C|X=3,H=2);
P(X=C) = P(X=C|X=1,H=2) P(X=1,H=2) + P(X=C|X=1,H=3) P(X=1,H=3) + P(X=C|X=2,H=1) P(X=2,H=1) + P(X=C|X=2,H=3) P(X=2,H=3) + P(X=C|X=3,H=1) P(X=3,H=1) + P(X=C|X=3,H=2) P(X=3,H=2);
P(X=1,H=2) + P(X=1,H=3) + P(X=2,H=1) + P(X=2,H=3) + P(X=3,H=1) + P(X=3,H=2) = 1;
therefore
P(X=C|X=1,H=2) = P(X=C|X=1,H=3) = P(X=C|X=2,H=1) = P(X=C|X=2,H=3) = P(X=C|X=3,H=1) = P(X=C|X=3,H=2) = P(X=C).
That is, the conditional probability is equal to the unconditional probability (in the symmetric case, of course). Boris Tsirelson (talk) 20:21, 7 December 2009 (UTC)
- Well, I don't hate words, but I do hate them when they hide an underlying problem. And, sorry, I do not understand what "semi-conditional" means. I gave before extensive solutions using Bayes and symmetry arguments. In both ways it is necessary to calculate a conditional probability. I would very much like anyone who claims there is a solution without such a calculation, demonstrate it, in proper terminology. Till now, no-one tried. Yes, in words, words, words. That's why. BTW: The above analysis, without words, also calculates a conditional probability. Nijdam (talk) 13:39, 8 December 2009 (UTC)
- "also calculates a conditional probability"? Ultimately, yes, of course. However, this way the fact that it equals to the unconditional probability becomes clear BEFORE we get the values of this or that probability. Not "numeric coincidence" but a logical necessity. Does it matter for you? Boris Tsirelson (talk) 16:53, 8 December 2009 (UTC)
- As another example, imagine you have to calculate the perimeter of an ellipse with major axis j and minor axis n. You are told that j = n. Is it better to do the calculation for an ellipse and then set j = n or should you observe that the ellipse is, in fact, a circle and do the calculation on that basis? Martin Hogbin (talk) 12:20, 13 December 2009 (UTC)
Theorem ( Boris Tsirelson, Dec 2009, published in the first time, all rights reserved :-) ) Let an even E and a random variable X be such that the conditional probability P(E|X) is a constant function of X. Then the conditional probability is equal to the unconditional probability: P(E|X) = P(E).
Eureka! Boris Tsirelson (talk) 17:38, 8 December 2009 (UTC)
- You puzzle me. What can I say more? Everything is said. What kind of point do you expect? Well, I repeat, but surely you will not be satisfied (and I do not understand why).
- First, I find the symmetric case more interesting. Second, I am glad to see that in this case it is rather easy to see (before specific calculations, just from simple general arguments) that the conditional probability (you know, of which event) is equal to the unconditional probability. I believe that this is a (or rather, the) mathematical formalization of a fact understood by many non-mathematicians by intuition and/or arguments about "no other useful information available" and all that. Boris Tsirelson (talk) 21:27, 8 December 2009 (UTC)
- I summarize my position in two points:
- 1. The symmetric case is more important for an encyclopedia than the general case. (Likewise, a circle is more important for an encyclopedia than an arbitrary curve.)
- 2. The coexistence of the conditional and the unconditional can be more peaceful. (Not just "numeric coincidence" in the symmetric case.)
- Boris Tsirelson (talk) 06:38, 9 December 2009 (UTC)
- I agree with you, and indeed is the conditional probability of the considered event not just coincidentally equal to the unconditional. But in my opinion it is necessary to mention the conditional probability as the probability of interest. That's why I asked you to formalize your reasoning. Nijdam (talk) 11:17, 9 December 2009 (UTC)
- For a better understanding of my position, I may add, that a lot of people, amonst them students, teachers etc., reason as follows: the car is with probability 1/3 behind the chosen door. Hence the probability to find it behind one of the two remaining doors is 2/3. After the opening of one of them this (!) probability now applies to the remaining unopened door. I consider this as insufficient. One needs to add here that the opening of one of the door "does not influence" the probability of the car being behind the chosen door. And what is the meaning of "does not influence"? Well that the conditional probability and the unconditional one are the same. Agree? Nijdam (talk) 11:26, 9 December 2009 (UTC)
- Agree! Boris Tsirelson (talk) 16:08, 9 December 2009 (UTC)
- I did not object against conditional probability.
- "After the opening of one of them this (!) probability now applies to the remaining unopened door." — For me this is the central (!) point of MHP. The only (!) point making it instructive to my students (in my opinion). A quote from #Boris Tsirelson's comments: "I spend 20-30 min on the Monty Hall paradox. I compare two cases: (a) the given case: the host knows what's behind the doors, and (b) the alternative case: he does not know, and it is his good luck that he opens a door which has a goat." I do so exactly for explaining why in (a) the 2/3 jump to the remaining door, while in (b) it scatters to both doors. Boris Tsirelson (talk) 16:28, 9 December 2009 (UTC)
- What makes things cumbersome? First of all, the lack of symmetry. And second, – to a much less extent, – conditioning. Boris Tsirelson (talk) 16:34, 9 December 2009 (UTC)
- I don't follow you any more. To put things straight: the probability 2/3 of the event that the car is behind one of the not chosen doors, differs (in type, not in value) from the conditional probability the car is behind the remaining unopened door given the other door opened. It concerns different probability measures. That's why I asked for formulas. Nijdam (talk) 11:51, 10 December 2009 (UTC)
- Nijdam, I completely agree with your words above. Why do you feel that you don't follow me any more? Boris Tsirelson (talk) 14:45, 10 December 2009 (UTC)
- Where you write: >>"After the opening of one of them this (!) probability now applies to the remaining unopened door." — For me this is the central (!) point of MHP. The only (!) point making it instructive to my students (in my opinion). << Because this is not in line with my explanation above. It is not this probability, but a different probability with the same value as the former. Nijdam (talk) 22:05, 10 December 2009 (UTC)
- (unindent) Ah, now I see; I was missing your point. Yes, of course. But on the other hand: in many cases people feel intuitively that some conditioning does not change some probability; and they are right often (probably, not always). There is a big difference between the approach of a mathematician (unless he speaks informally) and that of others. And I think, we should not be too insistent. I'd recommend a tone like that: ...thus the probability is 2/3. This probability is called unconditional, since ... In contrast, by the conditional probability one means ... It is worth to ask what is the conditional probability. It seems plausible that it is the same (2/3) since ...; and it is really the case. In general, conditional probability differs from unconditional probability, it is either bigger or smaller, depending on the condition; but on the average it is equal. In particular, in our case the conditional probability does not depend on the condition by symmetry. Thus it cannot be bigger or smaller, but only equal to the unconditional probability. However, without symmetry it is different; see Section "Asymmetric generalizations" below. Boris Tsirelson (talk) 07:11, 11 December 2009 (UTC)
- This has been Rick's and my opinion all along. We won't put much emphasize on it. But anyhow it should be mentioned that after opening of the door by the host a new situation has risen in which becuase of the symmetry the probability for the car being behind the chosen door is the same (has the same value) as before. This is not clear to many of the discussiants and some even fight this fact. Like in what is called the "combined door solution". I hope we may find you on "our" side? Nijdam (talk) 09:39, 11 December 2009 (UTC)
- It is a bit risky to say "yes" (or "no"), since we sometime misunderstand each other. Another example: let two fair coins be tossed; two independent events A,B of probability 0.5 appear. Now someone could say: P(B|A) = 0.5 = p(B), but this is a numeric coincidence only; P(.|A) is another probability measure different from P(.); never say "this 0.5", say "another copy of the number 0.5", etc. As for me, this is much too formal; definitely inappropriate when talking to non-mathematicians. It seems to me, you told this way. But maybe I again misinterpret your position. If so, then the better. Boris Tsirelson (talk) 11:39, 11 December 2009 (UTC)
- Nor Rick, me or Kmkmh have the intention to be formal (except of course in our minds). What we want is to be sure the distinction between the probability before the opening of a door by the host and the probability thereafter is somehow mentioned. I.e. we want to make clear or at least mention they are not the same. Why? Well, because some people, as I wrote before, reason as follows, or in a similar way: the probability of the car behind the chosen door 1 is 1/3. After opening of door 3 the probability for the car to be there is 0, hence (???) the probability for the other unopened door is 2/3. This "simple" explanation (solution?) is wrong. It turns up in many forms, all omitting the difference between the unconditional and conditional probabilities. This way of reasoning is copied by teachers, pupils, students and alas also by some mathematicians. It is the reasoning in the "combined door solution". It also turns up in many of the simulations on the internet. I suggested a very modest phrasing month ago: After the player has chosen a door, the probability it hides the car is 1/3. This probability is not influenced by the opening of a door with a goat by Monty, hence after Monty has opened a door with a goat, the probability the original chosen door hides the car is also 1/3. Because clearly the open door does not show the car, the remaining closed door must hide the car with probability 2/3. Hence switching increases the probability of winning the car from 1/3 to 2/3, which is formally correct. In a next, more formal section we could go into some more detail about the probability not being influenced. Of course this all is not Wikipedia:OR. It is found in many texts about the MHP, amongst them Morgan et al's. Nijdam (talk) 13:52, 11 December 2009 (UTC)
- Please pardon the intrusion. Nijdam, you wrote this above: "I consider this as insufficient." It seems to me that your opinion contrasts with many reliably published sources. I think Wikipedia's policy is pretty clear on this, and I imagine Rick Block would agree. Are you suggesting, insisting actually, that the article be edited to your opinion, rather than the reliably published sources? And although you did not ask me, I think your argument in the above paragraph is a rather weak insistence on the necessity of the so-called 'conditional' problem statement. How can a 'conditional' problem statement be a Monty Hall problem statement, when it doesn't even follow the 'words' of the actual problem statement from the published sources? Glkanter (talk) 11:52, 9 December 2009 (UTC)
- Boris, Nijdam - As Boris has suggested, the concept of a condition is to some degree arbitrary. What is the chance I will get a head if I toss fair coin given that I clap my hands before tossing it. Strictly speaking we could call that a conditional problem but nearly everyone would intuitively say that clapping your hands will not make any difference to the outcome so it can be ignored, and who can disagree? There has to come a time where it is permissible to say that a condition is unimportant and can be ignored. It is intuitive to most people that if a random legal goat door is opened it will not matter which one, this is also a true fact. It seems right to me, therefore, that we can ignore this detail in our initial presentation of the problem and its solution. Martin Hogbin (talk) 12:18, 11 December 2009 (UTC)
- (1) I agree with Martin (as usual).
- (2) About finding me on this or that side... Asked (by Nijdam) what is my point I have formulated it (in two concise items, you know). If each of us will do so, then it will become clear what are the possible coalitions (and the old "yes-no" list will become obsolete). I would be especially glad to reach a common position of Rick, Nijdam, Martin and myself. Boris Tsirelson (talk) 13:55, 11 December 2009 (UTC)
- Sorry Martin and Boris (I discoverd you reacted in the mean time). We have discussed these ideas over and over a long time ago. I'd rather see you coming to some insight. I write the conditioning completely out; chosen is door 1, door chosen by the host to be opened in boldface:
door* 1 door 2 door 3 probability car goat goat 1/6 car goat goat 1/6 goat car goat 1/3 goat goat car 1/3
- The host opens door 3 showing a goat:
door* 1 door 2 door 3 conditonal probability car goat goat 1/3 goat car goat 2/3
- I hope you admit there is factually conditioning and not some handclapping or whatever. Nijdam (talk) 14:25, 11 December 2009 (UTC)
- There is some conditioning in so far as a door has been opened, I think this is obvious to everyone. But we should allow the case that either door 2 or door 3 is opened to reveal a goat (even though in Whitaker's question he gives door 3 as an example), thus it is not important which door has been opened. Martin Hogbin (talk) 17:40, 11 December 2009 (UTC)
- I hope you admit there is factually conditioning and not some handclapping or whatever. Nijdam (talk) 14:25, 11 December 2009 (UTC)
- Okay, door 2 might be opened too. Here it comes:
- The host opens door 2 showing a goat:
door* 1 door 2 door 3 conditional probability car goat goat 1/3 goat goat car 2/3
- Again factual conditioning. You permanently "forget" that the player is on stage and sees which door is open. In fact the player might have chosen door 2 or 3 as her initial choice. Any combination of choice and opened door leads to the same analysis. That's why, the given combination serves as an example for the complete problem. Nijdam (talk) 09:20, 12 December 2009 (UTC)
Comment Boris
(I added a section break for easy editing)
- At the moment when X is already known (but H and C are not) we can say the following. If C=X then H is distributed uniformly on {1,2,3}-{X}.
>>>Okay, let us start from X given; then P(H=h|C=X)=1/2 for h!=X Nijdam (talk) 22:44, 14 December 2009 (UTC)
>>>>Yes; this is exactly what I wrote. To be uniform on a two-point set means just probabilities 1/2 at each of these two points. What is the problem here? Boris Tsirelson (talk) 05:59, 15 December 2009 (UTC) >>>>>No problem hereNijdam (talk) 14:22, 15 December 2009 (UTC) Otherwise, if C differs from X, then still, H is distributed uniformly on {1,2,3}-{X}. ("Given C != X", not "given C"!
>>>P(H=h|C!=X)=1/2 for h!=X ???? In my opinion: P(H=h|C!=X)=0 for h=C,XNijdam (talk) 22:44, 14 December 2009 (UTC)
>>>>No! Again: "Given C != X", not "given C"! It cannot be 0 for h=C simply because C is not given! This conditional probability does not depend on C. Boris Tsirelson (talk) 05:59, 15 December 2009 (UTC)
>>>>>Okay,I see what you mean.Nijdam (talk) 14:22, 15 December 2009 (UTC)
>>>>Use the force, do formulas not words! Words are sometimes ambiguous; formulas are not. The formula P(H=h|C!=X) means P(H=h|A) where A is the event C!=X. This event includes both values of C. In words: the condition "C != X" does not disclosure the value of C; it gives only a partial knowledge about C.
>>>>To be honest, these are not the true formulas, because one premise remains words: "At the moment when X is already known (but H and C are not)". But it is easy to get rid of the words completely, inserting the condition on X everywhere. That is, we consider P ( H = h | X, A ) where A is the event C != X. Or, if you prefer longer style formulas, P ( H = h | X=x, A ). In any form, it is important that the given condition leaves to C two possible values. Boris Tsirelson (talk) 06:54, 15 December 2009 (UTC)
The (3!) symmetry is already broken, but a (2!) symmetry persists.) That is, the (conditional) distribution of H does not depend on the event A = {C=X}. In other words, A and H are independent.
>>>>Before it was A={C!=X}?Nijdam (talk) 14:22, 15 December 2009 (UTC)
>>>I don't understand Nijdam (talk) 22:44, 14 December 2009 (UTC)
>>>>See above. And think again. Boris Tsirelson (talk) 06:38, 15 December 2009 (UTC)
>>>>It's clear now what you mean. Nijdam (talk) 14:22, 15 December 2009 (UTC)
>>>>You may also think about the following (rather standard) exercise. A fair coin is tossed 10 times. What is the (conditional) probability of "head" in the first trial given that there are exactly 7 "heads" in the total? Boris Tsirelson (talk) 07:10, 15 December 2009 (UTC)
Thus, the distinction between P(A) and P(A|H) is rather academical (like the distinction between (100-1)+25 and 100+(-1+25) in my example elsewhere); I think so. On the other hand, Rick says that somehow this distinction helps to many people to avoid errors. Maybe; this is beyond my expertise. Boris Tsirelson (talk) 12:40, 13 December 2009 (UTC)
>>>I did show in the above tables the factual conditioning. Where am I wrong? Nijdam (talk) 22:44, 14 December 2009 (UTC)
>>>>I never told there is no conditioning. I always agree that P(.|B) is not the same probability measure as P(.). I only say that the conditioning is ineffective on the considered event, due to independence. And the independence is not a numerical coincidence, but a consequence of the symmetry. Boris Tsirelson (talk) 06:43, 15 December 2009 (UTC)
>>>>>Okay, but I doubt whether these are the arguments the (I wrote adversaries, but meant) advocates of the simple solution have in mind or whether this makes it better understandable for the interested readers. I also cannot imagine you use this reasoning in your introductory course. Nijdam (talk) 14:22, 15 December 2009 (UTC)
- Well, we now agree on the mathematics (modulo a miserable problem of denoting by A different things on different days of discussion), and I feel my mission finished. About my introductory course: I had no reason to use this argument, but I could, and it would not make more troubles than other topics; however, this is hardly relevant to WP. About adversaries I do not know; I only feel that arguments of symmetry are quite easily guessed by many non-mathematicians (they do not vast effort to the trouble of formalization...). Boris Tsirelson (talk) 15:23, 15 December 2009 (UTC)
[outdent]I do not forget that fact. You seem to forget that we are addressing the formulation in which the host chooses randomly between legal goat doors. There is therefore a strict conditioning, just as there is in my urn problem, but it is quite obvious that this conditioning can be ignored. If the host chooses at random, it cannot matter which door he opens. It is just like my hand clap, is does physically occur but it is obviously irrelevant.
- It is relevant. Look at the probability the car is behind the door to be opened by the host. Before opening this equals 1/3, after opening 0. They are different probabilities. The same with de door left closed: before 1/3, after 2/3. How do we calculate these? By looking at the chosen door: before 1/3, after (also) 1/3, but although the same value, different probabilities! Before we have the (unconditional) distribution: 1/3, 1/3, 1/3; after (conditional): 1/3, 2/3, 0. Different distributions. If you're not convinced, please give a proper analysis, in formulas (with words to explain), not just in words. Nijdam (talk) 10:33, 12 December 2009 (UTC)
- It is not relevant to calculating the probability of interest, which is that the car is behind door 1. Martin Hogbin (talk) 10:51, 12 December 2009 (UTC)
Terminology and choice of sample set
The above heading is essentially for ease of editing - this section follows on from the thread above.
I do see your point, Nijdam, it is not easy to produce neat diagram (or formula) that proves my point, but I will have a go. To some degree the choice of initial sample set is arbitrary and this choice defines how the calculation proceeds. It is the choice of initial sample set that has to be defended with words. It may be that with your choice, my point is not easy to make. The problem is that other choices (say based on goat doors and car doors) may be difficult to justify. Let me think about it. Martin Hogbin (talk) 11:23, 13 December 2009 (UTC)
- But I took the challenge to answer to Nijdam in his own terms, see above. Boris Tsirelson (talk) 13:00, 13 December 2009 (UTC)
What about this? Not the terminology that you choose but it implicitly includes the symmetry of the situation in that the action of the host makes no difference to the answer.
1/3 | 1/3 | 1/3 | |||
Goat | Goat | Car | |||
The host opens door 3 to reveal a goat | |||||
Stick | Swap | Stick | Swap | Stick | Swap |
Goat | Car | Goat | Car | Car | Goat |
Martin Hogbin (talk) 12:14, 13 December 2009 (UTC)
- Hey, Nijdam, why don't you quit ignoring my edits and questions? It's pretty rude. If Huckleberry had used your method instead of Devlin's method, what would have been different in the game play that went on infinitely? Would he win a 'different' 2/3 of the time, or the 'same' 2/3 of the time? Wouldn't you agree it made no difference whatsoever to Huckleberry which door Monty opened? Isn't this known as 'indifference' in Mathematics? And, there is no Wikipedia policy stating story problem paradoxes must be solved using formal notation. Wouldn't be such a popular paradox in the real world with that requirement, would it? Glkanter (talk) 13:08, 12 December 2009 (UTC)
- There is also a words, not formulas explanation - which I think would be accessible to a layperson - at Talk:Monty Hall problem/FAQ. And, to address a point Glkanter attempts to interject above - this argument is firmly based on wp:reliable sources, not wp:or. Nijdam is reiterating what Morgan et al. and Gillman, and Grinstead and Snell, and most elementary probability textbooks (as claimed by Kmhkmh) say. Glkanter definitely, and Martin to a lesser extent, misinterpret the main point of the Morgan et al. paper. I believe Glkanter genuinely does not understand that there is a difference between asking about the probability of winning by switching (in general, for all players) and asking about the probability of winning given knowledge of which door the host opens, i.e. is having trouble with the basic concepts of conditional probability (see, for example, this edit). I think Martin understands the difference but insists that a complete explanation of the "notable" MHP can be made without mentioning this difference and it should therefore remain unmentioned until (essentially) a variant section. JeffJor understands this difference and insists that the MHP is explicitly asking about the former rather than the latter and therefore any source that says the MHP is asking about the latter is WRONG and should be excluded from THIS article.
- You told me not to speculate on your likely actions regarding this consensus. How then is it appropriate for you to speculate on my personal knowledge of the subject matter, given that we have had no communications whatsoever other than on the various Wikipedia pages? Glkanter (talk) 16:44, 11 December 2009 (UTC)
- I agree with both of Boris's "position points" above, but would add for the 2nd point that co-existence CANNOT mean excluding one in favor of the other (per FUNDAMENTAL Wikipedia policy, i.e. WP:NPOV). I've been referencing this version of the Solution on other threads as well, which I think is far closer to adhering to both of these points than what Glkanter, Martin, and Jeffjor are suggesting. -- Rick Block (talk) 16:24, 11 December 2009 (UTC)
- I guess I do agree with you, Rick, that I do not understand the main point of the Morgan paper, but that is because it is so badly written that it is impossible do divine what the main point is. I can assure you, however, that I do understand the subject and the issues involved. As you will see from my discussions with Nijdam, I accept that, strictly speaking and in some formulations, the problem is conditional, but in the symmetric case (host opens a legal goat door randomly) the condition (which door the host opens) is clearly irrelevant to calculating the probability of interest, thus it can be ignored. I also agree with Jeff, that the MHP can be reasonably interpreted to ask an unconditional question.
- Because of the both the facts presented above, I believe that the article should start with a complete section in which the problem is treated unconditionally (as in many reliable sources). After that (or even at the end of that section) I am happy to point out that even if the host chooses randomly the problem might be treated conditionally, with a reference to Morgan. This could then lead on to a variations section which would include the Morgan scenario (we know the door opening policy of the host). Martin Hogbin (talk) 11:09, 12 December 2009 (UTC)
- (unindent) back from weekend Here is a proof that 99+25=124:
- 99+25=100-1+25=124.
- Is it a correct, complete proof? Or should I show that I understand the conceptual difference between (100-1)+25 and 100+(-1+25) and use the associative law in order to overcome the difficulty? Boris Tsirelson (talk) 15:33, 12 December 2009 (UTC)
- People never write complete proofs. Yes, absolutely never. If in doubt, look at Mizar system; there you can find source texts and programs that allow to generate some complete proofs (but probably you'll never have enough paper in order to print such a proof).
- The more so, we should not insist on complete proofs in the encyclopedic context. Symmetry arguments are usually treated by non-mathematicians as too evident for being proved. Boris Tsirelson (talk) 15:40, 12 December 2009 (UTC)
- Thus, we'll never be able to convince the majority of our readers that the conditioning is really relevant in the symmetric case. Likewise, we could not convince children that 99+25=124 cannot be accepted if the associative law is not involved.
- And still, I think, it is not bad if conditioning will be mentioned in the article, as far as we'll not be insistent about its relevance in the symmetric case. Boris Tsirelson (talk) 15:48, 12 December 2009 (UTC)
- Isn't the "host forgets' case also symmetric? If we're using "symmetry" to say the host's action does not affect the player's initial chance of selecting the car, why is it that this same argument does not apply in this case? I know the reason and you know the reason but I'm willing to bet many people who "understand" the popular solutions to the MHP do not understand why these solutions DO NOT apply in this case. See, for example, this exchange on the arguments subpage.
- Rick, we've argued this many, many times. The problems are greatly different. Just one example, there are times the forgetful host doesn't offer the switch. Because he revealed the car. Two stated premises of the MHP are that he always reveals a goat, and always offers the switch. I'd hate to see any editor invest time by responding to this oldy, moldy filibuster. Glkanter (talk) 18:36, 12 December 2009 (UTC)
- (I've basically said this before - in this section, above) In a fairly recent column, vos Savant addresses the "host forgets" variant. In her analysis of this version she laments [1] [unfortunately now a dead link, and not available on the Internet Archive ] "Back in 1990, everyone was convinced that it didn’t help to switch, whether the host opened a losing door on purpose or not. ... Now everyone is convinced that it always helps to switch, regardless of what the host knows. But this is just as incorrect!" This is absolutely true. And, IMO, it's precisely because the popular sources do NOT address the "classic" MHP using conditional probability. By avoiding addressing the problem in this way, the popular sources have simply replaced one incorrect notion (two unopened doors always means each has equal probability) with another (whatever the host does he can never change the 1/3 probability of the player's initially selected door). As a featured article on Wikipedia, IMO this article must not make the same mistake. -- Rick Block (talk) 18:17, 12 December 2009 (UTC)
- Really? Well, if the conditioning helps to not make errors then of course it is useful. Then we only have to explain to the reader why it is important (namely, demonstrate him typical errors) before bothering him by conditioning; then hopefully he will not be disturbed. Boris Tsirelson (talk) 18:58, 12 December 2009 (UTC)
- Especially if we deal with the problem in more detail later on in the article for those who are interested. Martin Hogbin (talk) 13:25, 13 December 2009 (UTC)
Observation and suggestion
- I noticed that request for comments and recent posting related to it already follow the same pattern as the old discussion. The same old points get reiterated by the mostly by same people over and over (right now including me I'm afraid). Also there is tendency of explaining each's (personal) understanding of the problem rather than sticking to the sources and there is a rather endless arguing about relatively minor points ("which goes first"). This "game" is now played by the same actors (partially including me) for over a year without any productive result.
- Due to the observation above I'd like to repeat my recommendation above. The article is best served, if it gets a review by a group of knowledgeable, competent and "neutral" editors. All former editors and constant participants of this endless discussion should state their point in a single comment and other than that (voluntarily) stay out of the way. They should only give further comments if they are explicitly requested by the reviewers. The former editors that should stay out of the way explicitly include Martin Hogbin, Glkanter, JeffJor, Rick Block, Nijdam and Kmhkmh.
--Kmhkmh (talk) 13:13, 5 December 2009 (UTC)
- The MHP has been described as the world's most intractable brain teaser. It contains elements of both mathematics and philosophy. I think a suitable body of knowledgeable and neutral editors who will be able to understand to issues involved will be hard to find.
- As an alternative I would suggest some form of mediation, possible by someone who makes no attempt to address the problem itself but who mediates on general content principles and policies might help. Martin Hogbin (talk) 14:15, 5 December 2009 (UTC)
- Finding a group of people being better or at least as qualified as he current long members of the discussion is not that hard. The reviewers don't have to be the perfect mix but just good enough for the job. There real issue here is, if people are willing to step back and accept the sound judgement of others, even if it does not support their favoured solution. As far as mediation i have no objection if the other editors are fine with it, but i'm quite skeptical regarding the success.--Kmhkmh (talk) 15:06, 5 December 2009 (UTC)
- Where are they now then? Are you thinking of a bunch of mathematicians? Most of them will probably not be that interested, the maths is pretty simple, all the problems lie elsewhere. Martin Hogbin (talk) 16:42, 5 December 2009 (UTC)
- I think most third part expert (mathematicians or otherwise) are put off by the endless (and often pointless) discussion and they are definitely not interested in participating in such a thing over months or even years (neither am I). They might however be willing to help out with a review/3rd opinion provided this can be handled in timely fashion and that the results will be heeded. This means the first step would be that the "warring parties" explicitly agree beforehand to accept the result and refrain from further edits.--Kmhkmh (talk) 16:59, 5 December 2009 (UTC)
- Whichever side lost in that case would feel that they had not had their case taken seriously. I think mediation, where the current proponents can present their cases to a mediator, is a better bet. Martin Hogbin (talk) 17:02, 5 December 2009 (UTC)
- I think most third part expert (mathematicians or otherwise) are put off by the endless (and often pointless) discussion and they are definitely not interested in participating in such a thing over months or even years (neither am I). They might however be willing to help out with a review/3rd opinion provided this can be handled in timely fashion and that the results will be heeded. This means the first step would be that the "warring parties" explicitly agree beforehand to accept the result and refrain from further edits.--Kmhkmh (talk) 16:59, 5 December 2009 (UTC)
- Where are they now then? Are you thinking of a bunch of mathematicians? Most of them will probably not be that interested, the maths is pretty simple, all the problems lie elsewhere. Martin Hogbin (talk) 16:42, 5 December 2009 (UTC)
- Finding a group of people being better or at least as qualified as he current long members of the discussion is not that hard. The reviewers don't have to be the perfect mix but just good enough for the job. There real issue here is, if people are willing to step back and accept the sound judgement of others, even if it does not support their favoured solution. As far as mediation i have no objection if the other editors are fine with it, but i'm quite skeptical regarding the success.--Kmhkmh (talk) 15:06, 5 December 2009 (UTC)
This whole discussion could have reached an amicable end long ago
If the Morgan supporters would please read and respond to the existing section "Is The Contestant Aware?", and it's question:
- "Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?
- "It seems to me that this is a (unstated) premise of the problem, as both vos Savant (Whitaker) and Krauss and Wang begin the problem statement with: 'Suppose you're on a game show'. I read this as clearly stating it is only the contestant's point of view we are concerned about. And, being a game show, the host is prohibited from divulging to the contestant either where the car is, or where the car is not.
- "Is there agreement on this, or is this in dispute? Glkanter (talk) 11:28, 29 November 2009 (UTC)"
- http://en.wikipedia.org/wiki/Talk:Monty_Hall_problem#Is_The_Contestant_Aware.3F —Preceding unsigned comment added by Glkanter (talk • contribs) 16:15, 5 December 2009 (UTC)
- Yes this is in dispute. The view(s) that have to be described in the article, are the views taken in publication on the Monty Hall problem and there treatmeants which assume different perspectives. What we probably can be agreed on, that those different perspectives should be handled in a variations or generalization section and that they don't belong into the article lead. Also it is probably not appropriate to assume (specific und possibly) legal gameshow regulations. Since most of our readers, much of the original audience, potentially even whitaker himself and some of the people that have published on it, are possibly not aware of specific regulation and legal restrictions and definitely do not mention them. This means while the article can and should discuss how actual regulations (or the real monty's behaviour) might influence the analysis of the problem, but that definitely doesn't belong in the lead section nor can any such regulation simply be considered as an obvious "given" for analysing the problem.--Kmhkmh (talk) 17:01, 5 December 2009 (UTC)
- But it is indisputable. I don't care whether the problem starts with "Suppose you are on a game show," or what you think that implies. In every version of the MHP I have ever seen, the question is "Should the contestant switch?" It is not "What is the probability of winning the car if the contestant switches?" Or "Describe a parametric formula that allows the contestant, once given the proper data, to decide whether or not to switch." It is a yes/no question in the purest sense: only two answers address the question. "Yes, she should switch" and "No, she should not switch." If you are required to make assumptions to reduce your answer completely to one of those, there are acceptable means to do so. The point about the contestant being aware is indistutable because we have to make whatever assumptions are necessary to reduce the answer to "'yes' in all possibilities considered" or "'no' in all possibilities considered," which means that we have to apply the same knowledge we assume the contestant will use. When they address the actual MHP question, the sources that address the conditional problem make such assumptions; just not that specific assumption about how the host chooses between two goats. They do make the neutral contestant-knowledge assumptions for what I call "game protocol" (e.g., always revealing a goat) and placement bias.
- Morgan's point is NEVER that the answer to the MHP is 1/(1+q), or whatever. It is that one specific assumption is not really necessary when the problem is taken literally in conditional form (which itself is disputable: the sources that claim it is conditioanl misquote their sources in such a way that makes it conditional). You don't need to make the assumption that reduces 1/(1+q) to 2/3, since 1/(1+q) is always greater than 1/2. But that conditional form loses the real beauty of the MHP, that the seemingly paradoxical answer is correct. Far more sources discuss this aspect of the problem than address the "conditional solution," which is where some people's POV is interfering with their ability to approach the article objectively. Yes, the conditional problem is addressed by some sources. It is a minority, and neutrality means it should be treated as a variant. JeffJor (talk) 13:31, 7 December 2009 (UTC)
- I imagine Selvin explained his version of the game show this way: The contestant makes a random choice of doors. The probability she did NOT choose the car is 2/3. Monty reveals a goat, giving her no new information about either remaining door. She is indifferent to which door Monty opened. Being a sentient being, she says to herself, my selection's 2/3 chance of not being the car has not changed, therefore, on average, the remaining door has a 2/3 chance of being the car. I don't think I can do better than that. I'll take the switch! Glkanter (talk) 13:59, 8 December 2009 (UTC)
- These are the only 2 known English language Game Show situations (Whammy and the 1950s Scandal) in which the particular contestant had more 'information' than an 'average' contestant would have. Both situations were considered highly unexpected aberrations, and many steps were taken to prevent either from being repeated.
- So, yes, there is a commonly understood expectation of all game show watchers that each contestant has no information otherwise not available to other contestants, certainly not coming from the host (or the shows producers). And there's tons of laws and lawyers watching for this in the US.
- This is not ambiguous: 'Suppose you're on a game show' Glkanter (talk) 17:33, 5 December 2009 (UTC)
- You miss the point you are making assumptions based on your knowledge and perception of game shows. There is no evidence whatsoever that most of the readers have the same legal background as you have nor do most publications on subject deal with legal restrictions or "gameshow regulations". Basically this boils down to stick to the sources and no speculations by WP-Editors.--Kmhkmh (talk) 17:52, 5 December 2009 (UTC)
- Please see my response above. -- Rick Block (talk) 17:50, 5 December 2009 (UTC)
It is so pervasive, nobody bothered to mention it explicitly as an assumption. Everyone in the US understands this rule of game shows. The idea that one of the contestants on the screen has 'inside' knowledge? Folly. It's in the fine print that runs at the end of each and every episode. And I documented the only known times it improperly happened. This is not my opinion or interpretation. It's a defining characteristic of a game show, without which, it is no longer a game show. Your twisted interpretation more resembles a street hustler with a card table and a deck of cards. Or three shells and a pea, perhaps. Glkanter (talk) 18:11, 5 December 2009 (UTC)
- You make a bold claim (It is so pervasive, nobody bothered to mention it explicitly as an assumption.) without providing any evidence and moreover you miss the point again. The question was not posed to actual game show contestants (being most likely aware of all regulations) but to readers of weekly column and later through various (international) publications to a broader audience. Even if your most likely false claim was true for all readers of Marylin's column, it is certainly not true for the international audiences reading the various publications. I've sampled quite a lot of material on the subject in English and in German and on top of my head i cannot recall any of them mentioning legal restrictions and national game show regulations. In particular the German articles on the subject with a readership definitely being unaware of any such regulations do not mention any legal conditions whatsoever. The 2 cases you've posted above are not in question, however they do not prove your claim. What's in question here is your (false) line of reasoning.--Kmhkmh (talk) 18:35, 5 December 2009 (UTC)
- The MHP is really, as this article confirms, a probability puzzle. In such puzzles it is usual to make certain assumptions, which in the MHP would be that the cars are initially placed randomly, the player chooses randomly, and that the host chooses a legal goat door randomly. The assumptions are precisely those Glkanter claims above.
- If in the other hand, you want to claim that the puzzle is based on the question by Whitaker then I suggest that you first read the real Whitaker statement rather than the inacurate version misquoted by Morgan. In Whitaker's original question it is clear that the door number given for that initially chosen by the player and the door number opened by the host are intended to be examples rather than specified doors. I might add that, as no information is given in the question, we should still take all the human choices to be random by the principle of indifference. Martin Hogbin (talk) 11:27, 12 December 2009 (UTC)
Mediation?
Mathematics is the one area where there will be general agreement amongst those who understand the problem. The arguments are all about questions like: What is the MHP? What is the most notable aspect of the MHP? How should an encyclopedia article be presented? How should we use reliable sources? Which sources are the most reliable?
This list is not intended in any way to be exhaustive it just gives examples of the kind of thing that we are arguing over and both sides claim to have WP policies on their side. Does anyone think that we might benefit from mediation on the policies and principles involved here? Martin Hogbin (talk) 16:53, 5 December 2009 (UTC)
- Yes, I think a good mediator could be helpful in getting this long-festering discussion to converge sensibly. Dicklyon (talk) 17:23, 5 December 2009 (UTC)
- Sounds like a reasonable idea to me. -- Rick Block (talk) 18:23, 5 December 2009 (UTC)
- As a professional mathematician (indeed, a probabilist and statistician) I am excited there is a new serious discussion about what the MHP is all about. The problem with problems which appear mathematical problems, posed by non-mathematicians, is that the mathematician may well discover there are several ways to interpret what the original problem-poser meant. Mathematicians often work backwards. Someone asks a problem. The problem is in fact ambiguous. One discovers a way to disambiguate, which allows a pretty solution. Often, the disambiguation involves making some of the "hints" of the original poser much harder, and ignoring others. I was delighted to discover their are two ways to disambiguate the problem: one is the conditional version, one is the unconditional version. I was also delighted to discover that in order to get "the good solution" to the conditional problem, one needs to demand different conditions, than to get "the good solution" to the unconditional problem. I also discovered a game-theoretic resolution of the problem, and I discovered some more interesting literature, I'll add citations later. Since my own work (on my web page) is not yet converted into a regular mathematical paper, and not yet submitted to, let alone passed by, peer review, I don't suppose wikipedia can or should use my insights, for the time being! But I continue to follow the discussion with great interest and I'll certainly let you all know my five cents worth if I think I have few cents to chuck in. Gill110951 (talk) 15:40, 6 December 2009 (UTC)
- Just looked at the Morgan et al paper. It is published work in a well-known journal (but not a very high status one) and it is 20 years old, and there have been plenty more insights since then. Mathematics is never finished. Mathematicians will go on finding new variants of the problem and hopefully getting new insights about it. Also discussions between mathematicians and non-mathematicians about this particular problem will continue to expose new variants. Great!!! As a mathematician I am not particularly interested in what vos Savant really mean, what she thought was the solution, and whether or not it was correct. There are many nice problems here and many nice solutions. A mathematician's job is to simplify and unify and understand. Often, as I said, working backwards: to which questions is the answer "change doors" the (a) right answer? What I am saying is that the mathematician has somewhat different interests to the encyclopedia compiler. Also since mathematics is in principle more or less verifiable independent of knowing all kinds of facts about the real world, the idea that an encyclopeadia may not contain new results, is crazy. Finding new mathematics results is a creative process, but checking that they are correct is an exercise in logic, not a matter of citing authorities. Especially in this case, since understanding most versions of the MHP will presumably only involve elementary mathematics. I don't care too much about what "authorities" have written about this problem in the past. A mathematician is more convinced by a beautiful and transparent proof, with some new twists, of an a priori surprising result, than by a citation to some dusty printed matter. Gill110951 (talk) 16:27, 6 December 2009 (UTC)
- Finally for today, I actually edited the MHP page including the insights from my own researches from last March. Of course a wikipedia editor is not supposed to write about his or her own research. So you can all edit it all out again, but please think about what I am trying to say first. Maybe you can be convinced by the logic. If so you can decide whether notes for his students and for the many journalists who keep asking again about this problem, on the home page of a well known authority on mathematical statistics, who has been telling his students about MHP for years, is an authoritative enough source for wikipedia. If you like I can post the notes on arXiv.org so they will still be there after I'm retired... And maybe Boris Tsirelson can say if he agrees with my maths. I'll send him an email. Gill110951 (talk) 17:21, 6 December 2009 (UTC)
How about this as a start
Those in favour of change start a separate development article, no doubt based on the current one, that concentrates on the 'simple' problem and convincing explanations. Much of this is already in the current article. This would make clear exactly what changes are being sought. If the pro-changers cannot agree on what changes are wanted then the article stays as it is by default. On the other hand once we have an agreed alternative version, we can seek mediation on how to proceed from there. This will involve questions like: should we have one or two articles? What should they be called? How could the two versions be combined? What order should the sections be in? etc. The advantage of this approach is that it lets the mediator understand what the two sides are after. Martin Hogbin (talk) 12:41, 6 December 2009 (UTC)
- I'm sorry Martin, I think this suggestion just continues the horrid legacy of the 6 years and 9 archives (plus countless other argument and talk pages scattered throughout Wikipedia's servers, plus this HUGE talk page) filibuster engineered by Rick. You can spend all the time and effort you want, then in my opinion, Rick and Nijdam will ultimately exercise some sort of veto power, despite any majority opinion (including respected Mathematics professors) that would otherwise, by Wikipedia standards, be indicative of a clear consensus.
- For example, here's Rick's comment, before he decided to "reword to avoid potential misinterpretation as ownership".
- "(as if by convincing me that their POV is "correct" I would then agree to change the article as they wish)."
- Nijdam seems to think it's his way or the highway. So far, anyways, he's been right.
- No, you've got the attention of the whole Wikipedia Mathematics group. Follow through with gaining the consensus before wasting any more time on sandbox-style editing. Heck, if I recall correctly, there's already a 'new' page prototype somewhere under your own user name. I've contributed to it. And the only issue of contention is 'Is the contestant aware?'. Most everything else looks resolved, to me. Glkanter (talk) 13:10, 6 December 2009 (UTC)
- If not a development version the how about some kind of outline for the proposed article? I am trying to check that we are all after the same thing. Martin Hogbin (talk) 13:36, 6 December 2009 (UTC)
Martin, there's nothing ambiguous here:
Changes suggested by JeffJor, Martin Hogbin, and Glkanter
If you're here because you've been invited to comment, there are ,two,. three (related) suggestions.
- Glkanter's suggestion: Eliminate all 'host behaviour, etc' influenced discussion, save for the Wikipedia minimum necessary references to Morgan and his ilk, as the 'conditional' problem is the converse of "Suppose you are on a game show."
- JeffJor's suggestion: The so-called conditional problem needs to be a separate article, with "conditional" in its title.
- Martin Hogbin's suggestion: This article should concentrate on the unconditional solution with the Morgan's conditional solution in a variations section.
We've been clear and consistent throughout. And the consensus is really already here, save for whatever veto powers exist within Rick and Nijdam. Glkanter (talk) 13:46, 6 December 2009 (UTC)
- 8 to 4 (Henning Makholm's comments clearly indicate he's opposed to this change) with one side arguing a fundamental Wikipedia policy and the other arguing mostly WP:OR is not a consensus, and even if it were it cannot be used as a blank check to violate NPOV. -- Rick Block (talk) 15:30, 6 December 2009 (UTC)
- I wholeheartedly disagree with your characterization of this good faith discussion by many editors. Essentially, you're describing it as 'Good vs Evil', and 8 of us, so far are on the side of Evil. I'm not buying it. I reject your argument prima facie. Just more filibustering on your part. Glkanter (talk) 15:36, 6 December 2009 (UTC)
- Rick, perhaps you could ask Henning to put his name in the appropriate section and sign it.
Outline for an alternative article
This is where I would start, the current contents are given for reference. Martin Hogbin (talk) 16:12, 6 December 2009 (UTC)
Current
- Problem
- Popular solution
- Probabilistic solution
- Sources of confusion
- Aids to understanding
- Why the probability is not 1/2
- Increasing the number of doors
- Simulation
- Variants
- Other host behaviors
- N doors
- Quantum version
- History of the problem
- Bayesian analysis
- See also
- Similar problems
- References
- External links
Proposed
- Problem - Whitaker statement - Sourced unconditional restatement
- Solution - The unconditional solution, from one of many sources - clear diagrams.
- Sources of confusion - Much as it is now
- Aids to understanding
- Why the probability is not 1/2
- Increasing the number of doors
- Simulation
- Conditional Problem - Reasons the problem is strictly conditional.
- Variants
- Morgan Scenario
- N doors
- Quantum version
- History of the problem
- Bayesian analysis
- See also
- Similar problems
- References
- External links
I'd disagree with that. Morgan is not just variant but he explicitly deals with the original problem as well, so do other sources with a conditional problem. The conditional approach is not a different problem as the TOC seems to suggest now(?) but a different perspective or approach to the same (ambiguous) problem. The way you attempt to reframe, that in your table of content is clearly in contradiction to the treatment in various reputable sources and hence in this form nonnegationable. Not to mention it also contracts your own comments on Nijdam's recent posting ("No, your calculation is fine and as I have made clear before I accept that the MHP problem (as defined by the K & W statement in which specific doors are mentioned) is a problem of conditional probability."). If it this is just a case of ambiguous wording (i.e. I'm simply misreading you ) then please correct that before further discussion.--Kmhkmh (talk) 16:28, 6 December 2009 (UTC)
- I did not really expect you to agree as you are one of those who are against change here. This section was for those who want to change the article to check that they are all on the same wavelength. I will respond to your point about Morgan in a new section. Martin Hogbin (talk) 17:03, 6 December 2009 (UTC)
One time, in English please...
Would one of the Morgan supporters, using whichever published source you prefer, please finish this sentence:
Solving the unconditional K & W Monty Hall problem statement is not sufficient, and a conditional problem approach must be utilized due to the instance where the host chooses between 2 goats. Because...[Please continue here].
This is essentially your argument against the three similar proposals, right? Thank you. Glkanter (talk) 19:05, 5 December 2009 (UTC)
- Are you asking why this view should be in the article, or are you asking for an explanation of this view? If the former, see WP:NPOV. If the latter, see Talk:Monty Hall problem/FAQ or the Morgan et al., or Gillman, or Grinstead and Snell references. -- Rick Block (talk) 22:15, 5 December 2009 (UTC)
6 Years and 9 Archives Comes Down To The Definition of a Game Show.
"Suppose you are on a game show..."
That's how both Monty Hall problem statements begin in the article.
Call it whatever you want in Probability Speak, symmetry, indifference, random, unknown, or equal.
In game shows, it is understood that nothing will be communicated about this to the contestant. So he remains blissfully unaware of anything but symmetry, indifference, random, unknown, or equal.
This is not trivial, to be argued or interpreted away, it is the 5th and 6th words of the problem statement. GAME SHOW. Literate people the world over are expected to know and understand what they are watching on TV. It is not this article's responsibility to consider otherwise.
As part of the definition of Game Show, it was unnecessary to state outright as a premise: 'Host will not share location of car with contestant'. Because that's what you claim he 'could', bizarrely, illogically, somehow do.
So, go ahead and argue. Tell me, as always, that I am mis-interpreting, or simply wrong, or whatever. Do me one favor, though. Just tell the truth. Glkanter (talk) 22:27, 5 December 2009 (UTC)
- The explanation is given further up and btw i explained you that already almost half a year ago for the first time.--Kmhkmh (talk) 22:50, 5 December 2009 (UTC)
Why Is Mediation Even Being Considered?
I think a consensus already exists for the proposed changes. Or, it will exist soon enough, when everyone has had the opportunity to comment. Using the 80/20 rule, I figure we've already heard from most everybody that's going to weigh in.
I count:
- 3 editors for maintaining the status quo
- 8 editors who are in agreement with one of the 3 similar suggestions
- 1 editor who's preference is undetermined
- 10 'comments requested' editors who have not commented to date. I understand from my Wikipedia readings that ultimately 'silence implies consent'.
We know unanimity isn't required, or Wikipedia would call it 'Gaining Unanimity' instead of 'Gaining Consensus'. So unanimity isn't our goal. And certainly not going to happen.
And, while voting is improper, *counting* is necessary.
So, which is it:
- 'All editors are equal'
or
- 'All editors are equal, but some editors are more equal than others'?
Glkanter (talk) 14:34, 6 December 2009 (UTC)
- From WP:NPOV: "Neutral point of view" is one of Wikipedia's three core content policies, along with "Verifiability" and "No original research." Jointly, these policies determine the type and quality of material that is acceptable in Wikipedia articles. They should not be interpreted in isolation from one another, and editors should therefore familiarize themselves with all three. The principles upon which these policies are based cannot be superseded by other policies or guidelines, or by editors' consensus. [emphasis added]
- All 4 editors in the minority (Kmhkmh, Nijdam, Henning Makholm, and myself) have directly or indirectly cited this policy as the reason against the changes that have been suggested. I have made this point clear in my comments above and over and over and over and over again in response to your tendentious attempts to introduce this change. Although you know I am an administrator on this site (which means not that my opinion on content issues matters more than any other editor's, but that I am VERY familiar with Wikipedia policies), you apparently do not believe me. The point of mediation would be to bring in an outsider, also familiar with Wikipedia policy, who might help see some middle ground between what we 4 see as a direct violation of Wikipedia policy and the change that you're seeking to make.
- Is that clear? -- Rick Block (talk) 15:19, 6 December 2009 (UTC)
- I would point out first, that you, of all people, invoking Wikipedia's NPOV as a reason to maintain the MHP article is ironic and comedic. There's a consensus that doesn't agree with your interpretation of how to apply WP:NPOV. I see it as just more filibustering by you.
- Are you agreeing the article must be NPOV? If so, please explain how the changes you're seeking comply with this policy. I'm hearing what you're suggesting as "make the article take the POV that Morgan is wrong, or is addressing a different problem". If this is NOT what you mean, please explain what you DO mean. -- Rick Block (talk) 17:52, 6 December 2009 (UTC)
- Of course, NPOV is a requirement. I strongly and vocally for 14 months now have disagree with your assessment that the article currently satisfies WP:NPOV, especially for a Featured Article. I think it's drastically skewed to the conditional solutions, in emphasis, textual amounts, FAQs, additional optional narratives, etc. Doing the reader no good whatsoever. All I've said is give Morgan and his ilk the Wikipedia required reference for having been published. Not elimination, as you've mis-quoted me more than once. How much emphasis they receive remains to be determined by consensus. Glkanter (talk) 20:56, 6 December 2009 (UTC)
Why the Morgan scenario is a variant.
The problem statement in the current article is the unambiguous K&W version. In this, the host is stated to choose randomly when he has a choice of goat doors.
In the Morgan paper the authors assign a parameter q to the probability that the host will choose a specific door if the player has initially chosen the car. It is clearly envisaged that this parameter might have a value other than 1/2, thus Moragn are clearly considering a different scenario from that addressed by this article.
To be specific, Morgan are addressing the case that the car is initially placed randomly but the host is known to choose a goat door non-randomly when the player has initially chosen the car. They do not say this, in fact they say nothing about the initial car placement or the host's policy at the start of the paper. We are left to deduce the problem that they are addressing from the mathematics that they use to solve the problem. The fact that Morgan take the probability of the car initially being behind any given door as 1/3 tells us that they take the car to have been randomly placed at the start. This agrees with the K&W problem statement. The fact that Morgan take the probability that the host will open a specific door when the host has initially chosen a car tells us that the host is assumed to choose non-randomly in this case. This is not in agreement with our K&W problem statement.
The Morgan scenario is therefor a rather bizarre and unrealistic one in which we know that the car was placed randomly by the producer (or his agent) but we know that the host will choose a legal goat door non-randomly. Martin Hogbin (talk) 17:19, 6 December 2009 (UTC)
- Obviously Morgan paper (and other similar treatments) are a generalization of the non ambiguous case and as such can be seen as variant as well. However that does by no means change, that they use their generalization to deal with the non ambiguous case (K&W) too and explicitly state so themselves.
- That aside the attempt to frame the (K&W) formulation as "the" MHP problem is not appropriate either and not in line with sources. (K&W) is the most common approach to remove the ambiguity - not more, not less.--Kmhkmh (talk) 17:35, 6 December 2009 (UTC)
- As I have said before, I am not attempting to frame the K&W formulation as the MHP, others have done this for me, it is the formulation that a consensus of editors decided should represent the problem in this article. I do, in fact, support the K&W statement because it is based on their published and verified view of how most people interpret the problem.
- Morgan's problem statement, on the other hand, is criticised in the very journal in which it is published. Read Seymann's comment. It is about as critical as one can get whilst conforming to the strict protocol required for publication in peer reviewed journal. Why do you think it is there? It is quite unusual to have such a commentary immediately following a published paper
- K&W clearly state the problem that they are addressing, the only way to determine the problem that Morgan address is to work backwards from their calculations.
- Finally, to address a generalisation of a problem is to address a different problem. In Morgan's case it is an incomplete and unrealistic generalisation. Martin Hogbin (talk) 17:59, 6 December 2009 (UTC)
- I don't think the Morgan scenario is bizarre and unrealistic. I don't think the K&W version is a good version. There are different ways to disambiguate the original problem as posed by vos Savant; some are interesting in that they have interesting solutions. Why must we assume that the host chooses his door at uniformly at random, and so on? I think it is interesting that the unconditional problem has got the "paradoxical" (good) solution "always change" under rather weak assumptions: namely that the *player* chooses his initial door at random. He doesn't need to assume anything about the behaviour of the quiz team (where the car is located) nor of the quiz-master. Whatever probabilities are used for the quiz-teams and the hosts choices, the player will definitely win on 2/3 of the time. This is called an equalizer strategy in game-theory - the minimax solution is actually such that it doesn't matter a damn what the other player does, as long as you choose your equalizer strategy. The minimax solution or Nash equilibrium is the solution (which exists if you allow all choices of all parties at all stages to be random) such that for each player, if they use that strategy, they cannot do worse than the "value" of the game. ie if the player uses the solution "initial choice uniformly random, thereafter always switch" he is guaranteed at least a 2/3 (unconditional) chance of winning the car. If the team always locates the car uniformly at random and if the host always uniformly randomly opens a door revealing a goat and not chosen initially by the player, then the TV show is guaranteed at least a 1/3 (unconditional) chance of keeping the car. If either moves away from the minimax strategy, the other could in principle do better (by guessing the strategy of their opponent). So one might suppose that "reasonable" opponents will settle at Nash equilibrium. But why should anyone be reasonable? In practice, the players certainly weren't reasonable. Gill110951 (talk) 17:37, 6 December 2009 (UTC)
- Whether the K&W is a 'good' version is a matter of opinion but it is, according to the most reliable published source on this matter (K&W), the way most people actually interpret the problem.
- The MHP is notable because it is a very simple mathematical puzzle that most people get wrong. That is why there was such a furore about vos Savant's original and correct answer, most people did not believe it. It is the simple non-conditional problem that is notable and it is this that this article should initially address.
- Beyond this there are many ways to complicate the problem and I have no objections to these being discussed after the essential paradox has been dealt with. We can assume that the car is placed non-randomly, the player chooses non-randomly and that the host chooses non-randomly. With no further information, this makes the problem very uninteresting. There is no logical reason to take the host action to be non-random any more than there is to take the producer's initial car placement to be non-random, in reality they were probably both approximately random. We could envisage that the player has studied the history of the show. This would give information equally about the initial car placement and the host policy. There is no special reason to assume the car is placed randomly but host acts non-randomly except that it produces and 'elegant solution'. Martin Hogbin (talk) 18:18, 6 December 2009 (UTC)
- From the rejoinder to vos Savant's reply to the Morgan et al. article: "One of the ideas put forth in our article, and one of the few that directly concerns her responses, is that even if one accepts the restrictions she places on the reader's question, it is still a conditional probability problem. One may argue that the information necessary to use the conditional solution is not available to the player, or that given natural symmetry conditions, the unconditional approach necessarily leads to the same result, but this does not change the aforementioned fact."
- What I'm hearing you suggest is that the article exclude this POV, i.e. that the article should explicitly say (or implicitly imply, by omission) the "basic" MHP is NOT a conditional probability problem. Is that what you mean? -- Rick Block (talk) 19:54, 6 December 2009 (UTC)
- I accept that even with the K&W formulation, the problem is strictly one that requires conditional probability but that is not what makes the problem notable. Despite that fact, there are plenty of sources that treat the problem unconditionally and there are many reasons and justifications for us to do that here which we have been through many times.
- By the Morgan scenario, I mean the (rather unrealistic and somewhat contrived) formulation in which q can be other than 1/2. This clearly is not the K&W formulation that is stated at the start of this article. Thus Morgan address a different and (bizarrely) more general problem. This is what I call the Morgan scenario. Regrettable Morgan do not make clear the problem that they are addressing in their paper. Martin Hogbin (talk) 20:05, 6 December 2009 (UTC)
- Yes, there are plenty of sources that treat the problem unconditionally. But you're ALSO suggesting removing the conditional solution section and treating this as a variant (aren't you?). The article ALREADY leads with an unconditional solution. Excluding the conditional solution, or suggesting it applies only to a "variant" is what I'm objecting to. -- Rick Block (talk) 20:35, 6 December 2009 (UTC)
- Not quite. I suggest that after the unconditional solution we have 'Aids to understanding' and 'Sources of confusion' for the unconditional solution, then a bit to explain that in some formulations even the symmetric version should, strictly speaking, be treated conditionally but this makes no difference to the answer and there are good reasons not to do this. After that I would mention the Morgan scenario in which the host chooses non-randomly making a conditional approach essential. Martin Hogbin (talk) 21:48, 6 December 2009 (UTC)
- And, by doing this, aren't you making the article take the POV that the unconditional solution is complete and correct? YOU think that this is simply the truth, which is your prerogative. However, multiple sources (it's not just Morgan et al.) dispute this and say that the MHP is fundamentally a conditional probability problem, and that the unconditional solutions address a slightly different problem (that being the problem you think IS the MHP, but again this is your view). You want to elevate the unconditional approach to be the primary one, making the article NOT neutral on this issue. -- Rick Block (talk) 22:11, 6 December 2009 (UTC)
- I am making the point that the unconditional approach is the the notable one. If have accepted many times, even today, that for some problem formulations (such as the current K&W) the problem is, strictly speaking, conditional. The article is not currently neutral, it places far too much emphasis on the Morgan paper.
- 'Sources of confusion' and Aids to understanding' clearly belong to the unconditional section. Generally when people get to the stage that they realise that some problem formulations are conditional they are not confused and already understand the problem. Little of the the text in these sections relates to the conditional nature of some problem formulations. Martin Hogbin (talk) 22:36, 6 December 2009 (UTC)
- You say above that K&W is the "most reliable published source" on the matter of how people interpret the problem. From p.5 of the PDF version: "Although, semantically, Door 3 in the standard version is named merely as an example ("Monty Hall opens another door, say, number 3"), most participants take the opening of Door 3 for granted and base their reasoning on this fact. In a pretest we gave participants (N = 40) the standard version [the original Parade version], asking them to illustrate their view of the situation described by drawing a sketch. After excluding four uninterpretable drawings, we saw that 35 out of the remaining 36 (97%) indeed drew an open Door 3, and only a single participant (3%) indicated other constellations also remain possible according to the wording of the standard version. The assumption that only Door 3 will open is further reinforced by the question that follows: "Do you want to switch to Door Number 2?" Note that once formed, this assumption prevents the problem solver from gaining access to the intuitive solution illustrated in Figure 1."
- 97% drew an open Door 3. Are they thinking of the unconditional problem, or the conditional one? What, exactly, is confusing them?
- The unconditional solution is what is generally presented as the answer. But 35 out of 36 of K&W's test subjects try to solve the conditional problem. -- Rick Block (talk) 23:00, 6 December 2009 (UTC)
- What the unconditional solution says is something like: ha ha, the problem tricked you into looking at the problem wrong - what is really being asked isn't the conditional case you're trying to solve but the general chance of winning by switching which any idiot can see is 2/3. What the conditional solution says is: your approach was fine, you just didn't execute it quite right because you forgot that the host opens Door 2 sometimes when the car is behind Door 1. Both approaches are equally valid. That's what it means to be NPOV. -- Rick Block (talk) 00:31, 7 December 2009 (UTC)
- No, you are absolutely wrong about one thing, which is that any idiot can see that the chance of winning by switching for the unconditional case is 2/3. I have never seen that view expressed anywhere else, either here of in the literature. The average punter does not know what conditional and unconditional mean. I agree that the problem that they in fact address, when presented with the Whitaker question, is the problem that is strictly speaking conditional but there is no evidence that they see and understand and attempt to solve it this way. As you say above the unconditional solution is the most common.
- What the unconditional solution says is something like: ha ha, the problem tricked you into looking at the problem wrong - what is really being asked isn't the conditional case you're trying to solve but the general chance of winning by switching which any idiot can see is 2/3. What the conditional solution says is: your approach was fine, you just didn't execute it quite right because you forgot that the host opens Door 2 sometimes when the car is behind Door 1. Both approaches are equally valid. That's what it means to be NPOV. -- Rick Block (talk) 00:31, 7 December 2009 (UTC)
- What we agree on is that the unconditional solution is the most common and notable one. I also agree that, if the problem is presented in a conditional form, the solutions should strictly speaking be conditional, we agree that this is one of the valid points made by the Morgan paper, however the unconditional solution is by far the most notable one and it must be thoroughly addressed and explained in this article, including 'Aids to understanding' and 'Sources of confusion'. That is all that I am asking for. This may not be easy, whilst basing our article on reliable sources, but it is worth a try.Martin Hogbin (talk) 12:29, 7 December 2009 (UTC)
- Sorry - I was specifically talking about popular sources being the ones that are generally presented. The "any idiot" bit was meant to be part of what they say, not something I'm saying (they say any idiot can see that your initial chance of selecting the car is 1/3 so if you don't switch that must be your overall chance of winning). IMO, why people resist this solution is precisely because it violates their (conditional) mental model of what the problem is asking. To thoroughly address and explain this solution, you HAVE to talk about this.
- And, again, by presenting a more or less complete article (many sections) about the "unconditional" solution - in particular presenting a "solution" section that does not include a conditional solution - would make the article take the POV that the unconditional solution is the preferred (or most correct) solution. If you survey the breadth of the literature (not just the popular literature), this is a highly distorted view. -- Rick Block (talk) 14:24, 7 December 2009 (UTC)
Informal mediation requested
I have filed a request with the mediation cabal, see Wikipedia:Mediation Cabal/Cases/2009-12-06/Monty Hall problem. -- Rick Block (talk) 18:20, 6 December 2009 (UTC)
- What is the procedure now? Should we prepare a statement of what we definitely all do agree on? Martin Hogbin (talk) 18:56, 6 December 2009 (UTC)
- This mediation is completely informal and there's no guarantee anyone will accept this case (trying this first is generally a requirement before proceeding to formal mediation handled by the Mediation Committee). For now, I think we continue as best we can. -- Rick Block (talk) 19:17, 6 December 2009 (UTC)
The informal mediation request has been closed, with a recommendation for formal mediation.
Martin or Glkanter - would one of you like to file the request for formal mediation? See Wikipedia:Requests for mediation/Guide to filing a case. -- Rick Block (talk) 19:36, 17 December 2009 (UTC)
The Mathematics Rule I Am Properly Applying
Way back in junior high, we did some proofs or problems or something to do with absolute values. That's all I can remember.
But the thing I do remember is that after you 'solved' the problem, you had to go back and check each of the results to make sure it didn't violate the original problem statement in some way.
That's all I'm saying about Morgan and the rest. When you check your work with some 'host behaviour' variant, it no longer meets the original problem statement, "Suppose you're on a game show..." Go ahead and argue. Better you should save your breath. Hosts don't tell contestants where the car is.
So, as an encyclopedia, Wikipedia will properly refer to reliably published sources like Morgan. And Devlin. No problem.
But, as a self-appointed 'explainer' of all things MHP, I think the article improperly gives the conditional solutions way too much emphasis. Because it doesn't match the original problem statement any longer. Glkanter (talk) 21:47, 6 December 2009 (UTC)
- Often one solves a problem by saying something like "let x be the distance travelled". One converts the problem to algebra, and finds a solution "x=-2 or x=3". Lazy students quit there. But good students think. We go back and remember that we wanted a distance and it had to be positive, so the real answer is x=3. This is a fine problem solving strategy. One solves a relaxation of the problem, that is to say one solves the problem while forgetting about some of the constraints, finds some answers, and then looks to see if any satisfy the original problem. But I wouldn't call this strategy a "mathematics rule". I'm not sure it applies in this case, where the original problem is somewhat ambiguous. It turns out that there are a number of interesting MHPs. Gill110951 (talk) 05:19, 22 December 2009 (UTC)
- What I'm saying is that the so-called 'variants' do not satisfy the original problem. As I see it, the contestant being aware of any host bias is mutually exclusive with 'Suppose you're on a game show'. That's why the question the 'opposed' editors refuse to answer, 'Is the Contestant Aware...' is so pivotal to these discussions. A game-breaker, really. Glkanter (talk) 05:32, 22 December 2009 (UTC)
The Uber Reliably Published Source
Rick, you've used this argument before, and are likely to use it, again, soon.
So, to save us all a lot of time and typing, would you mind explaining why it's Morgan? Glkanter (talk) 00:39, 7 December 2009 (UTC)
Another five cents worth by a mathematician
Right now I agree that the conditional version of the problem is a "minor variant" and that the article should focus on the unconditional version. Here's some explanation for this opinion.
I think that the "nicest problem" is the unconditional problem which is solved by always changing doors, and in order to justify that, all you need is that the initial chance of choosing the good door is 1/3. This could be the case because the player chose a door uniformly at random, or because the quiz-team hid the door uniformly at random. A wise player will not allow the quizteam to fool him so he will choose his door uniformly at random. It's the minimax strategy (guarantees a win probability of 2/3) and even an equalizer strategy (win probability is 2/3 independently of strategy of the quiz team. [all probability statements here are unconditional ones]. I guess that Marilyn meant us to think about the unconditional problem because it is so beautiful, clean, paradoxical. But I haven't looked at all her writings (the obvious thing is to look at her answers, and see what problem she actually solves). This is a question of historical research and proper documentation, nothing wrong with that.
When the [unconditional] problem was first posed to me by my mathematician friend Adrian Baddeley I gave the wrong answer but afterwards could easily be convinced what was the right answer. My mother who is not a mathematician, indeed hardly had any schooling at all, instantly got the right answer by imagining the situation with 100 doors; you have chosen one, the quizmaster opens 98; do you switch? I think only a fool wouldn't.
But people (even Marilyn) feel it necessary to add to the question, "eg if you chose door 1 and the quizmaster opened door 2". This suggests to people who have heard about conditional probability that you ought to look at the conditional problem. Of course, nothing forbids you to study whatever problem you like. Then there is not a unique answer. I can tell you that if the player uses his minimax strategy (choose door at random, always switch) then there are strategies of the quizteam such that for some outcome of the first three moves (quizteam's car location, player's choice, host's choice) the car is certainly behind the player's initial door, for other outcomes of the first three moves the car is certainly behind the other closed door. So the conditional win chances, if your strategy is "switch, irrespective" vary from 0 through to 1. But on average you'll win 2/3 of the time. It's necessary to assume that the quizteam does not hide the car with uniform random chances for this situation to arise. EG the car is always behind door 1. You choose door 1. Quizmaster opens door 2. You switch. You don't get the car.
Things are not quite so bad if we insist that the car is hidden uniformly at random. Then, however you choose your door, and however the host makes his choice, your conditional probability of finding the car behind the other closed door is somewhere between 1/2 and 1. Again (of necessity) it averages out at 2/3.
At least we can say, in this situation it is never (ie, condionally) to your disadvantage to change.
But I repeat that in my opinion a wikipedia article on MHP ought to concentrate on a succinct expose of the unconditional problem, written in terms which will convince the man in the street (by its logic!) as well as satisfy the trained mathematician (who will know how to translate the english sentences into logical statements and mathematical expressions). Make some space for discussing variants.
The beauty of that is that one does not have to look at conditional probabilities, which as I mentioned could be anything, and in order to somehow force them to the nice answer you have to make assumptions about the host's behaviour. Why should you? All we know is that he always asks this question, every show, again and again. Such an article can be short and sweet and no maths elaboration is needed at all.
A section on variants of the problem could discuss the conditional formulation, and various answers to that under various assumptions.
The beauty of the simple unconditional pure MHP is that it is a true good mathematical paradox in the sense that everyone is first fooled but afterwards easily agrees what is the good answer (unless you are lawyers: in a survey at Nijmegen university, it was not possible to convince a lawyer that their initial answer was wrong). A paradox for talking about in the pub or at a party is not good if it requires tables of calculations and mathematical notation. The unconditional version is the version which you can solve by simulation, repeating over and over again (a splendid exercise, which forces one to fix each of the stages of the game). Gill110951 (talk) 04:43, 7 December 2009 (UTC)
- Yes; but also the conditional version can be "solved" by simulation. I always emphasize to students: you cannot enforce a condition to hold, but you can form a sub-sample by postselection! Boris Tsirelson (talk) 06:54, 7 December 2009 (UTC)
- There seems to be a bit of misconception though, if i read Gill's comment correctly. So to be sure here, vos Savant did not pose an (unconditional) problem, but she offered an (correct/reasonable) unconditional solution to a (conditional) problem posed by Whitaker. Moreover the original question goes back to Selvin, who suggested it for a statistical magazine and afaik as a conditional problem or with a conditional solution in mind. So while we can feature the unconditional solution first for its beauty/simplicity/accessibility and given the ambiguity as a perfectly correct solution, we cannot simply declare the problem itself unconditionally per se, since this mathematically speaking not quite true and grossly misrepresenting the bulk of publications/literature on the subject. So while you can argue that conditional perspective is not required, you cannot argue the conditional approach does not address the original problem (i.e. is mere variant). My 2 cents as a Wikipedian and fellow mathematician.--Kmhkmh (talk) 13:10, 7 December 2009 (UTC)
- You have been swayed in your opinion here by Morgan, who tried to take a question from an amateur in a popular general interest magazine to be a formal probability problem statement. What vos Savant did was to use her judgment to answer the question based on her interpretation of the intent of the questioner (as later suggested by Seymann). This is surely the better approach Martin Hogbin (talk) 13:41, 7 December 2009 (UTC)
- No i have not and i have told you so repeatedly in the past. In fact I haven't even read Morgan's paper. However I have read many other publications on the subject in German and English and as I've pointed out several times already almost all of them treat it at least as a conditional problem as well and sometimes as a conditional problem only.--Kmhkmh (talk) 15:00, 7 December 2009 (UTC)
- Or perhaps, Martin has been swayed in his opinion here by vos Savant. The relevant question here is not one of opinions, but about what the literature says. I hear Kmhkmh saying the literature as a whole does not give any preference to the unconditional solution. NPOV says then the article shouldn't either. -- Rick Block (talk) 14:39, 7 December 2009 (UTC)
- I guess his is where the mediation will come in. My opinion, stated many time before, is that Morgan et al. are experts on statistics, thus a paper published by them should be treated as a reliable source in matters of statistical calculation (after all they get most of this right). They have no more right to pontificate as to what 'The Monty Hall Problem' is than the average man in the street and considerably less credibility in this respect than vos Savant, whose job it was to regularly interpret vague questions from the general public. We say at the start of this article that the Monty Hall problem is a probability puzzle, and that is how it should be primarily treated here - as a simple puzzle that most people get wrong. Martin Hogbin (talk) 17:21, 7 December 2009 (UTC)
- You have been swayed in your opinion here by Morgan, who tried to take a question from an amateur in a popular general interest magazine to be a formal probability problem statement. What vos Savant did was to use her judgment to answer the question based on her interpretation of the intent of the questioner (as later suggested by Seymann). This is surely the better approach Martin Hogbin (talk) 13:41, 7 December 2009 (UTC)
Article removed from my watchlist.
This article keeps disappearing from my watchlist. Is anyone else having this problem? Martin Hogbin (talk) 12:48, 7 December 2009 (UTC)
Is This Chronology Correct?
1975 - Selvin poses the problem.
- It is solved unconditionally. It is hailed as a great Paradox.
Devlin and many others write articles and text books as reliably sourced references using only the unconditional solution. They make no mention of Morgan or conditionality.
- This might be a good time to explain why you put this here (and other places, multiple times, after being hinted that no Devlin paper earlier than 2003 is in evidence; and Morgan hasn't published his bit yet, so not mentioning it could probably go without saying at this date.). Dicklyon (talk) 07:10, 8 December 2009 (UTC)
1990 - Whitaker poses the problem to vos Savant.
- It is solved unconditionally. It is hailed as a great Paradox.
Devlin and many others continue to write articles and text books as reliably sourced references using only the unconditional solution. They make no mention of Morgan or conditionality.
1991 - Morgan 'restates and formalizes' the problem.
- It is solved conditionally, but only if the 'two remaining goat doors' constraint is fixed at 1/2. Morgan criticizes the unconditional solutions as being deficient in some way. Otherwise, Morgan offers no solution.
- You tell me. 16 years after Selvin, is the MHP still hailed as a great Paradox because of the 'old' unconditional solution? Or is it now considered a great Paradox because of the 'new' conditional solution, which requires working all the way down to the 'equal goat door constraint'? Or, is it no longer considered a great Paradox because of the 'new' conditional solution, which requires working all the way down to the 'equal goat door constraint'? My position? Selvin and the unconditional solution stand unbowed.
Devlin and many others continue...
Wikipedia editors decide that the conditional problem is why the MHP is a great paradox.
- The Wikipedia article is written to emphasize the lack of understanding of Probability by those we are satisfied with the unconditional solution.
Devlin and many others continue...
There are extensive arguments at Wikipedia, nearly always on unconditional vs conditional.
Devlin and many others continue...
Glkanter points out that the universal understanding of a TV game show denies any possibility of the conditional problem statement.
There are extensive arguments at Wikipedia, nearly always on unconditional vs conditional.
Is this roughly how we got here, today? Glkanter (talk) 14:05, 7 December 2009 (UTC) Glkanter (talk) 23:46, 7 December 2009 (UTC)
- Roughly. But if you try to keep it more accurate, you will do better making your point. Dicklyon (talk) 16:37, 7 December 2009 (UTC)
I see Dick Lyon discusses his numerous edit wars on his user page. I cannot compete with that. What is my best approach to take to return my interpretation of how events happened, which I want to share on this talk page, to it's original form? Thank you. Glkanter (talk) 17:24, 7 December 2009 (UTC)
- Look at this edit summary from Dicklyon:
- (cur) (prev) 11:34, 7 December 2009 Dicklyon (talk | contribs) (792,974 bytes) (Reverted 1 edit by Glkanter; Obviously incorrect, pointy addition, bordering on vandalism. (TW)) (undo)
- I don't deserve that. What, I vandalized a section I just created? Why would someone write that? Glkanter (talk) 17:30, 7 December 2009 (UTC)
Dicklyon, I don't want any trouble. I just want my interpretation of the chronology to be on this talk page. Unedited, but certainly commented on below. So, do you want this section, and I'll start a new one? You want to start a new one, for your interpretation, and I'll fix this one back to my vision? Just let me know. Thanks. Glkanter (talk) 17:47, 7 December 2009 (UTC)
- You can just write a sensible summary instead of interspersing repeated stuff as a roundabout way to make a point. And my user pages doesn't discuss most of my edit wars, just the "dramatic" ones. Dicklyon (talk) 17:50, 7 December 2009 (UTC)
- Dick, deleting material from a talk page is rarely justified and not so in this case. Maybe glkanter could have formatted his contribution better but that is no reason to delete it. It certainly was not vandalism. Martin Hogbin (talk) 18:44, 7 December 2009 (UTC)
- Thanks for that. Despite very strong opinions this discussion has remained completely civil, let us keep it that way. Martin Hogbin (talk) 19:18, 7 December 2009 (UTC)
Well, that's quite a Wikipedia education I was administered today. And I want to give thanks to the man who taught me this lesson, Dicklyon.
Thank you Dicklyon for:
- Vandalizing my edit
- Then, accusing me of being the vandal (That was outstanding. Really. Not many can pull this one off successfully! Kudos!)
- Calling my personal opinion on a talk page 'obviously incorrect'
- Lecturing me on the proper 'tone' to use on Wikipedia talk pages
- Denying in carefully chosen words that you did any of the above
- Making it necessary for another Wikipedian to defend me. That does wonders for my self esteem.
- Teaching me who the baddest edit warrior on Wikipedia is.
- Giving me permission to return the section I created to it's original status.
- Assuring me that you won't further violate Wikipedia rules regarding my edits in talk pages
- I'm sure there is much more I'm overlooking. I'm only human.
But anyways, an apology is in order to all you Wikipedia editors who were forced to sit through this.
In the spirit of the guy who got shot in the face by VP Dick Cheney, I apologize to Dicklyon for being the recipient of your unprovoked savage violations of my good faith edits to a Wikipedia talk page. Glkanter (talk) 21:49, 7 December 2009 (UTC)
- You really are a piece of work! Your current rant is no less disruptive than this edit that I reverted thinking it looked like vandalism. If it was not intended as disruptive, I misinterpreted your intention; is that my fault? Dicklyon (talk) 04:32, 8 December 2009 (UTC)
- Is it your fault? No, of course not. I read your edit warring exploits. It's never your fault. This time, it's my fault, like I said above. You just out of the blue decide to edit some guy's stuff on a talk page? How could you be to blame? I see you as the victim in this situation. In fact, I would speak on your behalf about how, despite your benevolence, Glkanter has wronged you. Glkanter (talk) 05:05, 8 December 2009 (UTC)
- To avoid further drama, I'll take this article off my watch list again, as I had the good sense to do last April when it was clear that progress would not be possible. I hope nobody invites me to comment again, as I might repeat my failing and come back for more. Dicklyon (talk) 07:34, 8 December 2009 (UTC)
Gentlemen. We do not need this. Despite strong feelings on both sides the discussion has remained civil. As I said above, I think Dick went a bit too far when he deleted text from the talk page, but he has said the he will not object if it is restored. Glkanter, why not just restore the text and leave it at that. Martin Hogbin (talk) 10:27, 8 December 2009 (UTC)
- That's too bad he's stopped watching this page. He's missing something extraordinary. Somebody should give him a head's up. Glkanter (talk) 22:45, 8 December 2009 (UTC)
What "the conditional problem" and "the unconditional problem" mean
I wasn't quite sure where to respond, so a started a section.
- There are multiple ways to address any probability problem. If I ask "What is the probability a die rolled a 3, if we know it rolled odd?" we can solve it conditionally or unconditionally. We can say there are six equiprobable outcomes, and the conditional probability is found by P(3|odd)=P(3 and odd)/(P(1)+P(3)+P(5)) = (1/6)/(1/6+1/6+1/6)=1/3, or that there are three equiprobable outcomes that are odd, so P(3)=1/3. The unconditional problem is isomorphic with the conditional one.
- The MHP IS a conditional problem, but depending on how you address it, you can solve an isomorphic unconditional problem in its place. Just like my trivial example. That is not what we want for the so-called "unconditional problem" that the main part of the article should address.
- What we want, is for the contestant to base her decision on P(switch to car|she choose a door and host opened another door) as opposed to P(switch to car|she choose door #1 and host opened door #3). And I worded that carefully, because the first option is EXACTLY what the Whitiker problem statement says, and what K&W admit is the sematic meaning of the statement. We call it "the unconditional problem" because it is not conditioned on knowing how the host treats the doors differently. The doors cannot distinguished from each other in terms how probability applies. That does not mean that they can't be distinguished, it just means it can't be used.
- The question in the classic MHP is "should the contestant switch." We want "the unconditional problem" because we are not told how the host treats the different doors in any statement of the problem that I am aware of. So there is no known source that uses the unique aspects of "the conditional problem" to DIRECTLY answer the MHP. None.
- The two sources that seem to be recognized as starting the "game show" variation of this problem (as opposed to the related and much older "Three Prisoners" problem and Bertrand's Box Paradox), that is to say Steve Selvin's 1975 The American Statistician article and Marilyn vos Savant's Parade article, both used names for the boxes/doors as examples. But neither used those names to treat them differently in the solution. So they are both addressing the unconditional problem. Marilyn vos Savant, at least, has clarified that she intended the the "unconditional" problem.
- There are only two aspects to "the conditional problem." (There are more that are what I call "game protocol," like whether the host always offers a switch. I disregard those, because we have to assume the game protocol is well represented in the problem statement.) There are two random chioces, not one, that are part of the uncertainty in the game and that require assumptions. Not all of the "conditional problem" sources use both (Morgan does not), but those that do universally make the assumptions about the car placement that they eschew for the host's choice. Tat is, they treat it like "the unconditional problem." No conclusions can be drawn, nor HAVE been drawn, about the answer to the full "unconditioanl problem."
- There are, in general, three types of sources for the MHP: (A) Those that address problem from a common-knowledge standpoint, and that universally use "the unconditional problem" WHETHER OR NOT THEY USE CONDITIONAL PROBABILITY in their solution method. (B) Those that address the cognitive issues that cause confusion in people when first presented with the problem, and that emphasize in non-intuitive aaspect of switching to improve probability. This has nothing to do with the "condititional problem." (C) Those that explore it from a more rigorous mathematical viewpoint. Some emphasize either the unconditional, or the conditional. Call them CU and CC. Now, I haven't read all the references, but I did a quick run-down of the ones listed in the article. Of those that I could find quickly, 14 were in category A, 5 in category B, 4 in category CU, and 3 in CC. That shows that most people view the MHP as unconditional problem.
- Nobody is saying to disregard "the conditional problem." It is just a variation. JeffJor (talk) 20:22, 7 December 2009 (UTC)
Well said! Are you a new editor or did you forget to log in? I removed the linefeeds to fix your numbering. Martin Hogbin (talk) 18:40, 7 December 2009 (UTC)
No, I do log in - but somehow in switching windows I keep ending up in a non-logged-in window, and I don't notice that when the edit comes up. JeffJor (talk) 20:22, 7 December 2009 (UTC)
- See also #Formulas, not words above. Boris Tsirelson (talk) 20:29, 7 December 2009 (UTC)
@128.244.9.7:
- I completely agree
- I agree for the most part, however the bold line is a bit iffy. Who is exactly is "we"?
- This is bit iffy. K&W don't admit but assess what the real meaning is according to them, other publications differ on that. Your reading of Whitakers intent is fine, but alas it is only valid option to read the problem. You can also see it as P(switch to car|she choose a door and host opened a particalur door). If you use particular instead of another you end up exactly where Morgan & bulk of mathematical treatments go and you do use the particular door, i.e. doors can be distinguished and used. Note that both version treat door 1 and door 3 as examples.
- Not knowing the host behaviour per se does not mean we can't model it nor that we cannot draw conclusion from it. In fact Morgan & Co argue that no matter what the host behaviour is, you are never worse off by switching.
- Your descripion is not quite correct here. Selvin the mathematician/statistician who coined the problem 15 years before the parade buzz provides a unconditional and conditional solution to it (see Rosenhouse). This btw also defeats a point Martin keeps rising for more than half a year now, i.e. that the problem is essential not a math problem, but the mathematical perspective is just a minor sideshow. If the original problem was posed, solved and named by mathematician in math journal and this is not supposed to to be a math problem, then frankly i don't what is. That vos Savant solved the problem as an unconditional one is fine, however since she did not pose the problem, this tells us nothing of the "real" intent/perspective of the question.
- Here i admit, i'm not quite sure what point you're trying to make. What exactly is the "full unconditional" and to what literature/publications areyou refering for having drawn no conclusion. I assume you are aware that by now almost any recent primer into probability (in various language) contains a conditional treatment of MHP.
- I disagree with your assessment of (B). One thing about conditional probilities is that they are considered "unintuitive", i.e. unintuitive aspects and in particular the handling a posteriori/additional knowledge for reasoning are all about conditional probabilities. As far as (C) goes I don't quite buy your category count at first glance but even if that turns out to be accurate it does match for overall treatment in literature to my experience. As i pointed out already almost any modern probability primer has a conditional approach to the problem.
- yes and no, that depends on reading between the lines and carefully reviewing all the statement made here over time. Also see the recent suggestion by Jeffor and Glkanter, which to some degree you can consider as an attempt to remove the conditional approach from this article. Another thing is here.how you read/use the term variation. In light what else has been said here recently, I'd rather argue if you consider 2 problems to be isomorphic they are the "same".
- One additional reminder: In our attempts to understand/explain to "what was really meant", we easily overlook that as far as WP is concerned (in particular for contentious issues), we are supposed to describe what various reputable sources say and not what we think is right.
--Kmhkmh (talk) 20:32, 7 December 2009 (UTC)
2. "We," I believe, are those who think the "conditional" problem is a variant. Sorry if it sounded to general.
3. K&W say that the doors numbers are examples only; that means it isn't intended literally. But AFAICT their treatment does not allow their participants to consider that the host is biased. Both are needed to make it "the conditional solution."
4. While you can model it, you can't use it to directly answer the question "Should the contestant switch?" because that question has to be answered from the contestant's SoK. It is just a coincidence that, for that one bias, it always goes up. For example, if the the host tells you the placement is biased 50%:25%:25% for D1:D2:D3, and that his opening bias (afte you choose Door #1) is either 25%:75% or 75%:25 for D2:D3, but not which? Then the probability of winning goes up in one case and down in the other, BUT IT IS STILL ADVANTAGEOUS TO SWITCH BECAUSE THE AVERAGE PROBABILITY GOES UP. The point is that it doesn't help the contestant IN GENERAL to say what the probability is based on hidden knowledge, if that knowledge is not also given.
5. Yes, Selvin did. See #2 above. Did you miss the part where Selvin said that the key to his solution was that he assumed the host chose randomly between two doors, making his solution a conditional approach to what we call the unconditional problem? Martin's point is entirely correct, and was the point of Selvin's and Savant's questions.
6. If door numbers are important, the solution has to treat the possibility of placement bias the same way it treats opening bias. If you do that, the question "Should she switch?" is unanswerable. THE CONDITIONAL PROBLEM CANNOT BE ANSWERED. So all the treatments eventually get around to assuming car placement is 1/3 to each door. The question they actually answer is halfway in between being the conditional one and the unconditional one.
7. Show me a survey where respondants took host bias into account.
8. Both approaches are to be included, even if separated into parts or even different articles. But any problem that does not address (1) The formal statement for the article or (2) The problem intended by either of the two articles that started the controversy, does not belong on an equal footing.
9. Agreed. In what way is the suggestion to keep them separate not? JeffJor (talk) 22:16, 7 December 2009 (UTC)
- Jeff - You've quoted the K&W statement about the semantic meaning of the problem before. I must assume you've read the rest of the paragraph. If not I've quoted the entire paragraph above (this edit). What they're actually saying is that although Door 3 is semantically used as an example most people (97% of their sample) don't treat it this way. You (and they) are taking this to mean that the semantically precise interpretation is the "correct" one and thus that most people are more or less tricked by this problem statement into trying to solve the problem conditionally (and when they do this they come up with the 1/2 answer). Given this source says most (nearly all) people interpret the problem conditionally (meaning they're thinking about the specific case where Door 1 is the initial pick and Door 3 is the door the host has opened), and that this conditional problem is completely isomorphic to what K&W claim is the actual problem, and that other sources say that this conditional problem is what is meant, wouldn't it be rather helpful to our readers to address this issue head-on by providing a correct conditional solution? This is totally aside from the issue of whether addressing the problem only unconditionally satisfies NPOV. I don't know if anyone exactly puts it like this (I have most of the sources cited in the article - I'll look at some point), but I very strongly suspect that MOST of the refusal to believe the unconditional 2/3 answers comes from people using an incorrect conditional solution (not from failing to understand that the problem is "supposed" to be interpreted unconditionally).
- As I put it above, the fully unconditional solution is essentially saying "Ha ha, the problem tricked you into looking at the problem wrong - what is really being asked isn't the conditional case you're trying to solve but the general chance of winning by switching. You can easily see this must be 2/3 like this: <your favorite unconditional solution here>." What the conditional solution says is "Your approach was fine, but it's not quite right because you forgot that the host opens Door 2 sometimes when the car is behind Door 1. You see the host open Door 3 all the time if the car is behind Door 2, but only 1/2 of the time if the car is behind Door 1. So, if you see the host open Door 3 the car is twice as likely to be behind Door 2." The segue into the "half conditional" problem happens quite naturally because of the "1/2" in this solution. Where exactly does it come from and what does it mean? It is of course the much maligned host preference. I would be fine deferring consideration of this "1/2" to a variant, but insisting that this "conditional approach" be treated as a variant or as addressing a "different question" seems absurd.
- If you reply to this can you please try to separate your opinions from what you think the preponderance of reliable sources might say? For example, do you agree with Kmhkmh's assessment that "almost any modern probability primer has a conditional approach to the problem" (I'm taking this to mean either only a conditional approach or both conditional and unconditional approaches)? If you agree with this, I think NPOV says we MUST treat the conditional and unconditional approaches equally without favoring either one. -- Rick Block (talk) 03:46, 8 December 2009 (UTC)
- 1. I do not agree. The conditional solution calculates the conditional probability in the unconditional probability space, but the so-called unconditional solution calculates an unconditional probability in a conditioned probability space. There always is a condition. Nijdam (talk) 14:09, 8 December 2009 (UTC)
Yes, Rick, I've read the whole paragraph. The only point I ever made from that quote is that it is valid (and in fact proper) to interpret the Whitaker/MvS problem statement as not requiring specific door numbers; that they were examples. I didn't try to address whether K&W were using the conditional or unconditional problem, since they solve the full conditional one later (more below), and addressing it could be seen as POV. The "clarification" you provided, which you interpreted according to your POV, only shows that K&W recognized that respondants treated the door numbers the same way MvS did; as examples. Not as implying any dependency on the actual door numbers. And it was, at least in part, because the non-required use of examples was reinforced by K&W (not Whitaker/MvS) in the form their next question took: "Do you want to switch to Door #2?" That can't be answered without using the examples. So K&W followed in Morgan's and Gillman's footsteps, in rewording the MvS problem statement to make it become the conditional problem. But that paragraph, and my excerpt of it, had nothing to do with how the problem is addressed anywhere.
Once again, since I am apparently still speaking in cat when I say this (or your POV gets in the way of comprehension): The MHP is a conditional problem, but depending on what you see in it, it can be solved unconditioanlly. It is valid to do this: see G&S, p138, where they choose not to make an assumption which they say could be made without loss of generality. Their Figure 4.3 illustrates what Nijdam described using his own POV, as "an unconditional probability in a conditioned probability space" (emphasis added). Assuming it is a conditioned space means you are assuming there is a bias in either car placement and/or host choice. But that treatment is not what I call the unconditional problem. The "unconditional solution" is represented on p139 in Figure 4.4, and is a conditional solution that is isomorphic with the fully unconditional one (the isomorphism comes from the generalization they declined).
The point we (well, at least I) am trying to make here is that this pair of solutions represents what most people see in the MHP. K&W bear this out. People can a take what appears to be an unconditional approach (Figure 4.3 in G&S), but that is more complicated and provides no more generality than the alternative. With the generlization that a single door used in the solution represents the uninformed (to the contestant) distribution of all possibilities, and so its probability distribution must be the average of those possibilities, they can use conditional probability to get the same result. (It is possible that biases exist in this system: I am not assuming them away. But they all get eliminated in either solution. And I have no reference for this, but it is the approach most people, who can answer simple probability questions, actually take. I've seen it refered to as "symmetry." The paradox in the MHP is that they overextend this symmetry, not realizing how the revealed information makes things assymetric.) What we want the article to separate out, is the solution G&S add after Figure 4.4, where suddenly a bias is assumed without valid reason. That is what I have called "the conditional problem," because it requires the use of conditional probability, and ignores the symmetry which K&W always apply to their formulation of the probability. They don't use the conditional problem - they allow for it but always put in parametric values representative of symmerty, and thus the unconditional solution. They even provide arguments for why people assume symmetry. And I can't speak for "most primers," but I would think that what Kmhkmh has noticed is a conditional probability not based on biases; or if it is, wheere biases get eliminated by symmetry. 128.244.9.7 (talk) 16:34, 8 December 2009 (UTC)
- As far as most primers are concerned, you get some impression via Google Books as well (or alternatively sample recent lectures on university websites or the classic libraries/bookstore approach). Also reading the beginning of your comment, there seems to be some misunderstanding. I'm not aware of anybody here (aside Nijdam maybe) claiming the unconditional would not be valid - certainly not me. As often in science there a various valid perspectives on a particular problem and various ways to model it accurately. There is also imho no issue with featuring the unconditional solution first and separate from the conditional one. There is however an issue with (subtle) attempts to redefine the problem as such in a manner, which is not in line with the bulk of the sources. This concerns in particular formulations insinuating, that the conditional approach is not solving the "real" MHP problem, that the MHP is not a math problem (but a layman's creation), that an "question-answering-expert" knows best what the problem "really" is and what the intent of the question being posed to her was, or that psychological study on how average people react or think of the problem, tells us what the problem really is. Because these various points are at best if not outright wrong the reflection of an individual publication and not the bulk of literature on subject. Hence the WP article cannot assume such a position. I'm not opposed to rearrangements/modification of the article as long as they don't create the critical points I've mentioned above. From my personal perspective the article could be arranged rather differently (starting with the ambiguous Parade formulation and the simple unconditional solution, after that separate chapters for mathematical analysis (conditional,conditional,criticism, subtle changes of perspective,various modelling approaches and the requirements, problem variants, etc.), psychological/cognitive analysis (K&M,Mueser, Granberg and others), occurrences of MHP in other areas/sciences and a historical overview/timeline (from Gardner until now). Having said that however in general i prefer a rather conservative editing policy for featured articles, because every major change requires a reevaluation of the features status. Also in my experience in contentious cases good articles can easily deteriorate with a lot disagreeing people editing around, that's actually one of the main reasons why I won't support a change until I see a feasible editor compromise emerge first, that a) provides a meaningful improvement rather than marginal changes only and b) is in line with WP guidelines.--Kmhkmh (talk) 17:13, 9 December 2009 (UTC)
- Jeff - can you be more specific about the exact changes you're talking about? Is it:
- Delete the entire section "Probabilistic solution"
- Delete the 4th paragraph of "Sources of confusion"
- Delete "Other host behaviors" under "Variants" (or only some)
- Delete "Bayesian Analysis"
- Or, are you suggesting something else?
- From your comments above, it sounds like you have no fundamental issue with a conditional analysis so long as it assumes no host bias - i.e. you're only insisting that it be taken by definition that the host open one of two goat doors with equal probability. This makes me think you should be OK with the "Bayesian analysis" section, and even the "Probabilistic solution" section with fairly minor changes (like deleting the last paragraph in this section about the "host preference" variant or moving this paragraph to a "variant" section) - or even a single "Solution" section more like what was there following the last FARC (i.e. this version) which made no mention of a "host preference" variant. I think how we got to where we are is contention over the transition paragraphs in this solution section (the paragraph starting with "The reasoning above ..." and the following one). Would you be OK with reverting to this version and working on these paragraphs? -- Rick Block (talk) 20:34, 9 December 2009 (UTC)
This Is All I Really Care About
If, after the Popular solution, these words were given great prominence before all that conditional stuff, etc., my main goals would be satisfied.
"The Monty Hall problem is unconditional. That is the whole paradox; the rest is the explanation; go and learn." Glkanter (talk) 23:13, 7 December 2009 (UTC)
Relevance of History
I found the history of the MHP enlightening. Now here's my point of view as to what that means for the article. The MHP became famous because of Marilyn vos Savant, and she has made clear that she was thinking of the unconditional problem, just as most people interpreted her words; that problem has a number of beautiful short and simultaneously rigorous solutions. It's a beautiful problem with a beautiful solution, that's why it has become folklore. (In my humble opinion) she mentions door 1 and door 2 merely as a pedagogic device, to make things concrete so you can imagine the situation better; IMHO I think she wants an answer which doesn't depend on which door is called door 1 and which door is called door 2. Door 1 *is* the door you first chose, whatever door that actually is; door 2 *is* the door the host chooses, whatever door that actually is. She wants a universal answer independent of the numbers of the doors; the answer "always switch" is the good unconditional answer as long as the door the player has chosen had 1/3 chance of the car being behind it, which is moreover what most people (IMHO) immediately imagine.
This unconditional game with its hidden assumption and the corresponding unconditional solution, I think, *is* what most people out there, whether mathematicians or plumbers, think of as *the* MHP.
Conditional versions of the MHP are interesting and it is enlightening to consider them. The "original" MHP might have been conditional. Who knows, maybe the Aztecs had an even earlier version which was unconditional. So what.
So today at 10:26 am CET I think the article should, for historical and cultural, and pedagogical reasons, start by elucidating and solving the unconditional problem, and then go on a) to describe the (pre)history of MHP (neutrally, objectively) and b) to discuss important variants, such as the conditional version of the problem (neutrally, objectively). Gill110951 (talk) 09:29, 8 December 2009 (UTC)
- I think the primitive chronology I created shows something similar. From Selvin, 1975 through vos Savant, 1990 it was consider a great paradox and had only been considered unconditional. In 1991 with Morgan, does this change to great patadox - conditional, stay as great paradox unconditional, or meh, if it's got to be conditional with the equal goat door constraint, it's not much of a great paradox anymore. I have my opinion, I'd like to hear the other side's response. Glkanter (talk) 11:20, 8 December 2009 (UTC)
- That is pretty much what I am suggesting and to some degree how the article is now. The change I would like is to link the 'Aids to understanding' and 'Sources of confusion' sections to the 'Popular solutions' (unconditional) section. If you read the 'Aids to understanding' and 'Sources of confusion' you will see that they relate far better to the unconditional problem than the conditional one.
- Following the unconditional solution I would like to have a section explaing why the problem, in certain formulations, is, strictly speaking, a conditional one but why this is not important for the symmetric (host chooses randomly) case. The Morgan scenario (host is known to choose non-randomly but car is placed randomly) I would like to relegate to the variants section. I am not going to try to hide my distaste for the Morgan paper. I think it did the whole MHP a major disservice by turning a simple problem that everybody gets wrong into a complicated problem that nobody cares about. Martin Hogbin (talk) 10:20, 8 December 2009 (UTC)
The unconditional problem
Let us formulate the unconditional problem: A player will be offered the choice of one out of three doors, one of them hiding a car. What is the probability she wins the car? Nijdam (talk) 14:13, 8 December 2009 (UTC)
- More OR? Ho hum. Let's focus on sources, and the commonly understood meaning of a game show. This is Wikipedia, and the MHP from Selvin (who may have originally called the puzzle 'Let's Make a Deal problem', the name of Monty Hall's show) through K & W begins 'Suppose you are on a game show...' Hosts don't tell contestant where the car is on a game show. Many reliable sources use logical notation, and/or words, to solve the problem. Is there a problem with that technique? Glkanter (talk) 14:27, 8 December 2009 (UTC)
- That is a very interesting question and it shows why it is not just mathematics that is required to answer probability problems. For anyone to answer they need to know exactly what you mean. Do you want the question answered from the point of view (state of knowledge) of the player. If so, are we to assume that she has no knowledge of where the car was placed?
- Perhaps you would prefer a more modern approach. In that case I would want to know the initial distribution of the car behind the doors and the distribution of the players door choice.
- The point is that without stating exactly what it is that you want to know your question cannot be answered. Martin Hogbin (talk) 15:45, 8 December 2009 (UTC)
A problem for you in return
There is an urn containing 9 balls numbered 1-9. You have to pick ball 9 to win. We assume that your picks are at random. What is the probability that you will win in one pick? I think we can agree that it is 1/9.
Now suppose that you have two picks and the ball from the first pick is not replaced before your second pick. You do not win on your first pick. What is the probability you will win on your second pick and how would you calculate this? Martin Hogbin (talk) 16:31, 8 December 2009 (UTC)
- I assume with pick you mean random pick.
- Yes that is why I said, 'We assume that your picks are at random'. This is, of course, generally assumed anyway in an urn problem.
- I have to calculate: with the outcome of the first and second pick respectively. The calculation may be with the use of the apropriate laws, but you may arrive at the same answer by using a symmetry argument, stating that the second pick is from a conditioned urn, with 8 balls with number 9 amongst them. Note, that you always calculate the conditional probability. Nijdam (talk) 21:01, 8 December 2009 (UTC)
- You can obviously see what I am getting at. Nobody would suggest that the correct calculation is to take all the possible conditions after the removal of the first ball (That is to say ball 1 missing, ball 2 missing...ball 8 missing) calculate the probability of getting a 9 separately in each case then combine these probabilities weighted according to the (equal) probability of each condition. We can just observe that, provided it is not a 9, the first ball picked makes no difference to the probability of picking a 9 on the second pick.
- So it is with the MHP. We can note that, if the host opens a legal goat door randomly, it makes no difference to the probability of winning by switching which door he opens. In other words the conditional problem can obviously be treated unconditionally. Martin Hogbin (talk) 22:01, 8 December 2009 (UTC)
- I think the problem here is the right wording. Of course one may use symmetry arguments, as BTW I did before, but you do this in order to calculate the conditional probability (for the car to be behind the picked door!) in an easy way, as its value is the same as the unconditional. Nevertheless the probability of interest is the conditional one!! Nijdam (talk) 11:09, 9 December 2009 (UTC)
- OK, I accept what you say but my suggestion is that we ignore the issue of conditional/unconditional for the first section where we treat the problem in the simplest manner possible, showing why the answer is 2/3 and discussing the reasons why people get it wrong.
- After that we could say that 'strictly speaking' the problem is one of conditional probability but for the symmetrical case we can ignore this fact. Finally, in in a separate section, we should discuss the necessarily conditional case, the Morgan scenario, where the player knows the non-random host door opening strategy (but not the initial car placement). 12:06, 9 December 2009 (UTC)
- Well Martin, as you may have noticed from our lengthy former discussion, neither Rick nor me insist on mentioning the word conditional, but we somehow want to make clear that after a door has been opened new probabilities are at stake. And that a simple solution, along the lines "the car is in 1/3 of the cases behind the chosen door, also with 2/3 it is behind one of the other doors and hence (???) after opening of one of them with 2/3 behind the remaining closed door", is not complete. Nijdam (talk) 16:09, 9 December 2009 (UTC)
- For the symmetrical case the probability that the car is behind the chosen door remains 1/3 after a legal random goat door has been opened. Most people assume this, and there are several-easy-to-understand reasons why it is so. In my opinion for the symmetrical case we can just state this fact in the starting section, to be discussed in more detail later. Martin Hogbin (talk) 17:55, 9 December 2009 (UTC)
- As I said about wording, what does it mean if you say: "the probability that the car is behind the chosen door remains 1/3"? There is your and many other's problem. What do you mean with probability? A probability always remains the same! What actually is the case - as we have discussed over and over - the probability changes, without changing value. Nijdam (talk) 11:44, 10 December 2009 (UTC)
The Meta Paradox of The Monty Hall Problem Paradox
Selvin poses the MHp. He solves it unconditionally at 2/3 vs 1/3 if you switch. The problem is hailed as a great paradox.
vos Savant prints a letter inspired by Selvin in a general interest USA Sunday newspaper supplement. She solves it unconditionally at 2/3 vs 1/3 both when you made your choice, and when the switch is offered. Because Monty's actions don't impart usable knowledge to the contestant. It's a sleight of hand. Nothing happened.
All heck breaks out. Tens of thousands of letters, including over 1,000 from PhDs tell her she's wrong. And they are certain!
vos Savant soothes the savage beasts with logic and smarts. The unconditional solution carries the day. The problem is, again, hailed as a great paradox.
This group, "J. P. Morgan, N. R. Chaganty, and M. J. Doviak are Associate Professors and R. C. Dahiya is Professor, all in the Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia" (284 The American Statistician, November 1991, Vol. 45, No. 4 (C 1991 American Statistical Association) develops the argument that the problem is only properly solved using a conditional problem statement. Their criticisms, etc. rest on this: That when faced with 2 goats, the host must decide which goat to reveal. This rests on the assumption (presumption, invention) that the contestant might somehow gain usable information as to the location of the car in this particular instance of the game by Monty's actions. It's left unstated whether Monty's actions would be shared with the contestant. And if they are shared, what method is used. But it's clear: in this instance of game play they claim, the subject contestant could be armed with more useful information that the average contestant.
- The only problem is, their 'assumption' is not consistent with the first words of the MHP problem statement: "Suppose you're on a game show...", as Hosts don't tell contestants where the car is hidden. Actually, some Wikipedia editors have found a math error in the paper, and are in communication with the publication. Oh, and "Richard G. Seymann is Professor of Statistics and Business Administration, School of Business, Lynchburg College, Lynchburg, VA 24501" (1991 American Statistical Association The American Statistician, November 1991, Vol. 45, No. 4 287) wrote a paper that spoke only about Morgan's paper. It was included in the very same issue of the journal. It's weird. Is it a disclaimer, a clarifier? It's sure not an endorsement.
Others come out with papers supporting Morgans criticisms, including Gillman in 1992 and Grinstead and Snell 2006.
Others continue publishing unconditional papers. (It seems likely that if 3 Wikipedia editors plus Seymann find fault with the paper, so too would members of the Professional Mathematics Community. And as professionals, they don't make a big stink about it. They just ignore the paper and continue publishing articles that rely solely on the unconditional problem statement.)
So, has the Professional Mathematics Community decided that Morgan is right, and Selvin was a hack? I don't think so. Before, during and after Morgan's paper, respected, credentialed reliable Mathematics professionals continued to publish articles solving the MHP unconditionally. I don't know that any of these professionals in either camp have attacked or counter-attacked anyone else's paper. It looks to me, that in the Professional Mathematics Community nothing happened. No usable information was gained. Perhaps Morgan's paper, like Monty revealing a goat is just sleight of hand, imparting no usable knowledge? It's possible. Most published MHP articles say nothing of Morgan or conditionality.
Which brings us, finally, to the Meta Paradox. The Wikipedia editors are arguing, essentially, over whether or not solving the unconditional problem is 'enough'.
Suppose you are given a story problem about a game show. The Professional Mathematics Community agrees heartily that this is a delightful paradox which can be 'proved' or 'solved' using an unconditional problem statement. Maybe not even requiring formal probability notation. Symbolic notation is often used. Then "J. P. Morgan, N. R. Chaganty, and M. J. Doviak are Associate Professors and R. C. Dahiya is Professor, all in the Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia" come forth and say it must be solved conditionally, based on the arguments set forth in their paper. You are then offered to stay with the unconditional solution being complete, or you may switch to the conditional solution.
Many people are fooled by this paradox, and accept the switch. Because they don't realize that like Monty revealing the goat, no new usable information has been revealed by this paper. Nothing happened. Glkanter (talk) 18:27, 8 December 2009 (UTC)
- I like this way of looking at things! Except that there is no law against studying conditional Monty Hall problems, and quiz-players should also realize what they get when they go with the unconditional solution. So I think that something useful did come of the Morgan et al. contribution. Now we just need a reputable mathematician to publish a peer reviewed paper on the Monty Hall paradox paradox, and then wikipedia editors can write articles on it. Gill110951 (talk) 05:36, 22 December 2009 (UTC)
- Glad you like it, thanks! It's a free country. They can do what they want. It just adds no value to those simply wanting to understand the MHP paradox. As the article is written, quite the opposite! There's an old salesperson's saying, 'Don't close past the sale' (it may actually be 'Don't sell past the order'). I may have had a professor say something like, 'Don't over-solve the test problems. Solve it and move on'.
- Maybe you could look at my 'Huckleberry' section and tell me how his approach was insufficient, and how his results would have been improved by someone explaining to him the 'equal goat door constraint'? Glkanter (talk) 05:48, 22 December 2009 (UTC)
OK, I'm Paraphrasing slightly...
When asked how he was able to sculpt the venerated 'David', Michelangelo replied, 'It was easy really. I removed everything that didn't look like David'.
A comment from a newcomer to this discussion
Wow, this thread is long! And I haven't even looked at the archive(s?). I thought it might be worthwhile to make a comment as a person who has not been following this thread before now. I just, in the last couple of days, read the article and a big portion of the discussion.
My comment is simple: Please, I am not attacking anyone; I am just making a general, honest, respectful IMO comment. (And yes, I am schooled in mathematics.) I agree with those that say the article is too long, very unwieldy, and often downright confusing. I think the article as it now stands is almost worthless. I agree with those who say: Just state the "standard" problem as most people assume it is stated, and give a simple explanation as to why it is correct. Then meander off into the conditional and unconditional ponderings, the Bayesian statistics, etc.
I started out reading the article, with expectation of fun. I was already familiar what the "Monty Hall problem", and I understood it, at least in its more obviously stated form (based on the generally accepted assumptions). As I read on I thought, "Whaaa???" Much of the -- sorry, but most -- of the article is a murky mess, and even those who are somewhat probabilistically astute I think would have difficulty making sense of some of it. I'll cite just one example: The section titled "Popular Solution" is, IMHO, poorly written and confusing. Frankly, it's not clear what the author is meaning to get across in several places (even though I understand exactly what it is that he/she is intending to say). It needs to be rewritten, as does much of the rest of the article. Not tweaked, but rewritten. This sort of muddled presentation is just not necessary, and it is not worthy of the standards of Wikipedia. This stuff is not string theory or Gödel's incompleteness theorems in ZFC. This is introductory-level probability, albeit a very subtly tricky example of it.
I never seen an article on Wikipedia that has created such a WikeWar as this article has. It apparently has no resolution in sight. Anyway, I'm all out of suggestions -- if I have even made any.
Finally, just for fun, I wanted to mention a somewhat similar conditional-probability problem which I haven't seen anyone else mention. (It is not relevant to this article, nor should it appear in it; it's just related.) You play a "flip three coins game". The person I am gambling with shakes up three fair coins in a canister and spills them onto the table top. I am not allowed to see the coins initially before I make my choice; the canister shaker (my opponent) hides the coins from me. The rules are, the shaker peeks at the coins on the table and he has to tell me what the "majority" coin is. There will be either a majority of heads (3 heads or 2 heads) or a majority of tails (3 tails or 2 tails). Then, having been told what the majority is, I must guess what the third coin is -- heads or tails. If I get it right I get paid a dollar by the shaker; if I get it wrong, I pay him two dollars. Most people would think this a stupid gamble on my part; they will assume that the guess as to the heads-tails of the third coin has a 50-50 chance of being right. But it's easy to see (though it is initially counter-intuitive to many people) that if you always guess the opposite of the majority, you will win 3/4 of the time. Just write down all combinations: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT, and it's obvious. That would make a cool bar game. Even offering your opponent the 2-to-1 payoff, you would still win, on average, in the long run 25 cents per play:
1/4*(-2) + 3/4*(1) = 1/4 of a dollar per coin-shake, in the long-run average.
Good luck with your little war. Worldrimroamer (talk) 23:53, 8 December 2009 (UTC)
- There's no war. Just good faith disagreement. Likely to end soon, anyways. Is there any reason this editor's comments should not be reflected in our quorum building? Glkanter (talk) 01:18, 9 December 2009 (UTC)
- I'm afraid I don't understand what you mean. Who is the "this editor" to whom you refer? I tried to state at the outset that I was not targeting my comments at any particular person. I did not compile a ledger of who wrote what. I was just making general comments, in the hopes that they might be a small but helpful contribution. —Preceding unsigned comment added by Worldrimroamer (talk • contribs) 03:10, 9 December 2009 (UTC)
- Er... the obvious guess is that the third coin is the same as the majority, which is a bloody good guess: it is correct 3/4 of the time, both conditionally on the majority being heads, and conditionally on the majority being tails, and unconditionally on the nature of the majority. I shall try this in the pub to see if my population is smarter than Worldrimroamer's. Gill110951 (talk) 05:45, 22 December 2009 (UTC)
Sorry for being so cryptic. 'The editor' is you. As we try to bring this discussion to a close, we are attempting to build a Wikipedia Consensus to make changes very much as you described. So, I was asking the other editors if there is any reason your opinion should not be considered as part of the 'let's change the article' consensus, of which I am a part. It's all good. I think you'll find the discussion of the last few weeks most meaningful. Some extra Wikipedia Mathematics Project people have begun contributing, by request, and it's helped move things forward a great deal. Again, sorry for being unclear. Glkanter (talk) 03:27, 9 December 2009 (UTC)
- Glkanter, thanks for your reply. I understand now what you meant. And by referring to "editing wars" I did not mean to be denigrating. Perhaps I should have used a different term. I just thought it was very interesting that this "simple" little topic has stirred up so much discussion. It does my heart good to see that there are people that care about esoterica like this (at least, in the eyes of the general public it would seem like esoterica). Best regards to all ... Worldrimroamer (talk) 17:48, 9 December 2009 (UTC)
- Worldrimroamer, thanks for you contribution. New thought is always welcome here. One thing I might explain is that the article as it is now (more or less) in not the end result of the pages of discussion that you have seen. It is little changed in principle from the original FA version.
- There are basically two broad factions of editors here. Those who want to keep the article as it is (or perhaps as it was about a year ago) and those who want to change the article to reflect more or less what you say, namely that The Monty Hall Problem is a simple mathematical puzzle that most people get wrong. What the article needs (according to the pro-change editors) is an initial section that concentrates on a simple description of the puzzle with normal 'puzzle assumptions' (all choices are random unless specified etc). This should be followed by some simple, convincing solutions that show why the player has a 2/3 chance of winning by switching.
- The anti-change editors believe that the proposed changes are not justified by the available sources and that making them would jeopardise the article's FA status. Of particular importance is a paper by Morgan et al. which claims that the problem must be treated as one of conditional probability. This has the effect of changing the simple problem that most people get wrong to a complicated problem that most people are bored by. I strongly dislike the Morgan paper, you should get a copy and see what you think.
- It the moment, apart from a few minor changes, the article is still built around the Morgan paper. The editors who want change have refrained from drastic editing as fought to get a consensus for change here. So far although there is a majority for change there is far from general agreement.
- There are some things that are generally agreed on such as the game rules (the host always offers the swap and always opens a unchosen door to reveal a goat) the fact that overall the player has a 2/3 chance of winning by switching, and that it is assumed that the player has not studied replays of old shows to gain statistical information. I might add that, although there are strong opinions on both sides discussion has generally remained civil and there has been no edit warring. Martin Hogbin (talk) 11:57, 9 December 2009 (UTC)
- Thanks, Martin. Yes, I understand what you mean about the basic disagreement. It sounds like a rather intractable situation to me. I hope you guys can work something out. As I told Glkanter in the post immediately above, I should perhaps not have used the term "edit wars". I just meant that it was impressive how much intense interest has been evidenced in this discussion. I think that's a good thing. I just wish that the article were not so ... opaque? IMO, Wikipedia should be accessible both to the experts in the field, as well as to the (curious and smart) not-so-expert people. There's room for both. I'll butt out now and wish you luck. You may need it. :o) Best regards. Worldrimroamer (talk) 17:48, 9 December 2009 (UTC)
__________________________________________________ —Preceding unsigned comment added by Worldrimroamer (talk • contribs) 17:51, 9 December 2009 (UTC)
- The links are at the top of this talk page - as of the last FARC the article looked like this. Since then the "Solution" section (that arguably took the POV that the "unconditional" solution does not address the question as asked) has been split into a "popular solution" and "probabilistic solution" in a more NPOV manner. Saying this is "relatively unchanged" understates the situation fairly dramatically. What this extended discussion is about is furthering this change, to make the article effectively take the POV that a "conditional solution" is an unnecessary nuisance - i.e. that the POV presented by the aforementioned Morgan et al. paper is invalid. Saying the article in its current form, or even as of the last FARC is "built around the Morgan paper" is (IMO) factually false.
- No article is ever finished and improvements are always welcome. This extended discussion is about whether additional changes are necessary to undo the POV some editors think was present in the version at the last FARC. -- Rick Block (talk) 15:35, 9 December 2009 (UTC)
Building Consenus - Mediation
At day's end, it will be 7 days since Rick requested comments from all the editors we could think of who had shown an interest in the MHP, plus a general request to the WikiProject Mathematics page. Three days ago, Rick requested mediation assistance. To date there have been no volunteers.
How and when do we keep moving forward toward a consensus? Glkanter (talk) 13:56, 9 December 2009 (UTC)
- If you look at the list of pending cases at Wikipedia:Mediation Cabal, the oldest was opened on Nov 20. I don't know if they treat the backlog as a strict FIFO queue, but it seems like 2-3 weeks might be a fairly reasonable amount of time to wait for a response. -- Rick Block (talk) 14:42, 9 December 2009 (UTC)
Rick, based on the following, I don't see this procedure as being any help whatsoever to this particular group of editors. Is there some other benefit to this that I do not understand? What other path can the clear consensus take toward gaining 'permission' from the minority view to move forward with editing the article?
- "The Mediation Cabal is a bunch of volunteers providing unofficial, informal mediation for disputes on Wikipedia. We do not impose sanctions or make judgments. We are just ordinary Wikipedians who help facilitate communication and help parties reach an agreement."
You made some claim a couple of days ago about this legitimately created good faith consensus violating NPOV, which I'm sure each member of the consensus would dispute strongly. Is this the issue you want mediated? The claim seems far fetched, certainly, to me. Or is this still an issue from your point of view? Glkanter (talk) 13:01, 10 December 2009 (UTC)
- I have two concerns that I believe mediation may help:
- 1. I am less than convinced that the users Martin has identified as "for change" agree with what you and Martin are thinking they agree with. For example, in the section below Colincbn says that the Solution section of this version "seems to have nothing in it about the conditional solution at all". I believe you and Martin disagree with this. So, is Colincbn for the change you're suggesting or not? To some extent, I think many of us are talking past each other and not necessarily understanding what others are saying.
- That point is rather more easily resolved. Why not ask those in the list if they are in the right section. I have asked people to sign to confirm that they are in the right section or to move themselves if they are not. You are free to ask any editor to check that they are in the right section if you think that I have got it wrong. Martin Hogbin (talk) 22:20, 10 December 2009 (UTC)
- Rick, re: 'talking past each other', I sure would appreciate some closure on the very first section I started when I returned: 'Is The Contestant Aware?' You're last response was 'yes, but', and I've asked you to clarify, as neither of us wants me mis-interpreting your intent. Thank you. Glkanter (talk) 04:45, 11 December 2009 (UTC)
- Martin - this is a perfect example. You seem to not be understanding my point, but I'm puzzled how to make it more clear. If I try to clarify I suspect you'll think I'm arguing with you about the "consensus". A mediator presumably wouldn't have this issue and might be able to convey the point I'm making in a way that you wouldn't take as an argument. Actually, I'm saying the same thing I've said in some other threads lately which is that without talking about specific changes it's very easy to miscommunicate. -- Rick Block (talk) 03:56, 11 December 2009 (UTC)
- 2. I personally have been trying to play two roles here, i.e. as a proponent for one "side" in this discussion (per my comment above, I'm not sure there are only two sides) and (since I am an administrator) as an authority on Wikipedia policies and procedures. You, in particular, seem to believe I am not acting in good faith and that everything I say reflects an advocacy of a POV.
- Well, it's evident now why these discussions take 6 years and never get anywhere. Rick, under who's auspices were you alone chosen to act "...as an authority on Wikipedia policies and procedures" for purposes of these discussions? Were the other editors advised of your dual role? I sure wasn't, and I am greatly distressed by this revelation. Is this common to have an entrenched protagonist also serve as some sort of 'junior mediator'? I can see all kinds of conflict from this, and have personally witnessed and been the recipient of this conflict of interest in your discussions for 14 months. Have you considered not continuing this dual role? More than ever, I'm certain we need to move beyond the mediator cabal level to declare the consensus in favor of the proposals. Glkanter (talk) 16:25, 12 December 2009 (UTC)
- Both of these are areas where I think an uninvolved mediator could help. -- Rick Block (talk) 17:26, 10 December 2009 (UTC)
- Please do not attribute motivations to me or Nijdam or anyone other than yourself. I am plenty willing to budge and have done so in the past. Whether you are willing to budge is up to you. Informal mediation is the next step in the dispute resolution process, see Wikipedia:Dispute resolution. Formal mediation comes next, but my understanding is informal mediation is generally treated as a prerequisite. -- Rick Block (talk) 17:58, 10 December 2009 (UTC)
- When I first read MHP and the explanations, in the first follow-up column in "Ask Marilyn" years ago, I believed she was right about the answer, but thought her explanations were ridiculous. The best apparent solution offered was to use a simulation--for example, with playing cards. I immediately grabbed a deck of cards and tried the simulation, and within a few minutes saw that it was obvious that the simulation would lead to a 1/3 stay, 2/3 switch ratio for winning over the long run. Obvious, obvious! That was obviously because, of the 2/3 cases in which the prize was not card #1 (i.e., behind door #1), it would be card #2 half the time, and card #3 half the time. So obvious!
- This explanation was very unsatisfactory because in the actual tricky puzzle, the actual door opened is actually identified. Door #3 is opened! What then? The simulation (like other explanations) did not specifically address this scenario, but instead included alternate scenarios in which a completely different door (#2) was opened. Sure, we can agree that if either of the non-chosen doors might be opened, the odds of switching are double the odds compared to staying--but what is true once one of the non-chosen doors is actually opened? That is what makes the puzzle an interesting, tricky puzzle! That's why people much smarter than I got fooled!
- Eventually I worked out a reasonable solution + explanation (c) and let it go. This Wikipedia article brought the irritation back. I contributed with some silly stuff, got bored, and left.
- I'm back in for a minute to argue:
- (a) even though Morgan et. al. are very wrong and beside the point overall, they ought not be ignored, because they made a nice (misguided, but nice) argument that the player can't be sure the odds of switching are 2-to-1, since for all we know, a host choosing between two losing doors might prefer one to the other. They rightly acknowledge that you should still switch. They don't add a lot to the discussion beyond that. Their contribution is mostly "too clever by half" and not useful. It is reasonable for the player to play with the assumption that the host is no more likely to open one losing door than another.
- (b) the usual explanations given for why you double your odds by switching are worded as if the actual puzzle said something like this:
- Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1. You know at least one of the other two doors has a goat behind it. The host, who knows what's behind the doors, opens one of the other two doors (No. 2 or No. 3),showing you a goat. He then says to you, "Do you want to switch to the closed door you didn't orignally pick?" Is it to your advantage to switch your choice?
- I don't enjoy the fact that this is treated as an argument about"conditional" versus "unconditional" thingies. Those words seem too complicated to me. Too, I believe Morgan et. al. are fundamentally wrong. But given all that, Rick Block is closer to reality on this topic than is the average person who fundamentally understands the problem. The usual explanations given are not explanations that fit the actual puzzle as it was actually worded! That matters! Simple31415 (talk) 03:24, 13 December 2009 (UTC)
- I agree with some of your ideas, disagree with some. Mainly whether "The usual explanations given are not explanations that fit the actual puzzle as it was actually worded! That matters!" As an analyst (problem solver), I'm always looking for shortcuts. And that strictly means find 'differences that make no difference'. I think the open door fits that description. Please look at the section I recently created, Let's Say Some Huckleberry Played Repeatedly and tell me where the contestant did not properly maximize his situation using a simple solution. Glkanter (talk) 05:17, 13 December 2009 (UTC)
- Fascinating! Is there any particular reason why you returned now to say, 'Rick Block is closer to reality on this topic than is the average person who fundamentally understands the problem? Perhaps you could tell us which aspect of his argument you support, particularly in view of your comment that Morgan's contribution is mostly, "too clever by half", which I agree with totally although I might add, 'and, in fact wrong'.
- Also, there is no "the question", although it is true that the most often quoted problem statement is that by Whitaker. Perhaps you should read it, it says, '...the host, who knows what's behind the doors, opens another door, say No. 3...'. Note the words that I have put in italics. It is quite obvious that it is not intention of the questioner to ask about the probability given that the host opens a specific door but to ask about the probability when he opens one of the other doors to reveal a goat. The door numbers quoted throughout the question are just examples. Nobody can imagine that Whitaker wanted to know about only the specific case where the player chooses door 1 and the host opens door 3!
- In fact the MHP is, as we say in the article, a probability puzzle. Its main interest is that it is a simple problem that nearly everyone gets wrong. We should therefore treat the problem and solution simply without even mentioning the terms conditional/unconditional, at least for the first section of the article. Martin Hogbin (talk) 10:26, 13 December 2009 (UTC)
I don't think the informal mediation is going to be of any value. It requires a volunteer mediator, of which none have come forward yet. It could takes weeks before one comes forward, it could be never. Whatever the mediator comes up with is non-binding. It has no teeth. In the meantime, all sorts of edits are being made to the article without any discussions whatsoever.
I suggest we request Mediation immediately, recognize the consensus for the proposed changes, and stop being hostages to this 6 year long filibuster. Glkanter (talk) 14:40, 14 December 2009 (UTC)
- I just read about Informal Mediation. If I read it correctly, any Wikipedia editor can offer to mediate. No prior approval of the volunteer mediators occurs. I just don't see where continuing to wait, perhaps forever, for this is of any benefit. Glkanter (talk) 14:46, 15 December 2009 (UTC)
Most recent FARC version
Can somebody please explain specifically what they don't like about the version of the article as of the last FARC, i.e. this version? I think the article is actually much worse now than it was then and it might be easier to proceed from this version than the current one. Just a thought. -- Rick Block (talk) 20:40, 9 December 2009 (UTC)
- I do not like any of the 'Solution' section including the diagram and the mention of conditional probability. We need a simple solution, followed by some convincing explanations. Martin Hogbin (talk) 21:45, 9 December 2009 (UTC)
- Up to "The reasoning above" does not mention conditional probability, and says something very similar to what is in the initial paragraph of the current version (doesn't it?). Do you like the large diagram in the current version? It seems isomorphic to me (it varies the player pick rather than the car location - but both are based on a specific concrete example). Are you thinking there's something wrong with the image in the FARC version? -- Rick Block (talk) 23:06, 9 December 2009 (UTC)
Rick Block states above: "I think the article is actually much worse now than it was then..." This was how the Solution section began as of the last FARC:
- Solution
- The overall probability of winning by switching is determined by the location of the car.
until it was deleted with this diff:
http://en.wikipedia.org/w/index.php?title=Monty_Hall_problem&diff=next&oldid=247025116
So, I think the newer version is a whole lot better.
But this new section is just another filibustering technique by Rick. Glkanter (talk) 12:29, 10 December 2009 (UTC)
- The excerpted quote above is a perfect example of the muddlement of this article. I refer to the quote: "The overall probability of winning by switching is determined by the location of the car." Huhh??? What does this mean? Again, let me emphasize, I have no idea who it is that was being quoted, which quote I just repeated. I don't mean to attack anyone in particular. But I have to ask ... First, what is the difference between the "probability" and the "overall probability"? Second, what does the probability have to do with the location of the car? I don't get it. The location of the car has nothing to do with the probability. If the writer has a point to be made, then he/she should try to state the point non-cryptically. Sorry for this rant, but this kind of thing is very frustrating ... End of rant. Shalom. Worldrimroamer (talk) 04:26, 10 December 2009 (UTC)
OK. So far, we have "delete the first sentence". Fine. Let's assume this change, which makes the Solution section look more like this version. And Martin's issue with the image (which he hasn't clarified) and his issue with anything that says "conditional probability". The referenced version has one sentence that says "conditional probability" in the 2nd paragraph below the figure basically identifying what the "subtly different" question is actually called. Possibly we could delete this sentence, although I'm not sure I see the point. Anybody else have specific gripes with this version? -- Rick Block (talk) 05:25, 10 December 2009 (UTC)
- Rick, I started discussing this in October, 2008. I have no interest whatsoever in comparing and contrasting to a May, 2008 version. I think it is unreasonable to request that we do so. It has the effect of negating discussions that led to all the edits to the article since the FARC version of May, 2008. Is that your intent? Glkanter (talk) 12:29, 10 December 2009 (UTC)
- Well, in-spite of my preference for the popular solution to be the main focus of the article I still feel that the conditional solution should be mentioned in the Variants section. The FARC version linked above seems to have nothing in it about the conditional solution at all. Also I like the current lead much better. Personally I think that simply moving the conditional section to the Variants heading would be the best compromise as that way no information will be lost from the article but the flow will still proceed from the most common understanding of the MHP towards more in-depth analysis. I would be more than willing to make the change myself as an editor who has only been part of this discussion for a short time. How does that sound to you guys? Colincbn (talk) 07:28, 10 December 2009 (UTC)
- Colincbn - You apparently mean something different by the "conditional solution" than at least Glkanter - which is why I would really prefer we talk about specific changes. I gather you are referring NOT to the entire "Probabilistic solution" section, but only the paragraph in that section starting with "Morgan et al. (1991) and Gillman (1992) both show a more general solution ..." and not the figure that follows. Is this right?
- Sorry about that, I'll try to be more specific but I might get my terminology wrong as I'm no mathematician (I'm a history, religion and biology guy). Basically I figure the conditional problem statements referenced to Morgan would be best explained in a single subsection under the variants section. ie: take the first paragraph of the Probabilistic solution section, that starts with "Morgan et al. (1991) state that many popular solutions are incomplete...", as well as the third paragraph that starts with "Morgan et al. (1991) and Gillman (1992) both show a more general solution ..." (as you said), and put them together in the new Conditional problem subsection. This might require some rewording of course. I might also suggest moving the probability tree image (File:Monty tree door1.svg) to the new subsection along with giving it any other useful visual aids or expanded explanations that are available. If the section gets too big we can add a "main article" link at the top of that subsection and continue expanding in a separate article that deals with the conditional problem extensively. I figure that way we get to keep all the information currently here (which I am in favor of) as well as restructuring the layout to flow from the most common interpretation to more advanced and in-depth analysis including variations and more focus on the Parade version of the question (which I am also in favor of). I think this is a fairly good compromise as no one can claim undue weight or anything like that, but we also get to cover the most prominent variants to the popular solution. Colincbn (talk) 15:52, 10 December 2009 (UTC)
- Colincbn - You apparently mean something different by the "conditional solution" than at least Glkanter - which is why I would really prefer we talk about specific changes. I gather you are referring NOT to the entire "Probabilistic solution" section, but only the paragraph in that section starting with "Morgan et al. (1991) and Gillman (1992) both show a more general solution ..." and not the figure that follows. Is this right?
- Colincbc, I think your suggestions have a lot of merit, at the appropriate time, as I posted just a few minutes ago. I would like to say however, I don't agree that the additional sections beyond the current 'Popular Solution' section are a "flow from the most common interpretation to more advanced and in-depth analysis". I consider those sections as unsupported criticisms and confusing different problems that might be of interest to some people. And they're not part of the Monty Hall problem paradox itself, but the published literature that only uses the MHP as a starting point. Glkanter (talk) 16:06, 10 December 2009 (UTC)
- While I can certainly see your point, my main concern is finding a compromise. I figure there are some things that will always be disagreed upon, and both sides will need to accept some things they don't like, but its the only way to really progress. The way I see it if neither side can have an article they think is 100% perfect both sides getting one they feel is 80% perfect is the best way forward. (Now we just need to figure out where that 80% is...) Colincbn (talk) 16:33, 10 December 2009 (UTC)
- Yes, a compromise is appropriate. Based on these discussions, I modified my proposal a couple of days ago. http://en.wikipedia.org/wiki/Talk:Monty_Hall_problem#Glkanter.27s_suggestion Glkanter (talk) 16:41, 10 December 2009 (UTC)
- Glkanter - I think comparing to a version that passed a Featured Article review is entirely appropriate. I'm talking about considering replacing just the "Popular solution" and "Probabilistic solution" (not the entire article) from the current version with the "Solution" section from this older version. I think it's clear not many people like the structure of the two current sections under "Solution" (even me). Choosing a different starting point might make progress easier. -- Rick Block (talk) 15:06, 10 December 2009 (UTC)
Colincbn - can you comment directly on the idea of using this version of the Solution section? I think it may well provide a reasonable compromise. -- Rick Block (talk) 18:04, 10 December 2009 (UTC)
- Correction: "good starting point for a reasonable compromise". -- Rick Block (talk) 18:20, 10 December 2009 (UTC)
- As a starting point definitely. I do like the Cecil Adams explanation and the two smaller images on either side of it in the current Popular solution section as a good "layman's" explanation though. And I imagine that we will need to move the paragraph that starts "A subtly different question is which strategy is...". Possibly replacing that paragraph with something that reiterates that the solution presented is only for the unconditional version and that the Parade version has spawned variants with a "see below" link. Colincbn (talk) 23:37, 10 December 2009 (UTC)
Who decides?
Consider 3 so-called 'variants':
- Instead of goats, behind 2 doors are cows.
- Instead of 1 car & 2 goats, it's 2 cars and 1 goat.
- The host may have a 'bias', or 'behaviour' or 'method' to determine which goat to reveal when the contestant selects the car. This may or may not be somehow made aware to the contestant.
Who decides when a 'variant' is no longer the Monty Hall problem? Glkanter (talk) 12:39, 10 December 2009 (UTC)
- Reliable sources decide. -- Rick Block (talk) 18:05, 10 December 2009 (UTC)
- In mathematics two problems are considered the same if they are isomorphic (see isomorphism). For example, the MHP and the Three Prisoners problem are isomorphic so even though they have different names and somewhat different descriptions they are actually the same problem. If you're asking for opinions you're on the wrong page (see /Arguments).
- The isomorphism with the Three Prisoners problem is an interesting point that has already been raised. The Morgan style argument is never heard concerning the TPP. It is never suggested that the warden may have had a preference for one of the prisoners. This is if course because Gardner specified that he chose randomly if he had a choice. He obviously did this specifically to avoid the Morgan style argument and to make the problem simple. That is exactly what we should do with the MHP, especially as the Morgan scenario is now shown to be just a contrafactual conjecture. Martin Hogbin (talk) 10:14, 11 December 2009 (UTC)
- No, what we should do is adhere to FUNDAMENTAL Wikipedia policy and neutrally represent what reliable sources say. To be clear, the specific point I"m talking about is whether a conditional probability analysis (using the assumption of equal probability of host choice between two goats) needs to be presented - more or less like it was in this version of the Solution section - not the generalization where the host preference is left as a variable. -- Rick Block (talk) 17:47, 11 December 2009 (UTC)
- That is absolutely fine and it is exactly what I want to do. We have a number of sources who treat the problem in a simple manner (Selvin, vos Savant, Devlin etc) and we have a number of sources that treat it in a more complex manner (Morgan etc). It is therefore quite reasonable, and advantageous for most readers, to start with a section that treats the problem and its solution and 'Sources of confusion' and 'Aids to understanding' in a simple manner. We can then treat the problem in a more complex manner for those interested in such things. Note that a simple treatment of a subject followed by a more detailed one cannot be described as a content fork, it is standard practice. The problem is that, when this has been suggested, you tell us that Morgan say that the sources that treat the problem simply are wrong or inapplicable, that they present 'false solutions'. This is an obviously pro-Morgan POV that should not be permitted.
- No, what we should do is adhere to FUNDAMENTAL Wikipedia policy and neutrally represent what reliable sources say. To be clear, the specific point I"m talking about is whether a conditional probability analysis (using the assumption of equal probability of host choice between two goats) needs to be presented - more or less like it was in this version of the Solution section - not the generalization where the host preference is left as a variable. -- Rick Block (talk) 17:47, 11 December 2009 (UTC)
- The fact is that we do have reliable sources that do treat the problem simply (do not distinguish between goat doors) and there is no reason not to represent these sources neutrally (not as false or incomplete or flawed) in the article. Martin Hogbin (talk) 18:33, 11 December 2009 (UTC)
- I do not like the diagram you refer to because it shows the choice of two goats as separate pictures, horribly complicating the problem for a beginner. We desperately need to keep it simple; the problem is plenty complicated enough for most people. If we do not do this we fail in our fundamental purpose of informing our readers.Martin Hogbin (talk) 18:33, 11 December 2009 (UTC)
- And I'm being somewhat non-responsive since I think one of the problems we've had on this page is too much focus on opinions and not enough focus on what reliable sources say. The three fundamental Wikipedia policies concerning content are WP:V, WP:OR, and WP:NPOV. If anyone hasn't read any of these they really should (how about right now?). What WP:V means is that everything an article says has to be sourced to a WP:reliable source. What WP:OR means is basically the converse of WP:V, i.e. you're not allowed to add content based on your own opinion. In the extreme, this is the case even if you know with absolute certainty that what you're saying is true but you can't find a reliable source to back you up. WP:NPOV means basically two things. First, that Wikipedia MUST fairly (neutrally) represent what reliable sources have to say. Second, if reliable sources disagree that Wikipedia articles are not allowed to take sides. In the aggregate, these policies mean that even though every editor has opinions and even though a collection of editors might have a collective opinion about content issues Wikipedia doesn't give a damn about these opinions. -- Rick Block (talk) 03:29, 11 December 2009 (UTC)
Repeated text
Why does the entire Krauss and Wang text appear twice in the first bit of the article? Isn't the article long enough without this repetition? RomaC (talk) 14:48, 10 December 2009 (UTC)
- Please see WP:Lead section. The lead is meant to be a concise, standalone overview of the entire article. The K&W problem definition was added to make the problem description unambiguous (even in the lead). I would be fine with deleting it and presenting only the Parade description. I predict others will object to this. -- Rick Block (talk) 15:44, 10 December 2009 (UTC)
- Actually I kind of like the FARC lead better, it clearly states that the problem as stated in Parade is ambiguous. The K&W problem is in the beginning of the Problem section as an example of an unambiguous way of stating the problem anyway so I don't think it is entirely necessary in the lead. But I would rather work on consensus of the other suggested changes above before diving into those waters... Colincbn (talk) 16:10, 10 December 2009 (UTC)
- Unlike the Parade statement of the problem, Selvin's statement is not very well known. It's also considerably longer. Because Jstor makes the first page of any reference available as a preview, it can be viewed online here. -- Rick Block (talk) 17:39, 10 December 2009 (UTC)
- I like the solution though. Selvin's answer is published in a reliable source and I would like to see it in the article (but in the form of a pretty diagram). It treats the opening of either of the two unchosen boxes by the host as equivalent. Martin Hogbin (talk) 22:28, 10 December 2009 (UTC)
- Unlike the Parade statement of the problem, Selvin's statement is not very well known. It's also considerably longer. Because Jstor makes the first page of any reference available as a preview, it can be viewed online here. -- Rick Block (talk) 17:39, 10 December 2009 (UTC)
- Isn't it funny how we've argued so much about vos Savant's failings and intents with the problem statement, and we never, since I've been around, discussed Selvin, et al's. I'm going to go read that page right now! I guess collaboration really can work. Who knew? Rick, thank you very much for the link! Glkanter (talk) 22:37, 10 December 2009 (UTC)
- It says it all - it gives a simple solution (showing that the contestant has a 1/3 chance of winning if they stick and a 2/3 chance if they swap) and notes that the contestant's chance of having the car in their original box is unchanged at 1/3 after Monty has opened a door. Not only that but it was published in the same peer-reviewed journal as Morgan but it is not followed by a highly critical comment. Looks like a winner to me. Martin Hogbin (talk) 23:43, 10 December 2009 (UTC)
- Isn't it funny how we've argued so much about vos Savant's failings and intents with the problem statement, and we never, since I've been around, discussed Selvin, et al's. I'm going to go read that page right now! I guess collaboration really can work. Who knew? Rick, thank you very much for the link! Glkanter (talk) 22:37, 10 December 2009 (UTC)
- From a sourcing perspective, it is important to note that this is a letter to the editor, not an article. Articles from peer reviewed journals are usually the most reliable sources. Letters to the editor are more like primary sources. -- Rick Block (talk) 17:18, 11 December 2009 (UTC)
- You might also be interested in Selvin's second letter in vol 29 #3. I can't find a pdf of the journal page online, but there's what appears to be a faithful copy here. Although I haven't compared it to the copy I have of the printed journal page, it's clearly missing a "/" on the 4th line of the conditional probability expansion. BTW - references to both of these have been in the article for several years. -- Rick Block (talk) 05:35, 11 December 2009 (UTC)
- Thanks for that. It does not, in my opinion, discount the solution given the his original letter. The interesting point, when you look at that history of the problem, is that Monty actually never offered the swap and, if Selvin's account is accurate, it was the contestant who suggested it. Martin Hogbin (talk) 09:56, 11 December 2009 (UTC)
- You might also be interested in Selvin's second letter in vol 29 #3. I can't find a pdf of the journal page online, but there's what appears to be a faithful copy here. Although I haven't compared it to the copy I have of the printed journal page, it's clearly missing a "/" on the 4th line of the conditional probability expansion. BTW - references to both of these have been in the article for several years. -- Rick Block (talk) 05:35, 11 December 2009 (UTC)
- The other point which becomes apparent is that Monty, who was the only person who knew for sure his door opening policy, has clearly stated that he knew the contestants chance of holding the car remained at 1/3 after he had opened his door. That seems to me to rule out Morgan's conjecture that he might have had a preference for one door or the other. Martin Hogbin (talk) 09:56, 11 December 2009 (UTC)
Thanks, Rick, for finding a solution that resolves my concern (an overly long intro) Butwhatdoiknow (talk) 00:02, 21 December 2009 (UTC)
There's A Difference Between A Logical Argument and OR
The first section I created after my return is called 'Is the Contestant Aware'. http://en.wikipedia.org/wiki/Talk:Monty_Hall_problem#Is_The_Contestant_Aware.3F
All I ask is this:
- "Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?"
It's a pretty simple, straightforward, and incredibly relevant question.
But neither Rick or Nijdam have given me a straight 'yes' or 'no'.
So, we're asking for mediation, but we haven't even stated our underlying reasons for disagreement. Because without their responses, any logical argument I might make to a mediator can be refuted, for most any reason.
And, as the section name indicates, using your noggin is not the same as OR. But that seems to be the reason for not replying.
So, enough of the 'well, I didn't say that per se', and let's get to the bottom of this. No mediator can help us if we're not being forthright. Glkanter (talk) 05:11, 11 December 2009 (UTC)
- Despite your claim to the contrary, this has already been answered. See, for example #Is The Contestant Aware?, above. And again (for what is it the 4th time?) this is a question for the /Arguments page, not here. Whether the editors of this article agree or don't agree about this should have no bearing whatsoever on what the content of the article should be. In fact, from a Wikipedia policy perspective an unreferenced logical argument is precisely WP:OR. The content of the article must be based only on what reliable sources say. Would it help if a mediator said this rather than me? -- Rick Block (talk) 06:25, 11 December 2009 (UTC)
- We disagree. And your choosing to not respond to my very relevant talk page questions is seen, by me, as dodging the truth. You interpret what all these Wikipedia policies & guidelines mean one way, sometimes I'll interpret them another. So, no, it will take the 'formalist' of procedures to convince me that 'endorsing wrongness' is a Wikipedia policy. The source material has conflicts. Editors resolve conflicts of content all the time in the real world. I'm just saying sometimes these conflicts are resolved using logic, often known as common sense. OK. Don't answer. Let's just move the process forward, and the mediation cabal has nothing to offer me. Can we move on to the next level? Glkanter (talk) 08:54, 11 December 2009 (UTC)
- Rick, despite your claims to the contrary, you have never answered the question. You have argued for why it shouldn't be answered based on Wikipedia policy, and why you think (and this is your POV) it doesn't matter, but you have not answered the question. Part of that is that it isn't quite the right question, but you have been evasive about why that is so. That is also POV.
- No reference uses non-standard, specific knowledge of the game protocol (whether a door is always revealed, and if it always reveals a goat), or a specific asymetric probability for the two uncertainties (car placement and host choice between two goats) to answer the question posed by the MHP. This cannot be denied. (Rick - feel free to try: just point out such a reference. If you don't, it validates my statement.) The so-called "conditional solution" includes terms to represent those concepts, but whenever they use that result to answer the question, they make at least three assumptions: A door is always opened, a goat is always revealed, and the car placement is treated as a uniform choice. For good reason - the question cannot be answered "yes" or "no" without those three assumptions. A brain teaser that asks a "yes/no" question has to have a "yes/no" answer. The fourth assumption - that the choice of the door to open is treated as a uniform choice - is made in some references, but not others. But those others don't use any knowledge of that probability to answer the question, either. They say it is irrelevant to the question. Either way, no knowledge that is not explicitly in the possession of the contestant is used. This is the point you need to agree to, or not: that every reference eliminates any reliance on knowledge that is neither specifically stated nor clearly assumable, before answering the question "should she switch?"
- Once you admit to that, and I don't see how you can deny it, we can see that the so-called "conditional solution" is only a tool that is used to generalize the MHP; it is not the MHP problem itself. Because the aspects unique to it are always removed before answering the MHP. Yes, this seems to contradict some sources (but not all, or even the majority - just those that try for a rigorous approach, and make a debatable interpretation about the meaning of door numbers). My point - and I hope Glkanter's, although he isn't phrasing it the same way - is that it only seems that it contradicts these sources. And that assuming there is a contradiction is POV because the sources do not (again, just find me one that uses it to answer the MHP, and I'll back off of this) use specific knowledge not available to the contestant. They either use the contestant's (assumed) knowledge, or say it is irrelevant.
- So it is not OR, or inconsistent with any source, to say the problem has to be addressed in the SoK of the player only. All of the sources do the same thing, after they have created that more general tool. They just don't come out and phrase explicitly that "We are now putting the solution into the SoK of the contestant because that is the only useful SoK." They just do it. All of them. The false impression that specific doors are meant never factors into any source's answer of the MHP itself. It never factors into any explanation about why the common intuition, that the remaining doors should be equivalent, is incorrect. So the conditional solution does not aid our readers in understanding why switching is the correct strategy, and it is not necessary to use it to explain the result. It does add extra depth to the various possibilities, but only after the reader has understood that revealing a door places conditions on the remaining door but not on the chosen door. It is not the MHP. It will not be diminished, or ignored, by moving it to after the discussion of alternate strategies. But it does get in the way of understanding, on an intuitive level, why switching works. JeffJor (talk) 18:02, 11 December 2009 (UTC)
- Nice job, JeffJor. But not quite perfect from my POV.
- Morgan uses some hazy allusion to contestant awareness to discredit the reliably sourced published 'simple' solutions. This is the nucleus of all of Rick's arguments, that Morgan says the 'simple solutions' are inadequate. No published source directly refutes Morgan, so, ipso facto, Morgan is the Uber Reliable Source. Which in Rick's world is a neutral POV. So, Morgan is relying on the contestant having some aura of knowledge. As someone else wrote on this problem, elsewhere in Wikipedia, 'I choose not to discuss a problem where the host and contestant are mind reading or are in collusion'.
- I get tired of typing it, and you guys have to be tired of reading it, but any 'host behaviour' is the opposite of the problem statement, 'Suppose you're on a game show...'. Now, you're on a street with a hustler and a card table, some shells and a pea. It is meaningless vis-a-vis the MHP paradox.
- There's little point in repeating and continuing the arguments. Both Rick and Nijdam have not given a straight answer to 'Is the Contestant Aware'. Which by the way, Mr. JeffJor, is phrased perfectly fine for my needs, thank you. I think I've asked courteously and tried to explain why it's important. When we get in front of someone to argue for the consensus, I am prepared to explain that this refusal to answer forthrightly (remember the 'meow, meow' answer to this very question?) is consistent with the filibustering, personal attacks on my grasp of the subject matter (Even got one of those from Rick just today. Everyone knows this has been decided in our favor for weeks, and he still says I don't know what I'm talking about. What does that tell you about his grasp of the subject matter, or perhaps his intellectual honesty?) and other passive-aggressive discussion techniques I've experienced personally for 14 months, and countless editors have put up with for 6 years or so. Glkanter (talk) 23:55, 11 December 2009 (UTC)
- Excuse me, but what question are you thinking I haven't answered? I answered Glkanter's questions about a week ago (with this edit), in the section I linked to above.
- If what I wrote before is not clear, what I think is that the contestant knows what is given in the problem statement, no more and no less. The specific problem statement varies, so what the contestant knows also varies. If, for example, we're given the K&W statement of the problem but without the host's protocol for choosing between two goats (I believe this is precisely the problem Morgan et al. call the "vos Savant scenario"), the contestant does not know the host's choice is uniform. This means a specific player who has initially chosen door 1 and has seen the host open door 3 does not have enough information to know her precise probability of winning by switching. It is something between 1/2 and 1, and it is unconditionally 2/3. I agree there is a sense in which this means the probability is 2/3, but I think this is actually the answer to a different question than what nearly everyone interprets the MHP to actually be asking and that this is the precise reason the MHP is a paradox. I also agree that most people who ask the question intend the answer to the "conditional" question (the one that most people think the question is asking) also be 2/3. IMO, this means the host's preference for choosing between two goats should be specified as part of the problem statement.
- However, in addition to the above I also believe that for editing purposes Wikipedia doesn't give a damn what I think and insisting that this question is important or that anyone answer such a question demonstrates a profound misunderstanding of Wikipedia's fundamental content policies.
- JeffJor - One reference is Morgan et al. Another one that uses exactly the same approach and makes exactly the same points is Gillman. Another one that distinguishes the "unconditional" and "conditional" questions is Grinstead and Snell. I definitely do not claim that the host preference must be treated as a variable (are you thinking I'm insisting on this?), but that the "Solution" section is really not complete without an analysis using conditional probability and that the distinction between conditional and unconditional probabilities is a central issue at the heart of the MHP. Martin has argued for a long time that a conditional probability analysis would be inaccessible to most readers - essentially that we need to "dumb down" the article. Per Wikipedia:Make technical articles accessible, we should start with an unconditional explanation, proceed with a conditional explanation, with a picture. Hmmm. This sounds exactly like the Solution section in this version. You have seemingly ignored the questions I asked you above at the end of #What "the conditional problem" and "the unconditional problem" mean. I am interested in your response. -- Rick Block (talk) 04:53, 12 December 2009 (UTC)
- Rick: Saying "Yes we are limited to the player's knowledge BUT the player can wonder about knowledge she doesn't have as though it were useful" is a non-answer. It is straddling the fence on exactly the point you refuse to acknowledge, by answering it both ways. If, for example, Morgan's formula had turned out to be 3/(4+3q) instead of 1/(1+q)? Such "wondering" would not be useful. Switching would be helpful with only some possible qs, but the question as it was asked would still have a definite answer. By assuming q=1/2 and P(C1)=P(C2)=P(C3)=1/3 because the player cannot assume anything else for them.
- Morgan, et al, say "In general, we cannot answer the question ... unless we know the host's strategy." So they don't answer the MHP question based on their (incomplete - ignoring P(C1)) formula. The only answer to the MHP question that they give is based on what they call "the vos Savant scenario" and the fact that q doesn't matter. Gillman wrote the assumtion of symmetry for car placement into their problem statement - at least they didn't ignore it like Morgan - but also only provide an answer to the MHP question based on standard game protocol and ignoring q. Grinstead and Snell don't address the MHP's question when they present the so-called conditional solution. So you have not provided the referencecs I asked for. Once again, there is no source that uses the unique properties of the so-called conditional solution to answer "Should the contestant switch?" Those properties only become important if non-standard game rules apply, or if assymetric probabilities are known by the contestant to exist.
- The article version you linked is still treating the possibility, that the doors could be treated differently, as something the player could use. It says "A subtly different question is which strategy is best for an individual player after being shown a particular open door." That answer is also "switch, based on an assumed q=1/2 and P(C1)=P(C2)=P(C3)=1/3" in every source that addresses it. Because they don't say that there are other values of those parameters that can be used. Any mention of it belongs after mentioning variants, and providing the "bayesian" solution. Because they don't address, in any way, the issues that make the MHP controversial. They only address ways that variations of the MHP, from what was intended by vos Savant and Selvin, can become more intersting as a mathematic (as opposed to logic) problem. JeffJor (talk) 20:46, 12 December 2009 (UTC)
- Jeff - Are you saying you consider any analysis using conditional probability to be a variant? Again, for about the 4th time in the last few days, I'm NOT saying we need to include the host preference (q) generalization in the initial analysis. I'm happy moving that to a variant section (I don't think there's anyone arguing against this, so maybe I'll do that right now). What I'm not happy with is introducing a POV which favors an unconditional solution by omitting any mention of conditional probability. A conditional solution assuming uniform car placement and random host selection between two goats belongs in the initial Solution section. Not doing this is what I'm saying would be counter to NPOV. IMO, the figure in the existing 'Probabilistic solution" section showing the symmetry belongs in the initial solution section. I think there should also be an explanation that there is a difference between the "conditional" question and the "unconditional" question very early, although perhaps not necessarily in the initial Solution section. -- Rick Block (talk) 00:31, 13 December 2009 (UTC)
- Rick - I never said that using conditional probability was a variant. I clearly said that using different probabilities based on door numbers is the variant. The problem can be solved with conditional probability, but by the tree G&S use as their figure 4.4. Morgan's criticism of the solutions from the Parade affair are mostly wrong, since they apply to the variant where different probabilities can be considered. Solutions that do not use conditional probability ARE NOT WRONG, as Morgan says; they are only wrong FOR MORGAN'S VARIANT WHICH IS NOT THE MHP. And those solutions Moragn dismissed, which are correct, are easier for lay people to understand. There is no need for G&S's conditional solution early, but it can be included. I don't think it helps anybody who needs to rely on the initial section. But there is no place for Morgan's formulation there.
- About my recent edits: It is you who is not sticking to what the sources actually said. MvS did not say that "letting the host choose a door with the car" was the only strategy that was not a part of her problem. She said it was the most significant, of of all the conditions she assumed were defined by her answers. That also incldues anything that would prevent her 2/3 answer from being correct. Every single one of her answers makes the assumptions that render Morgan's formula useless. You have to realize that she needs to compact her column into very few words, and so is not addressing the problem rigorously. And shouldn't be expected to. The only mention of anything having to do with assymetric probabilities CAN ONLY APPEAR in the "variants" section.
- Morgan, at al, do not, EVER, claim that their formula answers the question in the MHP. In fact, they say it doesn't. With emphasis added: "In general we cannot answer the question 'What is the probability of winning if I switch...' unless we either know the host's strategy of are Bayesians with a specified prior." This is why the question you keep refusing to give a direct answer to is important. It says we don't "know the host's strategy" and are not "are Bayesians with a specified prior." The direct answer you shodl give means that, by their own admission, MORGAN DOES NOT APPLY. Anything they conclude applies only to their variant, and only then if one of these conditions they describe holds. It does not help to "wonder" what the possibilities might be, except in the case where every answer to that wondering says "switch." WHICH WAS MORGAN'S THESIS. JeffJor (talk) 18:08, 14 December 2009 (UTC)
- Glkanter: I've given up on getting Rick to stop using Morgan. It is a reference, and it claims to address the MHP (it doesn't, it addresses what they changed the MHP into), so he will forever stand behind his "Wikipedia Policy" arguments to say it must be included. (I can find an internet reference that claims the answer is 4/9 and has not been discredited, mainly becasue it can't be understood, but we won't quote it because we know what is wrong with it. Why we can't do the same with Morgan, since they misquoted the problem into something which is documented to be not the intent, I can't fathom.) The fact is, it is an interesting treatment, but of a variation. And the only reason I said your question wasn't well-asked, was because it allowed Rick to give his non-answer. We need to include the fact that it is only useful to "wonder" about other knowledge if the contestant can actually get it. And just one last comment for Rick: although you will say this is OR that contradicts Gillman, it isn't. It is a fact that Gillman glosses over because of the altered nature of his MHP. The q=1/2 approach is not just the equivalent to "announcing the switch strategy before a door is opened," it is equivalent to "announcing the switch strategy without knowledge of how the host chooses a door." Under Gillman's modificaiton, "before" is the time that is not known, at least until he assumes q=1/2 so that he can answer the question. JeffJor (talk) 21:06, 12 December 2009 (UTC)
JeffJor, oh, yeah, you've made the right decision. We have clearly demonstrated a consensus. Rick is going to use every method he can devise to extend his filibuster. There have been a lot of folks prior to us who ultimately made the same decision, to quit arguing with Rick. But we have proven the argument unlike anyone before us. "Suppose you're on a game show..." End of discussion on who's POV for the doors, and no more host behaviour. They used to argue 'little green men from space's' POV as being the MHP. I kid you not. And, no more host behaviour means no more Morgan. Of course, kmhkmh is still arguing the definition of a game show, but won't answer the 'Is The Contestant Aware?' question.
As far as the details, Morgan and his ilk get mentioned, they're published. But no more bad mouthing the Devlin solutions. Did you read my Huckleberry section? Please re-read my modified proposal. So, we just have to navigate Wikipedia's consensus processes. Rick is the king of that crap, so we'll learn as we go up the chain, whatever it is. Do you have any experience with that? So, a few of us will continue working together the straight path to improving the article. Glkanter (talk) 22:53, 12 December 2009 (UTC)
Let's Say Some Huckleberry Played Repeatedly
Let's say some huckleberry played repeatedly. They play for $1 per play, rather than the car.
Huckleberry has figured things out using the Combining Doors solution. He doesn't understand, in fact, he is not even aware of the term 'conditional' as it applies in this instance, as he is less educated than others.
So, with just the brains and common sense he was born with, he wins 2/3 of the time. His method worked fine. Perfectly, really. Nobody could give Huckleberry ANY ADDITIONAL INFORMATION OF ANY CALIBER WHICH CAN PRODUCE A BETTER RESULT. And if this were repeated endlessly, it would always work out to 2/3.
There are other solution techniques which may produce the SAME result, but Huckleberry doesn't understand that the 'equal goat door constrain must = 1/2'.
It was an elegantly simple Paradox for Selvin, and vos Savant, and Huckleberry. 1/3 = 1/3. And always did. Glkanter (talk) 18:03, 11 December 2009 (UTC)
Parallel Universe Experience: Same random distribution of cars. Huckleberry makes all the same door choices. Monty picks the exact opposite equal goat door as he did above. Huckleberry always switches. There is NO CHANGE TO THE 2/3 OUTCOME, EVEN IF YOU PLAY FOREVER. And the individual play outcomes are identical. Huckleberry wins or loses the same exact instances regardless of which goat door Monty opens. Glkanter (talk) 14:50, 12 December 2009 (UTC)
I Guarantee It. Old Paradoxes Are Like New Technology.
“New technology goes through three stages:
First, it is ridiculed by those ignorant of its potential.
Next, it is subverted by those threatened by its potential.
Finally, it is considered self-evident.” –unknown —Preceding unsigned comment added by Glkanter (talk • contribs) 04:39, 12 December 2009 (UTC)
- Looks like I started this section just in the nick of time: Moved conditional analysis involving host preference q to variant section.
- There's an old saying: "If you see a parade, get in front of it." I think Rick must have heard this old saying, too. Maybe Rick will argue for the proposed changes when we get to mediation as well? Glkanter (talk) 05:23, 13 December 2009 (UTC)
Moved conditional analysis involving host preference q to variant section
I think this is at least one of the changes that has been argued for, and I haven't seen anyone argue against it (and those of you who think this is what I have been arguing against are simply incorrect), so I've moved the paragraph about the Morgan/Gillman generalization introducing the host preference as a variable q to the Variants section. If anyone is arguing about this, feel free to revert. -- Rick Block (talk) 00:51, 13 December 2009 (UTC)
- I think that is a good move but it still does not address the fundamental issue that many people here are concerned about, which is that the Monty Hall problem is a simple problem that most people get wrong. This article needs to reflect that fact, based on the many reliable sources that treat the problem simply (I am not going to mention the c-word). Martin Hogbin (talk) 17:50, 13 December 2009 (UTC)
- I understand your point, but based on other comments there seems to be some confusion about what those of us who favor presenting a "conditional approach" as an equally valid POV are saying. Rather than argue generalities, I think it is helpful to see specific changes. My stance is that a single solution section, more or less like (and about as long as) this one is sufficient. I think it would be very helpful if you (or anyone) could draft a specific proposal (not just an outline, but actual content). With the amount of text on this page, we could have 5 or 10 specific proposals by now. -- Rick Block (talk) 18:45, 13 December 2009 (UTC)
- Rick, you wrote this above: "..."conditional approach" as an equally valid POV...".
- Hey Rick, we talked about this before:
- "...The total probability must be 100%, and before the host opens a door it's surely 1/3 player's door vs. 2/3 for the other two doors (so 2/3 of the players who decide to switch before the host opens a door will win) but the only thing that keeps it that way after the host opens a door is that pesky equal goat door constraint. The best way I know to show that it CAN change is to contrast the problem as stated with a different problem (i.e. the aforementioned "host opens lowest numbered door possible" variant)...Rick Block (talk) 00:24, 27 October 2008 (UTC)"
- "False. The only thing 'keeping it that way' is that pesky law of probabilty that the outcomes must = 100%. Glkanter (talk) 03:46, 27 October 2008 (UTC)"
- "...The total probability must be 100%, and before the host opens a door it's surely 1/3 player's door vs. 2/3 for the other two doors (so 2/3 of the players who decide to switch before the host opens a door will win) but the only thing that keeps it that way after the host opens a door is that pesky equal goat door constraint. The best way I know to show that it CAN change is to contrast the problem as stated with a different problem (i.e. the aforementioned "host opens lowest numbered door possible" variant)...Rick Block (talk) 00:24, 27 October 2008 (UTC)"
- The more things change, the more they stay the same, eh? Still using some so-called 'variant' to explain away the simple solutions. That doesn't even make sense from a mathematical standpoint. Oh, but now (per your paragraph above) you say they're 'equally valid'.
- By now, your only remaining argument is that 'people could get confused', because the MHP only works with the particular set of premises given. The part about why it works may be true. The part about confused readers is your personal interpretation, and not supported by at least one editor who includes the problem in his course work. And it's certainly not Wikipedia's job to teach probability in the MHP Paradox article. Anything like that belongs in what I call the 'Diversions' section, if at all.
- And there is no point whatsoever in creating mock articles until the consensus for the proposals has been 'certified' to your satisfaction. They are not ambiguous in any way regarding the minimal value of the 'conditional' approach. Anything else is just a waste of time, and feeds your filibuster. Glkanter (talk) 12:46, 14 December 2009 (UTC)
- Hey Rick, we talked about this before:
Proposed unified solution section
Here's a proposal for a unified solution section that I suggest replace the current two solution subsections. I offer this partly as an example of what I mean by a specific suggestion, and partly to show what I think would be a sufficient, NPOV, solution section.
According to the problem statement above, a car and two goats are arranged behind three doors and then the player initially picks a door. Assuming the player's initial pick is Door 1 (vos Savant 1990):
- The player originally picked the door hiding the car. The game host must open one of the two remaining doors randomly.
- The car is behind Door 2 and the host must open Door 3.
- The car is behind Door 3 and the host must open Door 2.
Players who choose to switch win if the car is behind either of the two unchosen doors rather than the one that was originally picked. In two cases with 1/3 probability switching wins, so the probability of winning by switching is 2/3 as shown in the diagram below. In other words, there is a 2/3 chance of being wrong initially, and thus a 2/3 chance of being right when changing to the other door. This result has been verified experimentally using computer and other simulation techniques (see Simulation below).
Another way to understand the solution is to consider the two original unchosen doors together. Instead of one door being opened and shown to be a losing door, an equivalent action is to combine the two unchosen doors into one since the player cannot choose the opened door (Adams 1990; Devlin 2003; Williams 2004; Stibel et al., 2008).
As Cecil Adams puts it (Adams 1990), "Monty is saying in effect: you can keep your one door or you can have the other two doors." The player therefore has the choice of either sticking with the original choice of door, or choosing the sum of the contents of the two other doors, as the 2/3 chance of hiding the car hasn't been changed by the opening of one of these doors.
As Keith Devlin says (Devlin 2003), "By opening his door, Monty is saying to the contestant 'There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3. I'll help you by using my knowledge of where the prize is to open one of those two doors to show you that it does not hide the prize. You can now take advantage of this additional information. Your choice of door A has a chance of 1 in 3 of being the winner. I have not changed that. But by eliminating door C, I have shown you that the probability that door B hides the prize is 2 in 3.'"
Another way to analyze the problem is to determine the probability in a specific case such as that of a player who has picked Door 1 and has then seen the host open Door 3, as opposed to the approach above which addresses the average probability across all possible combinations of initial player choice and door the host opens (Morgan et al. 1991). This difference can also be expressed as whether the player must decide to switch before the host opens a door or is allowed to decide after seeing which door the host opens (Gillman 1992). The probability in a specific case can be determined by referring to the expanded figure below or to an equivalent decision tree as shown to the right (Chun 1991; Grinstead and Snell 2006:137-138). Considering only the possibilities where the host opens Door 3, switching loses with probability 1/6 when the player initially picked the car and otherwise wins with probability 1/3. Switching wins twice as often as staying, so the conditional probability of winning by switching for players who pick Door 1 and see the host open Door 3 is 2/3—the same as the overall probability of winning by switching. Although these two probabilities are both 2/3 for the unambiguous problem statement as presented above, depending on the exact formulation of the problem the conditional probability may differ from the overall probability and either or both may not be able to be determined (Gill 2009b), see Variants below.
A formal proof that the conditional probability of winning by switching is 2/3 using Bayes' theorem is presented below, see Bayesian analysis.
I have tried to remove any POV-ish statements in the above. If there's anything left that does not sound NPOV please either just fix it or discuss here. The idea is to present both a plainly correct unconditional solution (it's basically vos Savant's from her second column) as well as a plainly correct conditional solution, without expressing a preference for either treatment. -- Rick Block (talk) 19:12, 14 December 2009 (UTC)
- And I don't think you did a good job of removing your POV, Rick. It's all through the second paragraph, as you try to make it sound like what we want while leaving Morgan in it. My criticisms will address that, and hopefully suggest NPOV corrections, soon.
- The 3-point options at the start, and the following diagram, present the unconditional solution (e.g., Door #2 gets opened in some cases), yet seemingly claim to follow from the K&W statement. To use this approach (which we should), this part needs to attribute it to the MvS statement, and not use door numbers. (This is consistent with MvS's answer, which refers to "the first door" and "the second door".) The following preserves her use of examples while not requiring door numbers:
- According to the Parade problem statement, a car and two goats are arranged behind three doors. The player initially picks one, and the host always opens a different door with a goat, choosing at random if necessary (Seymann). (Aside: Seymann does not say the MvS problem statement is ambiguous, as the article currently claims. Seymann says the assumptions should be inferred from intent, and intent is quite clear from her solutions and following columns. Seymann is chastising Morgan, et al, for not using this clear intent. Seymann does say the host chooses randomly, as an "agent of chance".)
- The 3-point options at the start, and the following diagram, present the unconditional solution (e.g., Door #2 gets opened in some cases), yet seemingly claim to follow from the K&W statement. To use this approach (which we should), this part needs to attribute it to the MvS statement, and not use door numbers. (This is consistent with MvS's answer, which refers to "the first door" and "the second door".) The following preserves her use of examples while not requiring door numbers:
- The second paragraph does not match the decision tree it says it matches. It is trying to address the conditional solution without assymetric probabilities, yet duplicates the diagram's unconditional treatment. So, it should cite G&S only, not Morgan, because this "symmetric conditional" treatment is far closer to their treatment than Morgan's. I'm not going to try to draw a tree, but it is in G&S as Figure 4.4 (the existing tree misrepresents it). We can include the "disallowed" options that the existing tree does, opening Door #2, by making it clear how they are disallowed by G&S. Make them a different color, add a column that says "didn't happen." Alternately, it could just trim the diagram down, like G&S trim thier tree down.
- Another way to analyze the problem is with conditional probability, as the specific case where the player has picked Door 1 and the host has opened Door 3. This contrasts with the approach above which addresses the average probability across all possible combinations of initial player choice and door the host opens. The probability in this specific case can be determined by referring to the decision tree as (will be) shown to the right (Grinstead and Snell 2006:137-138). Considering only the possibilities where the player chooses Door #1 and the host opens Door 3, switching loses with probability 1/18 and wins with probability 1/9. But these possibilities comprise only 1/6 of the total possibilities, so their probabilities must be divided by 1/6 in accordance with Bayes Rule. Thus we get the same overall probability of winning by switching as before. And while these two probabilities are both 2/3 for the problem statement as presented above, if there is reason to beleive that the host prefers to open one door over another, the conditional probability may differ from the average probability that disregards which doors are chosen (Grinstead and Snell 2006:137-138, see Variants below.
- The second paragraph does not match the decision tree it says it matches. It is trying to address the conditional solution without assymetric probabilities, yet duplicates the diagram's unconditional treatment. So, it should cite G&S only, not Morgan, because this "symmetric conditional" treatment is far closer to their treatment than Morgan's. I'm not going to try to draw a tree, but it is in G&S as Figure 4.4 (the existing tree misrepresents it). We can include the "disallowed" options that the existing tree does, opening Door #2, by making it clear how they are disallowed by G&S. Make them a different color, add a column that says "didn't happen." Alternately, it could just trim the diagram down, like G&S trim thier tree down.
- I left it as "there is reason to believe" rather than the correct "the player has reason to believe" to keep what Rick thinks is POV out. The content of this section now matches G&S in every way, except it is a paraphrasal. G&S is aimed at students of probability, where we want to aim at non mathematicians. JeffJor (talk) 16:17, 15 December 2009 (UTC)
- The problem is that the diagram is still complicated by the fact that it shows the host opening one of two doors. You are stuck in the world of door numbers. I would prefer this diagram (with pretty pictures) and some explanation:
1/3 1/3 1/3 You choose a goat You choose a goat You choose a car The host opens a door to reveal a goat The host opens a door to reveal a goat The host opens a door to reveal a goat You Stick You Swap You Stick You Swap You Stick You Swap You get a Goat You get a Car You get a Goat You get a Car You get a Car You get a Goat
- So, you're OK with the words but not the diagram? Comments from other might be nice, but one thing I like about the diagram is that it SHOWS the symmetry (the diagram is symmetric). -- Rick Block (talk) 19:52, 14 December 2009 (UTC)
- I think this whole thing is a nightmare giant step backwards into an abyss. How did the Combined Doors solution get eliminated? Why? There is no need to include 1/6 in any explanation of the paradox. 'Simulations' as 'proofs'? Please. And I don't understand why editing is being discussed prior to the consensus being recognized. Just more filibustering. Enough, already! Glkanter (talk) 20:45, 14 December 2009 (UTC)
- Glkanter - Where did combined doors go? Well, I think vos Savant's solution is clear enough for nearly anyone. Most of the folks who talk about "combining doors" offer it as an aid to understanding, so it could certainly go there (and, let me just say, for one who is so against variants of the problem, your attachment to this explanation is incredibly incongruous). 1/6 is included because it is what the unconditional probability of the player's initially selected door becomes in the cases where the host opens each door, just like the sources say (the probability tree is directly from Chun, 1991). The forward reference to the Simulation section is there because some people trust simulations more than logic. If you don't want it, let's just delete it and see if anyone objects. I'm trying to work toward a specific compromise we can all live with. -- Rick Block (talk) 02:58, 15 December 2009 (UTC)
- Binary choices don't offer 'compromise'. The simple solutions are more than sufficient. Let's get the Formal Mediator to acknowledge the consensus to make the proposed changes. I think a lot of what is taking place lately does not represent a good faith effort at recognizing the consensus that clearly exists. Did Huckleberry get it right or not? Is the contestant aware? These are direct yes/no questions, not 'yes, but'. You are THE LAST person who should be leading the editing effort. Maybe you haven't accepted it yet, but your 6 year crusade was misguided. Simply wrong. Just like I've been trying to tell you for 14 months. I'd appreciate it if you'd stop all this mis-direction, and let the consensus edit the article as proposed, finally. Glkanter (talk) 03:25, 15 December 2009 (UTC)
Any other comments? I'm particularly interested in comments from the folks Glkanter is considering part of the "consensus", i.e. JeffJor, Colincbn, Boris, and Melchoir. -- Rick Block (talk) 14:29, 15 December 2009 (UTC)
- Which of your various roles are you in when you ask this question, Rick? Editor, Owner, FA Shepard, or Junior Moderator? Glkanter (talk) 14:37, 15 December 2009 (UTC)
- Glkanter - I see you've edited in the combining doors solution. Does this mean you're more or less OK with a single solution section? I'd prefer for the combining doors bit to be in an aid to understanding section (it's even phrased that way "Another way to understand the solution ..."). Do you consider this a deal breaker? I'm curious what others think about this. -- Rick Block (talk) 19:33, 15 December 2009 (UTC)
- Jeff - Aside from some minor quibbles (which I don't have time to enumerate at the moment - I will later, although perhaps not until tomorrow), I'm more or less OK with your version of the conditional analysis. Not minor - I assume you don't mean for the aside to be in the article. Please say what you think about including the "combining doors" bit in the solution section. -- Rick Block (talk)
- Martin - Is Jeff's version of the image more to your liking? -- Rick Block (talk) 19:33, 15 December 2009 (UTC)
- Anyone else have any comments about all of this? -- Rick Block (talk) 19:33, 15 December 2009 (UTC)
- Rick, you continue to be confused. There are no 'deals'. And you are NOT the deal maker. You chose to not be part of the consensus. It makes no sense, then, for you to be proposing what the consensus wants. I wish I had a way to just make you stop. I don't, of course. I have, over 14 months used the Combined Doors solution as the published simple solution to support my agrument with you and the others. You all continue to post, even today, that I don't understand the problem, or worse, actually. But, Devlin's solution was right and proper all along. And it's in the article that way now. Only YOU would think of removing it. Certainly not me. And I don't foresee a 'Unified solution' section, or a 'disclaimer' of any sort. It's not correct. That's why I proposed a brief section summarizing the 6 years and soon-to-be 10 archives of arguments, and why the consensus supports the simple solution. Glkanter (talk) 19:56, 15 December 2009 (UTC)
- Here's my interpretation of Wikipedia policy, speaking as an admin. Consensus applies to changes to articles (even specific changes), not to groups of editors. No editor is more or less a part of the consensus for any individual change than any other editor. Trying to cut off discussion about a specific suggested change is classic disruption. -- Rick Block (talk) 21:09, 15 December 2009 (UTC)
But Rick, you are not part of the consensus. You are against the proposals. Or has that changed, and mediation is no longer required? Glkanter (talk) 21:30, 15 December 2009 (UTC)
- And if you honestly believe I am attempting to violate disruption, I suggest you follow the procedures necessary to stop me. Otherwise, I'll consider the threat just one more of your endless filibustering techniques, not offered by you in good faith. Glkanter (talk) 21:40, 15 December 2009 (UTC)
Martin's suggestion
- I would like to see 'combining doors' back at the start of the article along with my suggestion above. I think that most people will agree that these are the most convincing explanations of the basic problem (where the issue of conditional/unconditional is deferred to a later section). These diagrams should be accompanied by new wording specifically relevant to the explanations shown in the diagrams. This should be followed by 'Aids to understanding' and 'Sources of confusion' relating to the simple solutions - this is what the Monty Hall Problem is all about.
- After the above we should explain why some formulations of the problem are, strictly speaking, conditional (with reference to the Morgan paper) and why this fact is not so important for the symmetrical case.
- Then we should have a section on variants, the most important of these being 'the host chooses any unchosen door randomly', but including the Morgan scenario, 'we are aware of the host door opening policy'. We can then have more on sources of confusion etc as it relates to the more complex cases. That is what I would like to see. Who agrees? Martin Hogbin (talk) 15:18, 15 December 2009 (UTC)
- Just to add, I would not object to a brief and mildly worded disclaimer at the start of the simple solution to the effect that some academic sources insist the problem must be treated conditionally but, for simplicity and clarity, these issues are discussed in a later section. Martin Hogbin (talk) 15:22, 15 December 2009 (UTC)
- The so called "combining doors solution" is in a certain sense the worst of all. Let me try to make this clear to you. Choose door 1 and start from this situation: then with prob. 1/3 the car is behind door 2 and also with 1/3 behind door 3. Formally: P(C=2)=P(C=3)=1/3. That's why we can say: P(C=2 or C=3) = 2/3 (doors combined). But there is no immediate logical reason why this should lead to the prob. 2/3 that the car is behind door 2 when door 3 is opened. We know P(C=3|H=3)=0, but this does not imply that P(C=2|H=3)= 2/3. Only if one reasons that from symmetry it follows that {H=3} is independent of {C=2 or C=3}, one may conclude that also P(C=2 or C=3|H=3) = P(C=2 or C=3) = 2/3, and hence P(C=2|H=3) = 2/3. Even then as you may note is the prob, of interest the conditional prob. I'm pretty sure however most of the readers do not understand this independency, but simply have no idea of the different prob.s and follow a wrong way of reasoning. BTW also some referred sources do! Nijdam (talk) 22:25, 15 December 2009 (UTC)
- I accept what you are saying. The issue is addresses in words by Adams, who says (without proof) '...as the 2/3 chance of hiding the car hasn't been changed by the opening of one of these doors'.
- On the other hand, three sources use this approach and I think it is intuitively acceptable to many people. What about having footnotes for the two simple solutions saying something along the lines of, 'There are some important complicating issues here which are discussed later but, as it happens, if the host chooses randomly they do not affect the result'. This has the advantage of not affecting the simplicity of the explanation but drawing attention the the complication issues for those interested. Again, this approach is not uncommon in maths text books. Martin Hogbin (talk) 10:00, 16 December 2009 (UTC)
- Being a lay person, I'm curious, what was the outcome of all the work Boris did? He said his mission here was finished. Devlin, Adams, etc. are published, and a consensus is that their approach has NOT been mathematically refuted by Morgan and the others. There's no complication with the MHP, the complication is with the 4 confused sources which do NOT address the MHP. I've suggested a brief paragraph after the simple (only) solutions that summarizes the lengthy discussions that have taken place as the appropriate way to recognize Morgan and the others. Glkanter (talk) 11:29, 16 December 2009 (UTC)
- Again, I'm not sure where to insert comments. (1) On my "aside": Yes, Rick, it was meant to be left here, not included in the article. That's why it is an aside. I didn't want you to change the text again to match your POV of what Seymann says, or to continue thinking than nobody had ever refuted Morgan's assertion that the door numbers were intended. That was refuted when Morgan published. (2) I'm not a great fan of the "combining doors" explanation, because it is a (very slightly) different problem. But not anywhere close to as different as Nijdam claims. The only difference is that you shold get the better of the two prizes, not both. (For Nijdam: the Host's required behavior, revealing an unchosen door with a lesser prize, is logically equivalent to the switching player receiving the better prize of the two unchosen doors. That is indeed an "immediate logical reason" for the 2/3 probability applied to the combination. But Carol Merrill should lead a goat out of the combined doors, in such a way that you don't know where it came from, for it to be the same problem. It is when we translate the combined-door probability back to the original, unopened Door #2 that it can be associated with one door only). I just don't know of a source that explains it my way. Delvin comes close, since he used empty doors instead of goats. But it is a good way over the intuition bubble, as Delvin explains; so the only reason to not use it is if we have too many explanations. I'm not sure which we are proposing keeping. We need at least one basic solution (as I did above), and one explanation for why it works (like combining doors). I don't consider the conditional probability one I listed "basic," but I'm not going to fight including it. That does, however, limit what other "basic" solutions we could use, because we lose readers if we over-explain. (3) We need a disclaimer with the full Morgan conditional statement. Not before. The disclaimer is that some advanced treatments consider it a Door 1/Door 3 issue; but that that was never the intent of the originators, has been denied by the originators, and is not even universally accepted. Then add that even those who use it remove the dependency on those door numbers before addressing "Should the contestant switch?" Essentially, the full conditional solution is a tool only. JeffJor (talk) 17:17, 16 December 2009 (UTC)
- Nijdam, how does this question differ from the issue you and Boris analyzed? Boris declared his mission finished. I'm beginning to think that like Rick, you are simply using various filibustering techniques to avoid recognizing the legitimate consensus for the proposed changes. Why don't you ever answer my questions? I imagine Socrates would have. Glkanter (talk) 12:08, 17 December 2009 (UTC)
OH MY GOD!!!
Holy cow. I went camping this weekend and I woke up in the tent in 33 degree F. cold trembling in a cold sweat, having a nightmare about goats and shiny cars and numbered doors and a genetic-cross monster whose body was that of a water buffalo and the head was that of Thomas Bayes, and everyone was throwing food about and nobody knew whether everything or anything is random or predetermined and people were capriciously changing their minds in the middle of the game conditionally and unconditionally, and I decided that I would just run off and try to re-read "Kant's Critique of Pure Reason", and then maybe shoot myself. That might be easier. :o) Worldrimroamer (talk) 02:04, 15 December 2009 (UTC)
- OK, folks ... please don't fuss at me. I was only kidding. Worldrimroamer (talk) 02:08, 15 December 2009 (UTC)
Why Wait To Edit The Article & FAQs As Per The Proposals Supported By The Consensus?
There's plenty of editing of the article currently taking place, including by Rick Block, who is not part of the consensus.
Why couldn't the consensus just go ahead and begin fixing the article, so that it is in line with the proposals? Glkanter (talk) 11:51, 15 December 2009 (UTC)
- I think the article is OK through the Popular solution. Then the Probabilistic solution begins all the trouble with the gibberish and double talk about Morgan's stuff.
- Let's put a paragraph right after the Popular solution that discusses why the Morgan solution, while published, is considered to be discussing a different problem.
- Then we replace nearly all the text of the Probabilistic solution, and replace it with how to do the conditional solutions.
- Beyond that, it's just chatter, that hopefully most readers won't need. Unless there's something in all that which denigrates the Popular solutions, I would just let it be for now. Glkanter (talk) 12:17, 15 December 2009 (UTC)
- The article is not protected. Anyone can make whatever change they'd like, and to a large extent making a change and seeing that it is not reverted is how consensus is demonstrated (see WP:BRD). I think you should also really read WP:BATTLE. -- Rick Block (talk) 14:46, 15 December 2009 (UTC)
How do we resolve the inconsistency between the Contestant's POV in the MHP, and the "unknowns'" POV in all the host variant stuff? Especially that large table. The whole thing makes no sense to me. Glkanter (talk) 15:08, 15 December 2009 (UTC)
- Are you suggesting this section is not a neutral recounting of what reliable sources say? Or is your problem that you don't understand it? -- Rick Block (talk) 15:40, 15 December 2009 (UTC)
- Like many things in the article it is inconstant from a logical standpoint. The MHP, as you know, is from the contestant's POV. All these so-called 'variants' are solved from some 'unknown' person's POV. So that's inconsistent. Even worse, since it's not the contestant's POV, it's not the MHP. I just don't see much value there. Maybe a section with links, and that's it.
What has to be considered the MHP??
I very much like this question to be answered first. (BTW several sources mention the MHP to be equivalent to the three prisoners problem.) Nijdam (talk) 22:35, 15 December 2009 (UTC)
- I created a "construction" site, where we may step by step build the article, until we come at a point we don't find agreement on. —Preceding unsigned comment added by Nijdam (talk • contribs) 16:41, 16 December 2009 (UTC)
- Nijdam, have you joined the consensus? Is mediation still necessary? What conclusions did you and Boris reach? Please address my points from this diff Huckleberry & Awareness and these questions before we invest all the time and effort. Glkanter (talk) 16:51, 16 December 2009 (UTC)
- I think everyone would like an answer to that question but it is unlikely that there will be agreement on the subject. I would say that there are several formulations of the problem. I would personally like the MHP be treated initially as a mathematical puzzle, formulated in such a way as to make the solution simple and not depend on conditional probability. In my opinion the MHP is fundamentally a simple problem that most people get wrong, this is undoubtedly its most notable aspect. This was obviously Gardner's intention with the TPP although you might still argue that this problem is still, strictly speaking, conditional. The difficulty with implementing my preferred approach here is that there are no sources that specifically treat the problem in that way, unless we can find one.
- Without doubt, the most notable problem statement is Whitaker's but, as you are well aware, this leaves so much unsaid that it can be interpreted in many different ways. In my opinion vos Savant answered the question correctly (but failed to make clear exactly what the question was) whilst Morgan interpreted the question too strictly (and still ambiguously) and then answered their interpretation (for the most part) correctly.
- The only unambiguous problem formulation that I know of in the literature is the Krauss and Wang version that we quote. This is exactly equivalent to the TPP but this suffers the problem that, although the effect of conditionality is negated by the host's random choice, it can still be argued (as you do) that the problem is still strictly one of conditional probability and thus, unfortunately, it is not amenable to a simple solution.
- So, I regret that I have failed to answer your very important question. The lack of an answer is the cause of much of the argument here. I think we just have to argue it out as best we can to create the best article possible. Martin Hogbin (talk) 11:00, 16 December 2009 (UTC)
- Here is the Monty Hall problem:
- "Suppose you're on a game show..."
- The symmetry is a premise. And needs no disclaimer or footnoting. Anybody that disagrees should tell me what's wrong with Huckleberry's approach and must answer, incorrectly, that 'the contestant is aware' to Is The Contestant Aware? Glkanter (talk) 11:22, 16 December 2009 (UTC)
- Here is the Monty Hall problem:
- From Wikipedia's perspective, the MHP is whatever reliable sources say it is. WE don't need to (in fact, we don't get to) decide. What we do need to do is say what reliable sources say about whatever they consider it to be.
- As I thought I made clear above, reliable sources do not answer the question of what exactly is the MHP. We have at least Selvin's original statement, Whitaker's question, vos Savant's partial formulation, Morgans misiquotation and subsequent incomplete formulation, and several formulations in K&W. The only one that is unambiguous is the one we quote from K&W, but K&W make no claim that this is The MHP. Thus reliable sources do not answer the question and we must decide what the subject of our own article is to be here.
- Martin - you keep saying it's a simple problem that most people get wrong and that approaching it as a conditional probability problem is (more or less) a nuisance. If it's so simple, why (in your opinion) do most people get it wrong? My answer to this is that they try to solve it conditionally (what was it in the K&W experiment - 35 out of 36 subjects consider only the case where the player has picked Door 1 and the host has opened Door 3), which means to me that we can't fully explain it without addressing this issue.
- Yes we can, there is no evidence anywhere that anyone considers it important which door the host opens. K&W's main point is that giving door numbers just confuses the issue, and I agree. Martin Hogbin (talk) 18:14, 16 December 2009 (UTC)
- Regardless of the above, since reliable sources address it both ways the article must also. Not to do so would imply a POV that the "unconditional" approach is better or more correct. Avoiding this POV is the reason I'm suggesting we go back to a single Solution section. JeffJor seems to be OK with this (since he's engaged in editing a proposal for such a suggestion). I can't tell if Glkanter is OK with this or not. You (Martin) are at least currently saying you're not OK with this (is that right?). So, directly, would both of you (Nijdam, too) be OK with a single solution section more or less like JeffJor's suggestion? -- Rick Block (talk) 15:04, 16 December 2009 (UTC)
- Rick, I didn't realize I hadn't made my position clear. The 'Solution section' will contain simple solutions only. No disclaimers, no footnotes, no 'buts'. This is what the consensus has agreed to. Your NPOV threats are just that. They are intimidation, and an attempt at prior restraint. Just more filibustering, as usual. And Rick, nobody is buying your NPOV act. The current article has such a heavy Rick Block/Morgan bias it's laughable. The FAQs alone make me want to vomit. Just getting to NPOV from your extremes will be an accomplishment. I can't conceive of the article ever being so POV-ed in the other direction as great as you have accomplished. I agree, as you posted on the Mediation Cabal page, this discussion is totally out of control. How many times does the consensus have to tell you your interpretations are inconsistent with the consensus? Glkanter (talk) 15:20, 16 December 2009 (UTC)
- Whoa, whoa, whoa. Rick, you have a sad way of imnposing your own POV in every possible way that lets you keep it in the article. It is downright insulting. I am not "OK" with considering the Morgan "conditional" approach as addressing the MHP. It does not. Seymann says it does not. Nobody, not even Morgan, says it does address the actual question. But I am a realist, and I recognize that you will not let it be separated AS IT SHOULD BE, because you feel justified in your POV because your misread those sources and think they say something they do not.
- So I am "OK" with trying to improve the article in a way that can be accomplished. That means inculding your un-intended, not-MHP, POV in the article. Specific door numbers were never intended to be important, and any possible importance implied by the "reliablle sources" you quote, who demonstratably misrpresented the MHP, was for rigor only. It gets removed by those sources before actually addressing the MHP with their formula. IT HAS NOTHING TO DO WITH THE MHP. IT IS NOT REPRESENTATIVE OF THE MHP. IT DOES NOT HELP IN UNDERSTANDING WHY THE MHP IS CONTROVERSIAL. All it does is extend the thought problem, in a way that is interesting only to mathematicians. The body of the article needs to address the problem as seen by the general populace, not pedantic mathematicians who proved one assumption (of two similar ones that are normally made) was unnecessary. While interesting, being unnecessary doesn't explain the unintuitive nature of the problem. JeffJor (talk) 22:46, 16 December 2009 (UTC)
- It seems like we must not be talking about the same thing here. I'm talking about a conditional probability analysis, like the one you wrote above (the paragraph starting "Another way to analyze the problem ..."). Which I think is exactly the way the general populace, not pedantic mathematicians, see the problem. What are you talking about? -- Rick Block (talk) 23:07, 16 December 2009 (UTC)
- Jeff - if you can reply I would appreciate it. My assumption was that since you wrote the above paragraph that you would be OK with putting it (or something like it) in the article. If that was not your intent I'm sorry to have misinterpreted. -- Rick Block (talk) 01:53, 17 December 2009 (UTC)
- Rick - Just to be clear, there are four categories that different sources have used to approach the MHP. I will call them "Unconditional Approach," "Symmetric Conditional Approach," "Asymmetric Conditional Approach," and "Reduced Asymmetric Conditional Approach." UA, SCA, ACA, and RACA for short. And there can be different kinds of RACAs, depending on what gets "reduced," by which I mean eliminating the importance of a condition.
- The UA supports either MvS's explanation, or Devlin's combined doors, or anything similar. It really doesn't utilize the doors in any specific way. It can be done rigorously (G&S do it in their first solution not attributed to MvS), but usually isn't. Non-rigorous solvers use symmetry to reduce G&S's twelve cases to four. The SCA looks at specific doors, but assumes that any uncertainty must be uniformly assigned as in G&S's second solution. Some UAs - those that mention door numbers - look like SCAs but are really using numbers only as examples. The "tree" version as it currently exists did this. You can tell because it has four cases that sum to P=1, not twelve cases as in G&S. ACA is stated in K&W (not Morgan, Gillman, or G&S since all ignore car placement bias) as Equation 1. But it is not useable to answer the question "Should she switch?" because of the placement bias. Morgan, Gillman, and G&S use an RACA where they make one reduction - they eliminate placement bias by assumption. K&W discuss several ways of reducing ACA by assumption - their no/one/two door solutions. Morgan's thesis is not that ACA is proper (although they mistakenly assume it), but that you don't need to assume anything to reduce the importance of host choice. They reduce it by making it unimportant to the quesiton, and that is still a reduction. Eventually, every source that considers an ACA reduces it to a question (not a probability value) that is answered independent of door numbers.
- So, what I think is that UA needs to be the primary focus of the article. SCA can be used, or not; but if you insist on it, it needs to be done properly as G&S did. That's why I changed the probabilities in the discussion I wrote of it. I don't think you will let anybody take it out of the body completely, so I left it and did my best to improve it. But it really does not help non-students of probability. They don't understand that conditional probability depends as much on what is removed as what is left. The terms in the SCA formula can show this, but the non-student will not understand how to read such a formula. So it really doesn't help them, it just makes the article look like a textbook that they don't want to read. And any ACA is addressing a variant of the MHP where asymetric probabilites need to be considered. Few people think the problem says that, and fewer still think they can be used. Certainly not any of the references, who always reduce ACA completely. So I don't see any benefit from that to the general public. There is benefit to students of probability, but that benefit is not directly related to the MHP itself. It only shows how you don't have to make assumptiosn to reduce all of the conditiosn that might affect a strategy. So it can be included AS LONG AS IT IS CLEARLY SEPARATED FROM THE ACTUAL MHP DISCUSSION. And it needs to be made clear that it is a non-standard interpretation of the problem, and that the parts that make it diffewrent are never used directly to answer the MHP.
- I hope this helps you and Nijdam. It is the ACA and RACA that belong as a variant. The SCA is "better" solution in the sense of rigor only; but it does not satisfy any need the article has, except rigor. Since the reason the MHP is enigmatic at all has nothing to do with rigor, but with intuition, we really should pay more attention to the issues that surround intuition. JeffJor (talk) 15:35, 17 December 2009 (UTC)
- I don't need help. What you call UA, suggesting an approach to the MHP, is just a solution of a specific simple version, not the K&W-formulation, and in my opinion also lacking the characteristics of the MHP. This is also admitted by G&S: This very simple analysis, though correct, does not quite solve the problem that Craig posed. The SCA and ACA are both solutions to the MHP (K&W version). Nijdam (talk) 12:55, 20 December 2009 (UTC)
- (1) Nijdam, elsewhere you had asked for my interpretation. When I "hoped it helped" you, I meant "helped you understand my interpretation." (2) But you apparently do need my help, because you think something other than UA/SCA is "the MHP." As has been clearly demonstrated through references,UA/SCA is indeed what was intended as "the MHP" by the originators, whether or not some others misinterpreted it. And all of the controversy in the general public surrounding those puplications stem directly from, and only from, it. Anything else is a distraction. The fact that some sources disagree on this point only proves that we need to handle that disagreement by separating the issues. Craig's problem can solved, as written. This is clearly admitted by Seymann. It is G&S's alternate interpretation that cannot be solved. They only supply a solution to an isolated example that they do not claim is as a valid solution the problem itself. And in fact, G&S is the model for how to do this. They separate their approaches the exact same way. JeffJor (talk) 13:04, 21 December 2009 (UTC)
- Nobody can solve the problem that Craig posed because it is not clear exactly what it is. Morgan interpreted it in one particular way that makes the problem clearly conditional. In fact he probably just wanted to know, 'What is the probability you win if your strategy is to switch?', as Morgan put it. Martin Hogbin (talk) 15:32, 20 December 2009 (UTC)
- Incorrect, Martin. Read Seymann's comment to Morgan. It was not supposed to be a rigorous prob=l;em statement, it was a "fun" puzzle in a newspaper. And there are clear assumptions that can, and should, be made; which were reinforced by MvS herself. The controversy had nothing to do with any of those possible ambiguities. JeffJor (talk) 20:55, 20 December 2009 (UTC)
- Jeff it would probably help our cause if you were to stop jumping down my throat at the first opportunity. I was the first, I believe, to bring to the attention of editors here Seymann's commentary, which points out that the problem can be interpreted in different ways. I am agreeing with you that Whitaker probably just wanted the simple unconditional problem answered, with normal 'puzzle' assumptions (as correctly made by vos Savant). That is why I quoted from Morgan's example of an unconditional statement of the problem. Even that 'most reliable of sources' gives the problem to which the simple solutions, including vos Savants are the answer; that is what I have quoted from. Martin Hogbin (talk) 21:23, 20 December 2009 (UTC)
- It would help if we were all on the same page. Any ambiguity in the Parade statement is unimportant to what makes the MHP controversial. Considering such is what leads some people to think the Morgan analysis should be part of the main issue. The point is, that the Whitiker statement is sufficient for the vehicle in which it was published. Seymann does not say it "can be interpreted in different ways." He says, and I quote, "Simply put, and quite clear considering her suggestions for simulation procedures in her two later columns, the host is to be viewed as nothing more than an agent of chance who always opens a losing door, reveals a goat, and offers the contestant the opportunity to switch to the remaining, unselected door." There is nothing vague about this. Craig's problem can be solved, without having to allow for any alternate host strategies.
- Rick (and others) keeps trying to inject his POV by suggesting Seymann thought it was ambiguous, and Seymann did not. And MvS said explicitly that the vast majority of the controversy in the letters had nothing to do with such possibilities. Your comment came in the middle of an "edit battle" where Rick reworded the article to inject that POV, and I removed it. As long as they keep trying, NPOV requires it be squelched. It is only this way that we can eliminate POV from the article. JeffJor (talk) 23:11, 20 December 2009 (UTC)
- I will leave you to your lone battle as you seem determined to pick fights with those who essentially agree with you. I was arguing that Craig's question should be treated in a manner sympathetic to its origin long ago. Martin Hogbin (talk) 00:12, 21 December 2009 (UTC)
- Incorrect, Martin. Read Seymann's comment to Morgan. It was not supposed to be a rigorous prob=l;em statement, it was a "fun" puzzle in a newspaper. And there are clear assumptions that can, and should, be made; which were reinforced by MvS herself. The controversy had nothing to do with any of those possible ambiguities. JeffJor (talk) 20:55, 20 December 2009 (UTC)
- Nobody can solve the problem that Craig posed because it is not clear exactly what it is. Morgan interpreted it in one particular way that makes the problem clearly conditional. In fact he probably just wanted to know, 'What is the probability you win if your strategy is to switch?', as Morgan put it. Martin Hogbin (talk) 15:32, 20 December 2009 (UTC)
- I'll note that I didn't "reword" anything. Butwhatdoiknow's complaint was the length of the lead. There was a pending suggestion on the talk page to drop the K&W problem statement in favor of the Parade problem statement. He dropped the Parade one. I simply flipped this to the (existing) paragraph about the (much better known) Parade one, with one minor change to make it less POV (in the same direction as JeffJor's subsequent edit). Here's a diff to an earlier version [2]. -- Rick Block (talk) 05:32, 21 December 2009 (UTC)
- Rick said "I didn't 'reword' anything ... with one minor change to make it less POV same direction as JeffJor's subsequent edit." Ignoring the self-contradiction, your rewording was in the opposite direction. You added "Some of the controversy was because the Parade version of the problem leaves certain aspects of the host's behavior unstated." You said this as though it was a significant portion of the controversy, which is what MvS specifically denied and what you have no support for. The only thing that contributes to the controversty soem have called "The Parade affair," which is the controversy meant here, is the unintuitive result. You left out the parts that said the problem statement was perfectly clear for the forum in which it was published, which is the "direction of my edits." JeffJor (talk) 12:46, 21 December 2009 (UTC)
- Sorry, wrong diff. The version I started with was this one. This is the correct diff [3]. I changed "A well-known, though ambiguous (Seymann 1991), statement of the problem was published in Parade magazine:" to "A well-known statement of the problem was published in Parade magazine:". The "Some of the controversy" sentence is in both what I started with and your subsequent edit. In general, references do not belong in the lead (see WP:LEADCITE). -- Rick Block (talk) 15:09, 21 December 2009 (UTC)
- I don't care what the diff is, Rick. If you edit a paragraph that has two known flaws in it, and take out only one flaw, you are tacitly approving the other. There is no documented evidence that the MvP statement generated that controversy. She denied it was a signifcant factor. Others inserted the possibility (not the controversy), by failing to interpret the Whitiker statement as it was clearly intended. That means no alternate host strategies, and no probabilities that differ by door number alone. So you can solve the problem by UA/SCA, which are equivalent as G&S said. The only place where any controversy has arisen surrounding anything besides UA/SCA is HERE. That isn't NPOV. It is OR. It does not come from any of your references, because they never said what q was. They only said there ewas a potential for it to make a difference, but it didn't affect the answer. JeffJor (talk) 17:29, 21 December 2009 (UTC)
- I agree intuition is the problem and that rigor is not the answer. IMO (this is WP:OR) the problem statement deliberately forces the reader to think about the specific conditional case where door 1 has been picked and the host has opened door 3, so the player is now looking at two specific closed doors and an open door. And, yes, this case is used as an example.
- The salient features of this case (which apply to any other) are 1) the player doesn't know where the car is with certainty, and 2) there are only two possible choices. The "equal probability" assumption (cf. Falk or Fox and Levav) strongly leads people to the conclusion that the odds must be 50/50 in this case, and therefore any other equivalent case. Note that this reasoning starts with a specific conditional case, and then extends to the unconditional answer not the other way around.
- The unconditional solutions ignore the specific conditional case the problem statement has forced the reader to think about, and jump straight to the (correct) unconditional answer. However, they NEVER reconnect back to the original conditional case—that is, these solutions do not address the mental model most people construct which led them so convincingly to their initial 50/50 conclusion. This is basically a bait and switch approach, leaving people with two choices - trust their "equal probability" intuition, or believe a solution that seems to be true but doesn't specifically address why or how their intuition failed. I think this is precisely what leads to many of the arguments over this problem. Most people are very reluctant to abandon what they see as an intuitively obvious answer. The unconditional solution approach tries to lead people to a different mental model. The other alternative is to address the conditional case head-on, and explain why even in this case the odds are 1/3:2/3. I would like the article to do both of these, in one solution section. As Boris says "The coexistence of the conditional and the unconditional can be more peaceful". -- Rick Block (talk) 21:27, 19 December 2009 (UTC)
I'm Ready for Formal Mediation
I suggest we quit waiting around for the informal mediation. There may never be a volunteer.
Formal Mediation is the next step, then arbitration.
Is there a second to my motion? Glkanter (talk) 15:26, 16 December 2009 (UTC)
Is Rick's diff out of control consistent with our efforts to find an unbiased informal mediator? Or has he made that impossible?
- "==Please help==
- "The situation at talk:Monty Hall problem is really getting out of control..." Rick Block (talk) 15:34, 15 December 2009 (UTC)"
Glkanter (talk) 20:22, 16 December 2009 (UTC)
- I've asked the chair of the mediation committee what a reasonable length of time to wait for an informal mediator might be before proceeding with a formal mediation request. See User talk:Ryan Postlethwaite#Mediation question. -- Rick Block (talk) 02:06, 17 December 2009 (UTC)
Wiki-Ego
I can see why someone who has a great deal of time, effort and pride invested in Wikipedia would be protective of his work. Especially if the only Featured Article which he personally 'sheparded' through the review process was at risk of being dramatically revised. Revised so much, that the FA designation would likely be at risk.
But there is a difference between understanding and condoning. In addition to all the filibustering that takes place on the talk pages, these phrases were used when requesting (supposedly) un-biased assistance as per Wikipedia policy:
- "...This is a featured article that has been through 2 FARCs..." - Request to Mediation Cabal
- "...The situation at talk:Monty Hall problem is really getting out of control..." - 2nd request to Mediation Cabal
- "...At least one of the other editors involved is agitating to proceed with formal mediation..." - 3rd request, sent to Ryan Postlethwaite, Chairperson of the Mediation Committee
So, I can see why an owner of an article would reject all good faith efforts at improved clarity. I just don't agree with the continual passive-aggressive intellectual dishonesty that I have witnessed throughout the 14 months I've been active on this article. I've got a lot of time, effort and pride invested in this article, too. And I'm part of a legitimate consensus for making the proposed changes. That's why I point out, without hesitation, when I think another editor is not behaving in good faith. These aren't personal attacks. They are a recognition of why the article has been so wrong, for so long, despite the efforts of so many 'agitating' editors who disagree with the "shepard's" POV.
Some people will argue that this discussion is out of line. But everything I wrote is supported by diffs. Why fear the truth? I would reply that the criticisms would come from those who favor the status quo for the article. Glkanter (talk) 13:05, 17 December 2009 (UTC)
- This rant is nothing but another in your continuing series of disruptive edits. Please stop. -- Rick Block (talk) 13:43, 17 December 2009 (UTC)
- Call it 'disruptive' if you want. I call it honest. There's been no edit warring by anybody. No profanity on the article's various talk pages. Nothing disruptive at all, other than your endless filibusters, despite a consensus to proceed. Just things you find uncomfortable. All these discussions show good faith by many people that they're looking for a proper Wikipedia solution. The only 'disruption' was Dicklyon's unprovoked savage violation of my MHP talk page new section. I noticed you didn't admonish him at the time. Not a peep out of you, the self-appointed Monty Hall problem Admin and Mediator. But you told a buddy, whom you may have been trying to recruit to mediate this dispute, that 'I chased him away'. Dicklyon takes no responsibility whatsoever for his own reprehensible actions, and you defend him. That's intellectually dishonest. And you know it. Glkanter (talk) 15:12, 17 December 2009 (UTC)
Does anyone object to Formal Mediation?
"Mediation is a voluntary process in which a neutral person works with the parties to a dispute. The mediator helps guide the parties into reaching an agreement that can be acceptable to everyone. When requesting formal mediation, be prepared to show that you tried to resolve the dispute using the steps listed above, and that all parties to the dispute are in agreement to mediate. Mediation cannot take place if all parties are not willing to take part. Mediation is only for disputes about Article Content, not for complaints about user conduct."
Please indicate below:
I am willing to take part in Formal Mediation
- Glkanter (talk) 15:03, 18 December 2009 (UTC)
- Rick Block (talk) 16:03, 18 December 2009 (UTC)
- Martin Hogbin (talk) 16:34, 18 December 2009 (UTC)
- JeffJor (talk) 16:35, 18 December 2009 (UTC)
- Gill110951 (talk) 13:22, 20 December 2009 (UTC)
- Colincbn (talk) 02:26, 22 December 2009 (UTC)
I am not willing to take part in Formal Mediation
Thank you. Glkanter (talk) 15:03, 18 December 2009 (UTC)
Nijdam posted an edit to this page nearly 24 hours ago. How much longer do we wait for him to indicate his decision? Glkanter (talk) 11:56, 21 December 2009 (UTC)
Nijdam posted an edit to this page over 48 hours ago. Still no comment or signature on this Formal Mediation. Can we move on without his signature? Is anybody else ready to move this forward? Glkanter (talk) 13:11, 22 December 2009 (UTC)
- I thought we were proceeding. This seems much more important than the RfC below. Martin Hogbin (talk) 13:15, 22 December 2009 (UTC)
- Well, we were. But there's no Formal Mediation unless everyone agrees to it. I see us as stuck, for no apparent reason.
- Yes, that RfC is a distraction. Unfortunately, for me, anyways, it has to be dealt with seriously. I knew this was coming the minute he vandalized my talk page edit, and begged Dicklyon to let me edit my own section as I had written it. All to no avail. So it goes. Glkanter (talk) 13:23, 22 December 2009 (UTC)
The parties that need to agree are the parties listed in the request for mediation. As far as I know this does not exist yet. After creating such a request you notify the named parties about it, see Wikipedia:Requests for mediation/Guide to filing a case. Pre-agreeing to mediation here is nice, but ultimately irrelevant. The parties named in the informal request were Martin, Jeff, Glkanter, Nijdam, Kmhkmh, Father Goose, and myself. I've been assuming Glkanter or Martin were working on a formal request. If this is not the case I'd be willing to write one up, although if someone else would prefer to do this that's fine with me. I might suggest Martin. -- Rick Block (talk) 17:09, 22 December 2009 (UTC)
- It would have been nice of you to present this view either when you signed, or instead of signing, back on the 18th. Why go to all the effort of creating a request until Nijdam, and Kmhkmh indicate they will go along with the decision? I have serious doubts that Nijdam will agree. How hard can it be for them to say? Just more stalling of the inevitable. Excellent job this time, Rick! Glkanter (talk) 17:23, 22 December 2009 (UTC)
- Based on the level of hostility you've exhibited toward me, I thought you would prefer someone else write up the mediation request. That's what this edit (from last Thursday) meant, where I provided a link to the appropriate procedure. I assumed you'd read this and that you or Martin were working on it. Per below, Martin is OK with me writing it up. Are you? -- Rick Block (talk) 20:09, 22 December 2009 (UTC)
- Rick, I am not familiar with the procedure, and would be quite happy for you to do it. If you want me to do it let me know and give me some clues what to do. Martin Hogbin (talk) 17:56, 22 December 2009 (UTC)
Informal mediation still a possibility
I note that User:K10wnsta has offered to serve as an informal mediator for the case at Wikipedia:Mediation_Cabal/Cases/2009-12-06/Monty_Hall_problem#Discussion.
If nobody had an objection to proceeding with the case at that venue, it would probably allow things to get under way sooner.--Father Goose (talk) 04:19, 24 December 2009 (UTC)
Re: Wikipedia:Mediation_Cabal/Cases/2009-12-06/Monty_Hall_problem
In my brief overview (I haven't delved into archives), the discussion appears to have remained civil and, more important, cooperative in seeking a means of negotiation. If everyone is willing to excuse the sluggish response to your request for informal mediation (blame it on the holidays ;) ), I'd be happy to work with you in resolving the dispute. However...
You waited over two weeks for assistance at MedCab and, procedurally, are justified in pursuing formal mediation. If someone has already applied significant effort in preparing for that, I understand if you wish to continue in that direction.
--K10wnsta (talk) 20:29, 25 December 2009 (UTC)
- There are 11 archives to this discussion, covering 6 years. While technically 'civil', it is very contentious. I appreciate your offer to help, but I'm not sure there would be a result worth the time investment you would need to make. Honestly, if I may, your original comment about this puzzle being 'mathy' did not create confidence in this reader. And I'm the least Mathematics educated person on this talk page.
- But, this is just one person's opinion, offered in good faith. I'd hope I can support whatever the consensus decides on your generous offer. Glkanter (talk) 17:33, 26 December 2009 (UTC)
- Hehe, well, the 'mathy' description stemmed from reading about a dispute involving 'mathematical sources' and 'conditional probability' in what appeared to be an article about the host of a campy game show. I couldn't fathom how the subjects were related (and even questioned my recall of the host's name). It was certainly not intended to express any personal disdain for mathematics - in fact, I enjoy and excel in most math-related fields (notably algebra, geometry, and statistics).
--K10wnsta (talk) 21:07, 28 December 2009 (UTC)
- Hehe, well, the 'mathy' description stemmed from reading about a dispute involving 'mathematical sources' and 'conditional probability' in what appeared to be an article about the host of a campy game show. I couldn't fathom how the subjects were related (and even questioned my recall of the host's name). It was certainly not intended to express any personal disdain for mathematics - in fact, I enjoy and excel in most math-related fields (notably algebra, geometry, and statistics).
- As one of the long term involved editors I would welcome some mediation. I am not sure what your understanding of maths and probability is like but the Monty Hall problem has been described as the world's most tenacious brain teaser. It will therefore be necessary for you to first get your head round the basic problem, if you are not already familiar with it. Note that nobody here disagrees with the basic numerical answer under the 'standard rules'.
- If you proceed with mediation, it would be interesting for you to start by reading the article (or possibly a previous version ) through to see how good an understanding of the problem it gives you before consulting other sources or talking to anyone about it, as the debate is essentially about how well this article addresses the basic problem. You currently have the advantage of seeing this article as a newcomer but once you have been drawn into the debate you will quickly lose that viewpoint. This suggestion is not intended to be an attempt to 'get in early' with my POV. Perhaps someone on the 'other side' could confirm that they would be happy for you to take this approach. Martin Hogbin (talk) 11:02, 28 December 2009 (UTC)
- I have read the article and now understand how Monty Hall could be associated with conditional probability (see my reply above). I haven't yet delved into the actual dispute as I prefer remaining sequestered from it until we get past the informal formalities (eg. all interested parties agreeing to participate in the mediation process).
--K10wnsta (talk) 21:07, 28 December 2009 (UTC)
- I have read the article and now understand how Monty Hall could be associated with conditional probability (see my reply above). I haven't yet delved into the actual dispute as I prefer remaining sequestered from it until we get past the informal formalities (eg. all interested parties agreeing to participate in the mediation process).
Kanov Is Wrong
The article states (without citation) that Kanov stated that in the "Ignorant Monty" case, swapping still yields a 2/3 chance of winning - but a quick simulation of all cases reveals this to be wrong: suppose I pick door 1, and Monty opens door 2 without knowing what is there but reveals a goat (all other permutations are equivalent to this): the car will now be behind either door 1 or door 3 with a 1/2 probability. --New Thought (talk) 09:44, 19 December 2009 (UTC)
- You are quite right. Because of all the argument here nobody has noticed a simple error. There seems to be a section based on a the supposed opinion of a mysterious Kanov. I will remove this unless someone can explain why I should not. Martin Hogbin (talk) 10:54, 19 December 2009 (UTC)
- I've wondered about this, going back to the summer. If I recall correctly, Marilyn vos Savant says it's 1/2 because of the plays that get eliminated by Monty revealing a car. I might suggest that once the contestant is faced with the two doors and a revealed goat, it's the same 1/3, 2/3 as the original MHP. Then I have to figure out how this is consistent with Deal or no Deal, which says there is no advantage to switching.
- Vos Savant is correct. If Monty chooses any unchosen door randomly you have to decide what to do if he reveals a car, asking the player whether she wants to change after a car has been revealed is pointless. Easiest would be to replay those games from the start. Games where Monty reveals a car are therefore discounted. These games can only be ones where the player has originally chosen a goat because, if the player has originally chosen the car, the host cannot reveal it. Thus in the 'Ignorant Monty' case we remove some games where the player originally chose a goat but none where she originally chose the car, thus her chance of having originally chosen the car goes up. Martin Hogbin (talk) 18:56, 19 December 2009 (UTC)
- I've wondered about this, going back to the summer. If I recall correctly, Marilyn vos Savant says it's 1/2 because of the plays that get eliminated by Monty revealing a car. I might suggest that once the contestant is faced with the two doors and a revealed goat, it's the same 1/3, 2/3 as the original MHP. Then I have to figure out how this is consistent with Deal or no Deal, which says there is no advantage to switching.
- Martin - I don't think you're addressing the issue. I believe the confusing scenario is a specific show, say last Tuesday's, where Glkanter was the contestant. On this show, he's initially picked a door, say Door 1, and Monty has forgotten where the car is. He says "Oh dear, I've forgotten where the car is. I hope this works out OK - Carol, please open a random unchosen door". And, fortuitously, the door that is opened, say Door 3, reveals a goat. There is no decision about what to do if the car is revealed, because the car simply wasn't revealed. Glkanter's initial choice has a 1/3 chance of having been correct. The chance the car is behind the open door is clearly 0. The other one must have a 2/3 chance.
- That is the issue I addressed. I explained why the probability that the player has chosen the car increases when Monty reveals a goat by chance. Even for the one-off case the fact that Monty has chosen randomly but in fact revealed a goat means that the player is more likely to have chosen the car. Martin Hogbin (talk) 20:15, 19 December 2009 (UTC)
- Martin - I don't think you're addressing the issue. I believe the confusing scenario is a specific show, say last Tuesday's, where Glkanter was the contestant. On this show, he's initially picked a door, say Door 1, and Monty has forgotten where the car is. He says "Oh dear, I've forgotten where the car is. I hope this works out OK - Carol, please open a random unchosen door". And, fortuitously, the door that is opened, say Door 3, reveals a goat. There is no decision about what to do if the car is revealed, because the car simply wasn't revealed. Glkanter's initial choice has a 1/3 chance of having been correct. The chance the car is behind the open door is clearly 0. The other one must have a 2/3 chance.
- Glkanter - is this more or less what you're thinking? -- Rick Block (talk) 19:41, 19 December 2009 (UTC)
- The original sources are correct. "Kanov" is presumably the name of the anonymous editor who put this in the article (yesterday). I've reverted this change.
- Glkanter - the probability this is talking about is precisely the one applying to the contestant faced with two doors and a revealed goat (in a case where the host has randomly, but successfully, opened a door revealing a goat). Perhaps Martin or JeffJor could explain to you why the "combining doors" solution (or any of the other unconditional solutions) do not apply, and why the probability is indeed 1/2 in this case. This is not a sarcastic suggestion - I could try to explain it but I doubt that you'd be willing to listen to me. -- Rick Block (talk) 17:24, 19 December 2009 (UTC)
Even though this problem is always described as "counter-intuitive", I find it interesting that EVERYONE on Earth understands the problem intuitively if you look at it another way: When you watch Deal or No Deal, the only reason it's suspenseful is because the person opening a case does NOT know if there's a big number inside that case. If you were on a Monty Hall Problem game show, and picked door #1, and the host said "I'm going to open a door now... hmmm... number 2" (ignorant monty - or at least from the player's POV, you must assume ignorant monty), you would be worried and suspense-filled that he might open the door with the car. When he doesn't, you feel relief. However, if Monty said, "Now, let me open a door with a goat in it... number 2" you would feel no suspense. He has told you the door has a goat, you know it's a goat, and it has no suspense. This is because there is no risk in him opening a door. He will always open a goat door. If your odds of having a goat behind your original selection improved, you'd be excited after he revealed a goat, but because he knows it's a goat, you feel no more excited about your first choice than before he opened the good. This is an example of how people DO intuitively understand this, but then don't recognize the ramifications of this feeling when offered the choice to switch observe below:
- Interesting. Martin Hogbin (talk) 20:19, 19 December 2009 (UTC)
Here is an analysis of all cases when the car is behind Door number 3 (logic dictates that there are tables for the car behind behind doors 1 and 2 that have identical probabilities (for the appropriate doors). The number at left is the door you choose; the number at the top is the door Ignorant Monty opens. The result is whether you should switch ("y" or "n"). "c" represents Monty revealing the car.
1 | 2 | 3 | |
---|---|---|---|
1 | y | c | |
2 | y | c | |
3 | n | n |
1,1 2,2 and 3,3 are greyed out, because he can't open the door you chose. As you can see, there are two cases where switching nets you a car, and two cases when it does not. There are also two cases where he reveals the car ("c") and you are (presumably) not offered a choice, as the car location is now known. Ignorant Monty has a 1/3 chance of revealing a car and ending the game. ONCE that does not happen, there are four possible cases left, 1/2 of which require switching to win, 1/2 of which require keeping to win. This is the conditional probability of "What is the probablity that switching will win GIVEN that Montry did not reveal the car?" The absolute probability is absolutely true - even with ignorant Monty, switching will win you the car 1/3 of the time - 1/3 of the time staying will win, and 1/3 of the time Monty will reveal the car, and you will not get the option.
Regular Monty has 0 chance of revealing a car. While regular monty has a decision to make SOMETIMES (if you select the car, he must pick which goat to reveal), as long as his pick is random, the result of his pick are both the same: you should still not switch, (so the conditional probability of winning by switch IF monty randomly selects one door or the other is 0 in both cases - you can't win by switching). Thus, if you picked right the first time, don't switch. If you picked wrong the first time, DO switch. Therefore, 1/3 of the time, don't switch, 2/3 of the time, switch.
This is true in the ignorany monty case also: If you picked wrong (2/3), do switch. If you picked right (1/3) don't switch. However, half of the time when you pick wrong (half of 2/3 = 1/3), Monty reveals the car, and you don't get to make a choice. Therefore, IF you get the option to switch (only 2/3 of the time will you get this far), then the odds are even between keeping (1/3) and switching (1/3) (the other third is monty reveals the car). TheHYPO (talk) 19:47, 19 December 2009 (UTC)
- As a PS: I thought I'd explain the difference in why one is conditional and one is not: remember that if you have four cases: in order to say that any of them has a 1 in 4 chance of occuring, there MUST be an equal chance of each occuring. In the original monty hall problem (let's say car is behind door 3):
- If you pick door 1 (1/3 chance), he MUST open door 2 100% of the time (thus, also 1/3 chance).
- If you pick door 2 (1/3 chance), he MUST open door 1 100% of the time (thus, also 1/3 chance).
- If you pick door 3 (1/3 chance), he could open doors 1 or 2 (if he picks randomly, 50% chance of either).
- As you can see, your choice of doors all have an equal 1/3 chance of occuring, there are four 2nd step cases ([you:1 monty:2], [you:2, monty:1], [you:3, monty:1], [you:3, monty:2] with DIFFERENT probabilities of occuring (1/3 each for the first two - both of which say "switch", 1/6 each for the second two - both of which say "don't switch"). Thus some people claim that logically, two of those four 2nd step cases say "switch" and two say "stay" - that's 50/50. But two cases occur half has often has the other two. In the Ignorant Monty problem, all 6 cases in my table above are equal probability (1/6). This is because when you pick a "wrong" door, he has two options, not one. so your 1/3 choice results in two 1/6 choices for Ignorant Monty (one of which reveals the car and ends the game). If he DOESN'T reveal a car, you're left with four cases with initial probability of 1/6, and thus, each case NOW has a 1/4 chance (two win by switch, two lose by switching, thus 1/2 chance of winning by switching.) TheHYPO (talk) 20:08, 19 December 2009 (UTC)
Technology required new section.
I believe I correctly summarized vos Savant.
Let's re-apply some things we've learned: 'Suppose you're on a game show...' Still true? Contestant's SoK? 'Random' would equal Deal or No Deal. 'He's drunk' or 'forgetful' might not be communicated to the contestant. Then it's still the MHP from the contestant's SoK.
What exactly is the revised problem statement? —Preceding unsigned comment added by Glkanter (talk • contribs) 20:04, 19 December 2009 (UTC)
- The probability from the contestant's point of view depends on the contestant's knowledge of the game rules. If the contestant is told the host knows what is behind the doors and will always choose a goat then the probability of winning by switching is 2/3 from the contestant's POV (SoK). If the contestant knows the host is choosing another door randomly (and then is relieved to see a goat revealed - see comment above) the probability of winning by switching is now 1/2. Is that your understanding? Martin Hogbin (talk) 20:28, 19 December 2009 (UTC)
By 'random' I mean 'car or goat revealed by Monty'.
I don't thìnk your summary or Rick's summary reflect my thoughts on this puzzle. Have I been obtuse? Why summarize me at all? —Preceding unsigned comment added by Glkanter (talk • contribs) 21:03, 19 December 2009 (UTC)
- It was meant to be an explanation of why the probability of winning by switching is 1/2 if the host chooses an unchosen door randomly (that is to say he might choose a car or a goat). You seemed uncertain as to whether you agree with this statement. Do you agree? Martin Hogbin (talk) 21:16, 19 December 2009 (UTC)
- If the contestant is informed (that is, it's a premise of the puzzle) that the host is opening doors randomly, and may reveal a car, then it's Deal or No Deal. Rick had a very elaborate scenario for the 'drunk' or 'forgetful' Monty. What is communicated to the contestant prior to his decision? Is this still a game show, then? How is it stated as premises?
- I'm just pointing out that 'random', or 'forgetful' still require 'formalized' problem statements, which may be different. Absent that, either, or any answer may be correct. I'm not real good at multi-tasking. I just had some thoughts that could have developed into something. But until we have the underlying MHP squared away, I find this personally distracting. Glkanter (talk) 21:27, 19 December 2009 (UTC)
Many more words
Until now only Boris has shown the derivation of a solution in formulas, using symmetry. This leads to the conclusion - as I BTW showed a million comments ago - that the conditional probability we are interested in is equal to the unconditional and hence may be easily calculated. It doesn't show the conditional probability is not needed. All others come with words, words, .... Nijdam (talk) 17:25, 20 December 2009 (UTC)
- Yes, Nijdam, too many words. How about your signature agreeing to Formal Mediation? Glkanter (talk) 17:33, 20 December 2009 (UTC)
- There are so many words here because people don't explicitly state the assumptions they use to get their solution. Conditional and unconditional probabilies are equal and easy to find by symmetry in a special, nice, symmetric case. I took a look at the Selvin paper. I like the intro very much indeed, I don't like the solution. He does not say in advance what assumptions he is making, you can only guess them by studying his proof. He enumerates the cases and solves the problem by counting. This means that he is assuming that all cases are equally likely. This means that he is assuming the car-key is hidden uniformly at random, that the quiz-player chooses a box uniformly at random independently of the location of the key, and that the quiz-master opens a box uniformly at random out of those available to him, given the previous two choices. Why I don't like Selvin's solution? Because it depends on his strong assumptions. We only need to assume that the first box you pick has 1/3 probability of having the key, in order to guarantee that always switching gives you 2/3 probability of ending with the key. Proof: everytime you would have got the key without switching you don't get it with switching, and vice-versa. I guess that most players think that they have a 1/3 chance of picking the right box first time. Whether or not this is true could be empirically verified. This is both real and theoretical game theory. Gill110951 (talk) 06:06, 22 December 2009 (UTC)
How About A Temporary Editing Freeze On The Article
I don't understand why all this article editing is taking place without being discussed.
While we 'old guys' are working towards a formal WP solution, newer people are editing at will.
This seems unproductive, not good for the article or readers, and distracting.
Any support for a temporary freeze? Is this even plausible? Thanks. Glkanter (talk) 12:25, 21 December 2009 (UTC)
- I think it would be wise for editors to wait, as after mediation there may well be major changes (I hope) and they would the be wasting all their effort. Martin Hogbin (talk) 15:31, 21 December 2009 (UTC)
- I guess I am "a newer person, editing at will". My excuse: when I see factual incorrectness or incompleteness in the existing article I make small edits - I don't touch the main structure. I obviously won't/can't object if those contributions get thrown out later. What I do like is the draft construction page, http://en.wikipedia.org/wiki/Talk:Monty_Hall_problem%5CConstruction That seems to me to be a very useful step: make a fresh start aiming to accomodate the various opinions which are around. It is precisely because there are so many different ways to formulate a Monty Hall problem that it is so attractive. Gill110951 (talk) 19:35, 21 December 2009 (UTC)
Rick Just Filed This RfC On Me
He's expecting Dicklyon to 2nd it. I see a lot of unintended irony here. I had just created a new section on the talk page with 3 edits. Then, here's what I call Dicklyons's unprovoked vandalism on my talk page edits:
It's all right here: Is This Chronology Correct?
So, if anybody wants to put in a good word for me, I'd be much obliged. Please note, I'm pretty sure I will get this promptly dismissed, but any support is appreciated. Glkanter (talk) 04:25, 22 December 2009 (UTC)
- Anyone unfamiliar with this process might want to review Wikipedia:Requests for comment/User conduct/Guidance2. It is certified now which means it won't be closed until the criteria at Wikipedia:Requests for comment/User conduct/Closingis met. Anyone is welcome to comment. -- Rick Block (talk) 05:41, 22 December 2009 (UTC)
On this RfC/U, Rick Block and Dicklyon are trying to make a case that I am disruptive, don't edit the article often enough(?),incivil, interrupt consensus building, chase other editors away, contribute nothing of value, too aggressive with my POV, have bad breath, etc. I'm holding my own on the RfC. It's gotten pretty ugly. So, if anybody would like to drop a supportive word about good ol' Glkanter, now would be a good time. By reading the RfC, you will also learn a lot about the inner thought processes of some well known editors. Thanks. Glkanter (talk) 22:22, 26 December 2009 (UTC)
Please see this new section on the Arguments Page
Thank you. Glkanter (talk) 15:10, 23 December 2009 (UTC)
I'll Bet That 'Paradox' and 'Game Theory' Are Mutually Exclusive And Opposites
I think there are 2 POVs regarding how to 'cherish' the MHP paradox.
Some of us, including myself, love the simplicity. Nothing happens. Heated Arguments over 1/2 vs 2/3 ensue. More than once, even.
Other people like the complexity, and 'what ifs' that the MHP could be with just a little tweaking. The permutations can approach Game Theory scenarios.
Since it was a great paradox before Morgan and conditional, I consider the 'simplicity' people the ones who accurately support how Selvin's MHP paradox should be presented in the Wikipedia Article. Glkanter (talk) 15:35, 23 December 2009 (UTC)
There's Only 2 Things Being Debated Anymore
1. The simple solutions are not solving the correct problem.
2. Morgan's paper, published in 1991, can claim to recognize and describe the Monty Hall Problem Paradox, first published by Selvin in 1975, equally as well (and equally importantly) as Selvin's original paper, which relied only on simple solutions.
I'd like to see the people arguing in support of those 2 arguments come out and directly say it. Once you clearly state your positions, the other editors, using reliably published sources can then address your objections to the proposed changes. Glkanter (talk) 18:32, 23 December 2009 (UTC)
This Is Why They Can't Be Represented in the Article 'Equally'.
Simple solution is not a solution at all
"This is the same topic discussed in more detail three sections down (about the subtly different question), and indeed Morgan et al. argue the "simple" solution is not a solution at all." -- Rick Block (talk) 16:28, 26 October 2008 (UTC) —Preceding unsigned comment added by Glkanter (talk • contribs)
- Many sources do give simple solutions but you try to use one source to veto all others by saying, 'Morgan et al. argue the "simple" solution is not a solution at all'. This is your POV but it is not what sources (note the plural) all say. Some sources give the simple solution as the correct one. These sources should be properly represented in the article. Martin Hogbin (talk) 13:42, 24 December 2009 (UTC)
- I have no idea why Glkanter reposted this old quote. I think we all agree the POV of the article should not be that the unconditional solutions are incorrect. On the other hand, the article does need include the POV expressed by Morgan et al., and Gillman, and Grinstead and Snell (this is their POV - and whether any editor here agrees with it or not is completely irrelevant) that the unconditional solutions are addressing a slightly different problem than what they think the problem is. I think the only question here should be how best to do this in an NPOV manner. What I hear you (and Glkanter and Jeff) arguing is that they're wrong (sorry, per WP:OR and WP:V Wikipedia doesn't care what you think about their POV), or that their POV should be excluded (sorry, per WP:NPOV Wikipedia must include all significant views). I'd be delighted to work toward a more NPOV treatment. -- Rick Block (talk) 17:27, 24 December 2009 (UTC)
- Is it typical for a FA article to need a '...more NPOV treatment.'?
- This is yet another example of exactly the kind of disruptive behavior Wikipedia:Requests for comment/Glkanter is about. I'm offering to help you achieve your goal. What would you say you're doing? I'd call it trolling. I can't speak for anyone else, but I'm extremely tired of it. Please stop. -- Rick Block (talk) 19:53, 24 December 2009 (UTC)
- Actually, Rick, this is another example of you claiming whatever fits your current needs. Here you sound the alarm about the potential for the article to have a POV Last Paragraph. As if the Wikipedia world as we know it would collapse if that happened. But, when you acknowledge that the article currently has a Morgan POV (above), you're not quite as concerned about fixing it in a timely manner. Glkanter (talk) 22:47, 24 December 2009 (UTC)
@Rick, you seem to be putting up an Aunt Sally (Strawman argument). You seem to be implying that I want to remove the POV of Morgan and others who agree with them from the article. That is not the case. I have always suggested that the article should start with the simple non-conditional solutions and then, after discussing these thoroughly, move on to the conditional case discussed by Morgan and others. It is clear, from your reposted quote above (I had not noticed that it had been reposted) that you believe that the Morgan paper should somehow veto or overrule all other sources no matter what they say. Martin Hogbin (talk) 22:54, 26 December 2009 (UTC)
- Am I somehow not being clear here? What I believe is that the article should represent as a POV what it is that Morgan et al. (and Gillman, and Grinstead and Snell) say. What they all say is that the unconditional solutions don't exactly address the problem. Morgan et al. go so far as to say the "simple" solution is a "false" solution. In the quote above, I'm saying that Morgan et al. say this, not that I think this POV should veto or overrule all other sources. Whether you agree with what they say or not, do you at least agree that this is what these sources say? I assume you understand that saying that these sources say the simple solution is no solution is not the same as the article taking this as its POV. -- Rick Block (talk) 00:21, 27 December 2009 (UTC)
I Guess I'd Better Start Editing The Article
In the RfC that Rick Block and Dicklyon filed on me RfC Glkanter one of the 'complaints' was that I argue on the MHP talk pages too much, at the expense of actually editing the MHP article. The associated 'remedy' was that I modify the MHP article more frequently and discuss my reasons for doing so less often.
Now, that's no reason to slap me with an RfC, but the point is well taken. I've asked for a 'freeze' on the article of some sort at least twice in the last couple of weeks. Meanwhile, some editors just make edits without discussing them first.
So, consistent with my stated understanding of the various literature on the MHP, and in accordance with Rick's criticism/suggestion as conveyed via Wikipedia's formal RfC procedure, I will begin to thoughtfully edit the article as I understand the consensus has approved. Glkanter (talk) 15:51, 24 December 2009 (UTC)
How about I start with the FAQs on the talk page? That looks like pure Morgan POV, a clear violation of NPOV. Anybody want to clean it up, or should I just delete it? Glkanter (talk) 16:47, 24 December 2009 (UTC)
- I'm sorry, but how do you find the FAQ a "clear violation of NPOV"? Would it help if it said "according to these sources" a couple of times? There is no particular requirement that talk page FAQs adhere to NPOV, but I'd be happy to work with you to make this more NPOV if it bothers you (which it clearly seems to). -- Rick Block (talk) 17:39, 24 December 2009 (UTC)
- No thanks. I'm going to use the RfC as an opportunity to learn. Since you feel I should be sanctioned because I've only made '6 article edits out of about 1000 talk page edits', I'll go it alone, without all that 'discussing' you find so offensive from me. Glkanter (talk) 22:39, 24 December 2009 (UTC)
Here's another one. Id like to change the 'Simple solution' heading to something like 'Original Paradox explanation' or 'Selvin's Proof' or 'vos Savant's Popular Solution'? I'd like to get the point across concisely that it was this level of understand from which all the excitement about the paradox came. Not to be confused with the 'conditional solution' or, non-solution without the equal goat door constraint being equal to exactly 1/2, that came out some 15 years later. Glkanter (talk) 16:03, 25 December 2009 (UTC)
Then a transition section that says 'For many people, this is all the understanding they need, and was Selvins and vos Savant's point. Others may want to continue further into this article...' And as long as there's no bad-mouthing the 'original' solutions, you 'conditional' guys can pretty much do what you want with the article from there. Glkanter (talk) 16:10, 25 December 2009 (UTC)
FAQ page boilerplate
This is most of the 'greeting' to the talk page of the FAQs. Probably only seen by other editors.
- "This page is an FAQ about the corresponding page Monty Hall problem."
- "It provides responses to certain topics being brought up again and again on the talk page, sapping many editors' time and energy by forcing them to respond repeatedly to the same issues. The FAQ addresses these common concerns, criticisms, and arguments, and answers various misconceptions behind them."
I think this can be improved. Anybody mind if I take a shot at it? Glkanter (talk) 23:32, 26 December 2009 (UTC)
- This is standard boilerplate from Template:FAQ page. Are you suggesting changing the standard boilerplate (used on over 100 pages) or replacing the standard message with something custom for this page? -- Rick Block (talk) 00:29, 27 December 2009 (UTC)
- It's a variant of template:FAQ, one of many templates intended for use on talk pages. See Wikipedia:Template messages/Talk namespace. Template:FAQ2 is another version. - Rick Block (talk) 01:58, 27 December 2009 (UTC)
- No, there is no rule against changing the text in templates. I'm sure there are plenty of people who have these templates on their watchlists. If you make a change anyone objects to they'll revert it. Whatever change you make will show up on every talk page the template is used on, so don't change the text to be less generic. -- Rick Block (talk) 05:01, 27 December 2009 (UTC)
- I'm saying if you edit template:FAQ page the text will appear on any page that transcludes this template (anything marked as "transclusion" here). This is a feature of the MediaWiki software used to run this site. This is both theory (in the sense that it is a known feature of the software) and something I have personally experienced, many hundreds of times. -- Rick Block (talk) 18:50, 27 December 2009 (UTC)
I appreciate your help with this, Rick. I'm suggesting we would edit . What then? Glkanter (talk) 19:27, 27 December 2009 (UTC)
- If you edit Talk:Monty Hall problem/FAQ and change
<noinclude>{{FAQ page}}</noinclude>
- to something else, e.g.
<noinclude>blah blah blah</noinclude>
- "blah blah blah" will only show up on the MHP FAQ page. However, because the text you're talking about is inside the "noinclude" tags it does not appear when you're viewing this page (Talk:Monty Hall problem), even if you click the "show" link at the top of this page (scroll up to the top of this page and try it!). The bottom line is you only see this text if you're editing the FAQ page (and previewing your edit), or directly viewing the FAQ page as opposed to the talk page (there's no link to it, so I'm not sure how this would happen). I might suggest that whatever you think of this text, it's not worth worrying about. -- Rick Block (talk) 19:56, 27 December 2009 (UTC)
Variants
Variants - Slightly Modified Problems section.
Since the MHP is from the contestant's POV, there should be some narrative about what the POV's in this whole section represent. Are they the contestant's? Is it a premise in each different problem that it's no longer the contestant's POV? What about addressing the Monty Hall problem from 'not-the-contestant's POV' for comparison purposes? This would be beneficial to the readers, I believe. Glkanter (talk) 16:55, 27 December 2009 (UTC)
- Can we call this "state of knowledge", not "POV" (to distinguish from the local Wikipedia meaning of POV)? In all cases what is meant is the probability given everything included in the problem statement. This is perhaps most literally the SoK of the puzzle solver, but presumably matches the contestant's SoK as well. The "MHP" is also from the puzzle solver's SoK, so there's really no difference. If this is not clear it wouldn't hurt to try to clarify it, but I don't think anyone should be confused about this since it is how mathematical word problems are universally treated. If it's important to the problem to take some particular perspective, the problem says to. For example, in vos Savant's "little green woman" scenario [4] if the player has picked door 1 and the host has opened door 3 we (the puzzle solver) know the probabilities are split 1/3 (door 1) and 2/3 (door 2) but the question is what are the little green woman's chances of randomly picking the door with the car, not what is the probability the car is behind door 1 or door 2. -- Rick Block (talk) 19:28, 27 December 2009 (UTC)
- In Selvin's and vos Savant's MHP, what the reader knows and what the contestant knows are both consistent with "Suppose you're on a game show..." Every host/producer decision is described as 'random'. That's no longer true with the 'variants' where the reader becomes aware of some host bias. The contestant, of course, cannot. Hence, I disagree with your above explanation.Glkanter (talk) 19:37, 27 December 2009 (UTC)
Editing the MHP FAQs
Rick, the current text includes this:
- "The point of introducing this variant is to show the difference between the unconditional and conditional questions. In this variant, these questions have different answers exposing the difference between unconditional and conditional solutions."
I still disagree that using a different problem is a means of challenging a particular problem. Originalists would argue that all you've demonstrated is the difference between puzzles with different premises. I would further argue that with the contestant being aware of Monty's left door bias, this is no longer the MHP about a game show that Selvin and vos Sovant made so famous. Glkanter (talk) 06:38, 27 December 2009 (UTC)
Are there 3 published solutions?
Selvin's - simple: 2/3 & 1/3, always switching doubles your likelihood of getting the car
Morgan's - conditional, no symmetry: between 1/2 and 1 (?), never to your disadvantage to switch
Morgan's - conditional, with symmetry: 2/3 & 1/3, always switching doubles your likelihood of getting the car
Have I summarized the above properly? Glkanter (talk) 11:03, 27 December 2009 (UTC)
If so, maybe the article could transition from:
Simple, to conditional - with symmetry (they are equivalent), to conditional - no symmetry (leftmost door variant). Glkanter (talk) 11:30, 27 December 2009 (UTC)
- This is the current structure of the article, so I don't get what you're suggesting (change the article to be like the article?). The conditional with symmetry solution dates to Selvin as well. -- Rick Block (talk) 20:24, 27 December 2009 (UTC)
- Rick, rather than make controversial changes or deletions to the article's text, I am trying to make it clearer to the reader how the whole 'Morgan' controversy started. I thought Morgan's whole point was that Selvin and vos Savant overlooked something? So, I'm just suggesting to actually add a 3rd solution section, for increased overall clarity. Glkanter (talk) 20:41, 27 December 2009 (UTC)
Some questions for you all
In an attempt to see exactly who thinks what I have set up some questions on User:Martin_Hogbin/Monty_Hall_problem/dissenters. Everyone is welcome to add their answers. Please comment briefly only in the comment section and have discussions about the questions on the associated talk page.
Whether we have external mediation or not I am sure it will help if everyone answers the questions on this page. I am trying to determine of we have two distinct camps, a single axis of opinion, or just randomly scattered views on the subject. Are there any other questions that editors feel will help sort out the differences of opinion here? I have just added a few extra ones. Martin Hogbin (talk) 11:36, 28 December 2009 (UTC)
Is the term 'Variant' as used in the MHP a common usage?
I disagree with your recent reverts to the article, Rick.
I just checked the Morgan paper, and they do not use the word 'variant' or any derivative of it when describing the problems.
I think this is an uncommon usage, and does not clearly indicate to the reader exactly what is being described. I don't think adding 'Slightly Modified Problem' to a heading, and replacing 1 instance of 'variant' in the article also with 'slightly modified problem' is 'pointy'. Different than your POV, perhaps, but that does not necessarily make it, or any other edits I may make in good faith, 'pointy'. Glkanter (talk) 19:45, 27 December 2009 (UTC)
- Yes, "variant" is common usage. Pointy was referring to the dates. Sevlin's 2nd letter has a conditional solution, so saying the "probabilistic" solution dates from 1991, or is not the "original" solution, or addresses only a variant is a clear attempt to diminish this solution which is not only a violation of NPOV but is factually false. I would appreciate it if someone (anyone) would revert this change. If Glkanter reverts again without further discussion here I'll report him to Wikipedia:Administrators' noticeboard/Edit warring for edit warring. -- Rick Block (talk) 20:15, 27 December 2009 (UTC)
What's Wrong With Adding Dates For Clarity?
Explain the problem to me please. Glkanter (talk) 01:45, 28 December 2009 (UTC)
- Per above, the date you're adding for the conditional solution is simply wrong. Both the "popular" (unconditional) and conditional solutions date to 1975, both to Selvin. If you're going to add dates, you need to add 1975 for both which makes it completely redundant. You seem to be trying to insert your completely made up chronology (#Is This Chronology Correct?) into the article. -- Rick Block (talk) 04:31, 28 December 2009 (UTC)
- So this whole time 'conditional v unconditional' has really been 'Selvin v Selvin'? No way. That's never been your stated intent. Or the way your POV article is written. It's always been 'Morgan v Selvin/vos Savant'. Adding two simple dates to two headings makes it clear where it all started. And it wasn't with Morgan. So the dates help the reader, and do not hurt the article. Your response is not credible based on your previous arguments for many years. Glkanter (talk) 12:56, 28 December 2009 (UTC)
- Rick, you have already acknowledged a Pro-Morgan POV in the article. And offered to help me edit. Why must you continue to confound my very modest efforts at improving the article by removing this POV? Clarifying what a so-called 'variant' is and adding dates are non-antagonistic efforts to improve the article. They just happen to be different than your preference. Your actions, especially calling out to 'anyone' for 'revert' help seem to show ownership issues. Just let me edit out the POV in good faith, OK? Glkanter (talk) 13:20, 28 December 2009 (UTC)
- This whole time, conditional v unconditional has always been different. In Selvin's second letter (which I've previously pointed you to, to refresh your memory there's a copy here), he says he received "a number of letters" including several who "claim my answer is incorrect". Like vos Savant, he says "The basis to my solution is that Monty Hall knows which box contains the keys" but unlike vos Savant he goes on to say "and when he can open either of two boxes without exposing the keys, he chooses between them at random" (emphasis added). Also unlike vos Savant he goes on to present a solution using conditional probability which he calls an "alternative solution" to the solution in his first letter "enumerating the mutually exclusive and equally likely outcomes".
- The issue Morgan et al. address is that vos Savant's solution, and her subsequent defense of it, and the popular discussion at the time (1990 and 1991), completely overlooked the critical assumption that makes the unconditional and conditional solutions the same, i.e. that the host must choose between two boxes (two goats doors) randomly. Selvin knew this and acknowledged it in his second letter. Martin Gardner knew this and addressed it in his version of the Three Prisoners problem. vos Savant blew it, and both Morgan et al. and Gillman called her on it. That's what the "Morgan controversy" is about. Morgan et al. and Gillman both examine the consequences of omitting this assumption, in the process showing why it's critical and how the unconditional and conditional solutions are different. The unconditional solution is NOT saying that every player who switches has a 2/3 chance of winning, but that the average across all players is 2/3. The conditional solution shows that the chances are the same (2/3) for each player only if the "equal goat" assumption is made. Even without it, players who switch will win on average 2/3 of the time (and if they switch they're never worse off), but to say a player who picks door 1 and sees the host open door 3 has a 2/3 chance of winning by switching is a conditional statement and requires this assumption. The assumption can be explicitly part of the problem description (as per the Krauss and Wang version) or implicitly assumed because of symmetry or the principle of indifference, but the statement is still a conditional statement. I don't think ANYONE here (other than you) has ever argued against any of this.
- What I said was "I'd be delighted to work toward a more NPOV treatment". This is not saying that I think the article has a pro-Morgan POV, but that I'm acknowledging that you think it does and I'd be happy to work with you to make it address whatever concerns you have. Adding dates (even if they weren't wrong) is not improving the article or addressing any POV concern. What you seem to be doing is trying to introduce an anti-Morgan POV. That's not how it works. Please read WP:NPOV again, specifically WP:STRUCTURE. You have said repeatedly [5] you want an unconditional solution first and foremost, followed by a disclaimer like "The Monty Hall problem is unconditional. That is the whole paradox; the rest is the explanation; go and learn." This would be sort of the exact opposite of editing out POV. -- Rick Block (talk) 19:08, 28 December 2009 (UTC)
- Let the readers decide amongst the published papers. Just don't cloud the story with unnatural euphemism's like 'variant'. The 3 separate solution sections approach accomplishes my goal, along with dates clearly highlighting the history and clarifying what the heck a 'variant' is. That's not a POV, that's shedding light on the controversy. It just weakens your POV, so you demonize it.
- But anyway, you argue whatever side of the coin is convenient for you each day. What's the point in going around further?
- I've asked user:K10wnsta to drop by and say what he might be able to do as an informal mediator. I'll wait to see what he says. Do not in any way take the fact that I haven't reverted your change (again) to mean I accept it or agree with it. I think it might be helpful if some other folks would comment on this specific change as well. -- Rick Block (talk) 23:39, 28 December 2009 (UTC)
Any support for Arbitration?
I think Rick Block and Nijdam are fillibustering and ownershipping against beneficial changes to this article.
I see no point in waiting for either form of mediation unless Nijdam indicates he will accept the findings.
Rick filed an RfC against me last week, the first item of which is 'only edited the article 1 time.' Now, as you've seen yesterday, every edit I make, he or Nijdam at his request, reverts.
If at least 2 people are with me, I'll proceed. Glkanter (talk) 17:31, 28 December 2009 (UTC)
- Be aware that mediation does not produce "findings". Its purpose is diplomatic -- to help the parties to find points of agreement. In the best of cases, the parties can agree upon a full course of action, thereby resolving the dispute.
- I am almost certain that the Arbitration Committee would not accept the case, since this is primarily a content dispute, and ArbCom rejects content disputes flat out. Just FYI.--Father Goose (talk) 06:20, 29 December 2009 (UTC)
- Father Goose, this is from the Formal Mediation page:
- "Mediation cannot take place if all parties are not willing to take part. Mediation is only for disputes about Article Content, not for complaints about user conduct."
- Since I asked on December 18th if there were objections, and would everyone indicate their agreement to take part, Nijdam has not responded. I felt that rendered the exercise useless.
- Father Goose, this is from the Formal Mediation page:
- The arbitration request would allege that Rick Block and Nijdam are using various filibustering and ownershipping techniques (for example, not replying to the Formal Mediation question) against beneficial changes to the MHP article, as desired by the consensus of editors many weeks ago. I fear some of them have lost interest because of the filibustering. Glkanter (talk) 11:01, 29 December 2009 (UTC)
Mediation
Case link
I've re-opened the case at MedCab and volunteered to assume the role of mediator in a discussion aimed at resolving an on-going dispute here. Additionally, I've issued invitations to participate in the discussion to all involved parties listed in the mediation request. While anyone is welcome to offer input, I ask that those who participate do their best to be concise and refrain from assumption/presumption regarding other's perspectives.
As mediator, my primary goal is to step in as an uninvolved party and help find some common ground from which to proceed. It is not my task to pass judgment on anyone's opinion in the discussion and there is no 'right' or 'wrong' beyond that which is dictated by Wikipedia policy.
I have read the article and understand its subject matter and all it details. As I begin delving through the talk page archives, I'll open the discussion with a call for opening statements. If you feel any archived passages are significant in summarizing the situation, it would help to include links, but please conclude your first post with a Summary of Position (your opinion as it relates to the matter). And remember...concise ;-)
--K10wnsta (talk) 05:28, 29 December 2009 (UTC)
Nijdam's position
I want the article clearly mention the remark made by some sources that the so called "simple solution" is not complete. It doesn't need initially mentioning the technical term "conditional probability". To make my point clear: the following resoning:
- The player, having chosen a door, has a 1/3 chance of having the car behind the chosen door and a 2/3 chance that it's behind one of the other doors. Hence when the host opens a door to reveal a goat, the probability of a car behind the remaining door must be 2/3.
is not complete and better should read:
- The player, having chosen a door, has a 1/3 chance of having the car behind the chosen door and a 2/3 chance that it's behind one of the other doors. When the host opens a door to reveal a goat, this action does not give the player any new information about what is behind the door she has chosen, so the probability of there being a car remains 1/3. Hence the probability of a car behind the remaining door must be 2/3.
Something alike holds for the so called "combined doors solution" and most of the other simple ways of understanding. That's all.Nijdam (talk) 08:36, 29 December 2009 (UTC)
Martin Hogbin's position
The MHP is essentially a simple mathematical puzzle that most people get wrong. At least the first part of the article should concentrate on giving a simple, clear, and convincing solution that does not involve conditional probability. All diagrams and explanations in this section should not show or discuss the possible difference that the door opened by the host might make, although I would be happy to include, 'this action does not give the player any new information about what is behind the door she has chosen' as in Nijdam's second statement above. The first section should give aids to understanding and discuss why many people get the solution wrong, without the use of conditional probability. The first section should be supported by sources which do not mention conditional probability
The simple solution section should be followed by an explanation of why some formulations of the problem require the use of conditional probability, with reference to the paper by Morgan et al. and other sources. It should also include the various variations of the basic problem and other, more complex, issues. Martin Hogbin (talk) 10:19, 29 December 2009 (UTC)
Glkanter's position
I want the article to clearly mention that the remarks made by some sources, that the so called "simple solution" is not complete, is not shared by all sources. It need not mention "conditional probability" beyond saying that due to the symmetry forced by being a game show, the simple solution is equivalent to the symmetric 'conditional solution'.
I think I agree with Nijdam on the text, although they are both OR. It's consistent with my 1st talk page edit, using an IP address in October, 2008:
- Monty's Action Does Not Cause The Original Odds To Change.
- When Monty opens a door, he doesn't tell us anything we didn't already need to know. He always shows a goat. It makes no difference to this puzzle which remaining door he shows. So it starts out as 1/3 for your door + 2/3 for the remaining doors = 100%. Then he shows a door, but we knew in advance that he was going to show a goat. The odds simply haven't changed following his action. They remain 1/3 for your door + 2/3 for the remaining doors (of which there is now just 1).
I'd like to see 3 solution sections: Selvin's simple solution of 1975, transitions to Selvin's symmetrically equivalent conditional solution of 1975 (where the discussion of the simple solution's criticisms occurs), transitioning to Morgan's conditional non-solution of 1991.
I'd like to see the word 'variant' either stricken, or augmented by 'slightly different problem'.
I'd like to see a lot of 'blather' removed from the article. Too much time and effort is spent in the various remaining sections explaining the conditional solution, for no real reader benefit. Glkanter (talk) 10:39, 29 December 2009 (UTC)
And the 'Variants - Slightly Modified Problems' section needs work. The MHP is from the contestant's state of knowledge (SoK). The versions in this section are not. This needs to be normalized for the reader in a few possible ways: An explicit statement that the contestant is aware of these new conditions (in which case these are no longer game show problems), or the explicit statement these problems are not from the contestant's SoK, and a comparison of the MHP from a non-contestant's SoK. Glkanter (talk) 13:14, 29 December 2009 (UTC)
Rick Block's position
First, I think the basic issue is an NPOV issue. The primary question is whether the article currently expresses a "pro-Morgan" POV, i.e. takes the POV of the Morgan et al. source that "unconditional" solutions are unresponsive to the question and are therefore "false" solutions - and, if so, what should the remedy be.
There are a variety of sub-issues we need to discuss but I think the main event is how the solution section is presented. I strongly object to splitting the solution section into separate sections (this was done some time ago, well after the last FARC), which inherently favors whatever solution is presented in the first such section. I mildly object to including the "combining doors" explanation in the solution section rather than in a subsequent "aid to understanding" section.
What I would like is for the article to represent in an NPOV fashion both a well-sourced "unconditional" simple solution (e.g. vos Savant's or Selvin's) and a well-sourced conditional solution of the symmetric case (e.g. Chun's, or Morgan et al.'s, or Gillman's, or Grinstead and Snell's) in a single "Solution" section, more or less like the suggestion above (see #Proposed unified solution section - somewhat modified just now). This follows the guidelines at Wikipedia:Make technical articles accessible, specifically most accessible parts up front, add a concrete example, add a picture, and do not "dumb-down".
Once we address this basic issue I think the other issues will be easier. -- Rick Block (talk) 19:43, 29 December 2009 (UTC)