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That's 100%. [[User:Glkanter|Glkanter]] ([[User talk:Glkanter|talk]]) 23:04, 26 June 2010 (UTC) |
That's 100%. [[User:Glkanter|Glkanter]] ([[User talk:Glkanter|talk]]) 23:04, 26 June 2010 (UTC) |
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And it '''SURE DOESN'T say MUST BE DOOR #3''', does it? [[User:Glkanter|Glkanter]] ([[User talk:Glkanter|talk]]) 23:05, 26 June 2010 (UTC) |
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===Unreferenced Figure In The Conditional Solution section=== |
===Unreferenced Figure In The Conditional Solution section=== |
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Don't test me on this, Rick.
Or anyone else. Glkanter (talk) 18:31, 14 June 2010 (UTC)
Each column of that table comes directly from either the premises or the solution's text immediately above it. There is no need to reduce the clarity of the solution. That does not benefit the reader. Glkanter (talk) 18:37, 14 June 2010 (UTC)
The idea that you would intentionally not account for the car and both goats the player's choice of door #1 and the host having revealed door #3 is foreign to me, and to nearly every other word of the article. Your motivation, therefore, escapes me. Glkanter (talk) 18:40, 14 June 2010 (UTC)
- What are you talking about? The text doesn't say anything about door 1 or "original" and "unchanged" probabilities. The whole point of this paragraph is its extreme simplicity. Simplifying the table to match the text is entirely appropriate. For reference, what I changed it to is this:
Player's door | Door the host opens | Remaining Door | Probability |
---|---|---|---|
Car | Goat | Goat | 1/3 |
Goat | Goat | Car | 2/3 |
- And what I changed it from is this:
Door 1 | Original Probability | Host Has Opened Door #2, or Host Has Opened Door #3 | Remaining Door | Probability (unchanged) |
---|---|---|---|---|
Car | 1/3 | Goat | Goat | 1/3 |
Goat | 2/3 | Goat | Car | 2/3 |
- I really don't understand what you're so upset about. It's virtually the SAME table. It presents the same information in a slightly more compact form (that is a better match for what the text says as well). -- Rick Block (talk) 18:47, 14 June 2010 (UTC)
The other column and heading text are from the premises. The solution is not required to restate them. Glkanter (talk) 18:48, 14 June 2010 (UTC)
I hope these aren't your reasons for making that unnecessary revert:
- You want this new table to resemble the other (criticized) simple solutions, and look less like the 'more rigorous' conditional solutions
- By eliminating the door #s, this table wouldn't exactly match the problem
- By making the above changes, your arguments about simple solutions mis-leading readers and violating NPOV could conceivably have any merit whatsoever
- You do not want the original 1/3, and the ending 1/3 on the same line, or the original 2/3, and the ending 2/3 on the same line as this is definitive proof that Nijdam's never-ending '1/3 <> /13' argument is wrong.
- You are willing to sacrifice the readers' understanding and comprehension of the problem and it's solutions in order to further your POV
Please tell me that I'm wrong, Rick. What are your reasons for changing that table? Glkanter (talk) 19:19, 14 June 2010 (UTC)
The column you removed entirely, the inital 1/3, 2/3 column, was in fact prompted by one of your earliest suggestions:
- "The probability is obviously 1/3 BEFORE the host opens a door, but what this table is now saying is that this is the probability AFTER the host opens a door." - Rick Block
And it's peculiar, Rick. That is the 5th different table you have proposed since I first introduced this table early in the morning on June 12th. What is so wrong with this interpretation of the 'you get the opposite' solution that you would propose 5 various alternative tables to a solution you don't even think has merit? It really makes me question your motives, Rick. Glkanter (talk) 19:39, 14 June 2010 (UTC)
- This table is in support of the solution consisting of the following sentence:
- An even simpler solution is to reason that switching loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3.
- This sentence says nothing about door numbers, but the table does.
- This sentence says nothing about "original" or "unchanged" probabilities, but the tables does.
- What's peculiar here is your insistence to include a complicated table in the simple solution section. My motive here is to have the table match the solution. Reading between the lines, even Martin doesn't like this table. -- Rick Block (talk) 00:57, 15 June 2010 (UTC)
Rick, I would not call a 5 column, 2 row table 'complicated', as you did above. That's BS. As I pointed out to you earlier, whatever is in that table, or the column headings, that does not come from the solution comes from the problem's premises or narrative. There is no gain from repeating them in the solution. Nor is there a requirement to do so. There is nothing in this table that is OR. How could the reader possibly benefit from the remove of vital door # information from the headers, Rick? And without the door #s, which come directly from Whitaker's letter, the solution no longer matches Whitaker's letter. Is that intentional on your part, Rick? "Reading between the lines", it sure looks like that is your motive.
I would suggest you let Martin speak for himself. Or is speaking for others, like reliably published sources, a uniquely special skill you possess? I have found Martin very capable of expressing his thoughts. Haven't you? Glkanter (talk) 01:30, 15 June 2010 (UTC)
- As I said, my motive is to make the table match the solution. That's it. What I, and Glopk, are trying to change the table to matches the solution nearly word for word. -- Rick Block (talk) 01:54, 15 June 2010 (UTC)
Glopk
Maybe you know the way to force the column headings to wrap? Then the table wouldn't be so wide. Yes, I should make the headings consistent either with '#' or without them. Which way do you have a preference for? Why? An 'unchanged' probability? Please, I defer to Martin and Nijdam for that. But that fraction looks the same on the left and the right. Plus, the reader might be surprised that the values didn't change. It is a famous paradox after all. With 'unchanged'; in the heading, they will understand that that's not a typo, or due to any editorial confusion. Glkanter (talk) 20:21, 14 June 2010 (UTC)
Glopk, I have incorporated 2 of your suggestions. I removed the '#'s from the column headings, and I changed column widths for improved readability. I left 'Original' and 'Unchanged' in the column headings for the reasons described above. They are neutral, probability-oriented terms. Thanks. Glkanter (talk) 02:01, 15 June 2010 (UTC)
- @Glkanter. Hmm, let's see. First, I have no idea of what you mean by "neutral, probability-oriented" term. And "original probability"? As good'ol' Inigo Montoya would say: "You keep using that word. I do not think it means what you think it means". My instinct would be to map it to "prior probability", which is a term of art. But the story of this page painfully tells us that you refuse to entertain the notions of prior and posterior probabilities, at least in the context of the MHP. So what is it? And what is "unchanged" about the latter probability, if it's not a posterior (i.e. conditional) probability? Mind you, am not saying that a solution for the "canonical" (original, Vos Savant, call it what you want) case of the MHP cannot be expressed coherently within a frequentist formulation of P.T., and without recourse to conditionals - although one really has to twist oneself in knots to do it correctly (as opposed to "intuitively"). Here I am simply objecting to your use of an unnecessarily verbose and ultimately obscure table format. Rick Block's form is faithful to the source (which, I assume, does not show a table itself), clean and complete. I am reverting to it.glopk (talk) 04:50, 16 June 2010 (UTC)
- I see, another 'literalist' unconcerned about the readers' experience. So you disagree with the values in the end column? How? Why? Isn't that exactly the point of the 'you always get the opposite after Monty opens door #3' solution? And part of the lovely confusion of the paradox: 'But isn't' it 1/2?' Give me a break. But most importantly, 'Who cares'? It's from Carlton's reliably published solution. I didn't make it up.
- Here's Rick's comment to Martin within the last few day:
- "Basically, it's a "choice centric" view rather than "car centric" view (very similar to vos Savant's table). The new table Glkanter has added shows the 1/3-2/3, switch and you get the opposite solution (which didn't have any corresponding graphic)."
- You see? Rick finally 'got it' that he needed to 'change his view 90 degrees.' Of course, this simple representation of Carlton's elegant solution conflicts with Rick's 5 year mission to defend Morgan, et al. A mission even they have, at last, given up on. So, where's the beef? How else to show that the probabilities didn't change in a reader-friendly table format?
- It's nice to see that Rick is on-board with the table's usage. Just for fun, here are his response to that exact same 5 column, 2 row table when I first interpreted it, just a few days ago (well, it was just 4 columns, Rick actually suggested that the probability needed to be in there twice. Quite the irony, eh?):
- about the table, ::::I wish I could explain to you the difference between unconditional and conditional
- what I think the table should look like (Oh, maybe he's seen the light?)
- which source are you talking about? (Eh, maybe he hasn't seen the light.)
- I assure you I am acting in good faith (sadly, not very competently)
- Or, here's the whole section.
- Of course, Rick never says, 'By golly, you (and Martin) are right. I guess I was mistaken about the clock being right only twice a day. Thanks for staying with me until I got it.' No, he just decides that the table needs his special brand of "fixin'". Thanks, Rick. Thanks for nothing. You think it's fun to have to respond to all of Rick's nonsense for a 5 column, 2 row table that interprets a reliably published solution? It's not fun at all. It's downright tedious.
- What about the column headings with the door #s? You didn't even mention those in your comments above. Why can't they reflect Whitaker's question? Only because it's disadvantageous to Rick's and Nijdam's arguments on what order, and with what caveats and weasel words, will the simple solutions be presented in the article. There can be no valid claim that the reader benefits from the loss of specificity to the table, or that the column headings don't come from Whitaker's question.
- "...unnecessarily verbose and ultimately obscure table format..." More BS. It's 5 columns and 2 rows. 3 columns of where the car and 2 goats can be, 2 columns showing the probability before and after the goat is revealed. Period. Nothing else.
- It's odd that after such a long break from any commentary, you come in with your guns a-blazin'. How did that come about, anyways, Glopk?
- Save your techno-mumbo-jumbo and big words for some other sucker. This is a simple solution, as reliably published by Carlton. Got a problem with it? Take it up with him, not me. I ain't buying' it.
- What was that other thing Inigo Montoya used to say? Figuratively, with regard to your specious arguments regarding the appropriateness of this table as it represents Carlton's solution, I suggest the same. Glkanter (talk) 07:16, 16 June 2010 (UTC)
What do you all think of this one?
You pick a car | You pick a goat | You pick a goat | |||
---|---|---|---|---|---|
You swap | You stick | You swap | You stick | You swap | You stick |
You get a goat | You get a car | You get a car | You get a goat | You get a car | You get a goat |
For what purpose? As a substitute for the new supporting table in the article? Or in support of which other reliably published source? How does it address Whitaker's door 1 and door 3? Do you see a problem with the new table that supports the 'you always switch' solution? Please share your concerns. Glkanter (talk) 00:47, 15 June 2010 (UTC)
- I just put it up for general comment as a possible table for the simple section because it seems simple and intuitive to me. Martin Hogbin (talk) 08:59, 15 June 2010 (UTC)
- I do not want to address Whitaker's door 1 and door 3 for two reason:
- It makes the problem clearly conditional thus raising the issue of how to treat the host policy. (We have argued about this for years and although I agree with you that, in the absence of knowledge to the contrary, it is correct to treat the host goat door choice as uniform at random, others do not agree so I am trying to circumvent this contentious point for the purpose of clarity)
- K&W have shown that people understand the problem better if all references to door numbers are removed. Martin Hogbin (talk) 11:20, 15 June 2010 (UTC)
- Martin - isn't that roughly the same as this version?
Player's door | Door the host opens | Remaining Door | Probability |
---|---|---|---|
Car | Goat | Goat | 1/3 |
Goat | Goat | Car | 2/3 |
- I assume you would prefer this to Glkanter's version. Is that right? I find your version a little hard to interpret (why are there two identical columns? why are there 4 cells under each heading? - this just makes it hard to understand). -- Rick Block (talk) 13:29, 15 June 2010 (UTC)
- There are three columns to represent three equally likely possibilities. Maybe you could combine the lower cells vertically. I am just offering this as possibility. To me it seems the simplest to understand. Martin Hogbin (talk) 22:30, 15 June 2010 (UTC)
- Yes - I puzzled this out. Basically, it's a "choice centric" view rather than "car centric" view (very similar to vos Savant's table). The new table Glkanter has added shows the 1/3-2/3, switch and you get the opposite solution (which didn't have any corresponding graphic). Direct question - of the three tables we're talking about here, which would you consider to be a more faithful representation of the solution we're illustrating? -- Rick Block (talk) 23:50, 15 June 2010 (UTC)
Yeah, Rick, you 'puzzled this out'. Brilliant! It's exactly the table I initially proposed, right down to originally only showing the probabilities once. Of course, you've obscured the door #s to make it look less similar to Whitaker's letter.
"== Please Criticize This Re-interpretation of vos Savant's or Carlton's Solution =="
Host always offers the switch, host must choose randomly
...so, if the contestant chooses door #1, and the host opens door #3, the contestant will win the car twice as often if he switches.
Door 1 | Door 2 or 3 | Remaining Door | Probability |
---|---|---|---|
Car | Goat | Goat | 1/3 |
Goat | Goat | Car | 2/3 |
Boy, that sure looks familiar!!! Wait, it gets better. Rick even had the balls to write this:
- I assume you would prefer this to Glkanter's version.
What a skeeve. Kudos, Rick! Glkanter (talk) 11:20, 16 June 2010 (UTC)
- @Martin - of the three tables we're talking about here, which would you consider to be a more faithful representation of the solution we're illustrating? -- Rick Block (talk) 14:00, 16 June 2010 (UTC)
The differences between the tables
There are 5 columns, and just 2 rows in the table I designed to support the 'you always get the opposite' solution in the 'Simple solution' section of the article.
Of those 5 columns, Rick changed the column headings on the first 3, and eliminated the 5th entirely.
These are the changes to the wording Rick felt necessary without prior discussion:
Original Text vs Rick's Substitute
Door 1 was replaced by Player's door
Original Probability was replaced by Probability
Host Has Opened Door 2 or Host Has Opened Door 3 was replaced by Door the host opens
Additionally, column 5, Probability (unchanged), was removed entirely. This column held the same values as the 'Original Probability' column. A pretty important column, as we'll discuss later.
Of less significance is Rick deleted bolding on the words 'car' and 'goat' in the 2 columns which were used to indicate the 'opposites' on both rows of the table.
That is the totality of the changes made by Rick.
Notice that each of the eliminated words and values cause the table to look less and less like Whitaker's original question, making it more difficult for the reader, and other editors, to associate the solution and table to Whitaker's question. I believe this was Rick's primary intent. This is also an important item, and will be discussed later.
Finally, nothing in this table violates any WP policies. Each column heading and the values in the column come from either the problem statement, the premises, or the solution. There is no OR or POV in this table.
Glkanter (talk) 03:46, 15 June 2010 (UTC)
- Well, actually, the policy that is most relevant is Wikipedia:Verifiability. The source for this solution is Carlton. Here's what he says (section 5):
- Before presenting a formal solution to the Monty Hall Problem to my students, I find that it helps to give an intuitive explanation for the 1/3 - 2/3 solution. Imagine you plan to play Let's Make a Deal and employ the 'switching strategy.' As long as you initially pick a goat prize, you can't lose: Monty Hall must reveal the location of the other goat, and you switch to the remaining door - the car. In fact, the only way you can lose is if you guessed the car's location correctly in the first place and then switched away. Hence, whether the strategy works just depends on whether you initially picked a goat (2 chances out of 3) or the car (1 chance out of 3) [bold added].
- There's no mention of "original probability" or "probability (unchanged)". If anything, per the bold sentence the "probability" column should perhaps be changed to "initial probability" - but I'm not insisting on this. Changing the other labels from "Door 1" and "Host Has Opened Door 2 or Host Has Opened Door 3" to "Player's door" and "Door the host opens" is partly editorial, but also makes the table more accurately reflect the reference as well. Although you added this table, you don't "own" it - note the disclaimer in small text below the text entry form ("If you do not want your writing to be edited, used, and redistributed at will, then do not submit it here"). Edit warring to preserve your version is ridiculous. Seriously - I don't there there's anyone here other than you who prefers your version (if there is anyone, please speak up). -- Rick Block (talk) 14:28, 15 June 2010 (UTC)
- It can certainly be 'verified' that everything in that table is either a premise of the puzzle, from Whitaker's letter, or from the solution. This literalness of each word stuff on your part is once again, a difference in interpretation of Wikipedia policies from mine. And the way you state your interpretation as 'fact' is another difference from myself. You often leave meaningful aspects of your sources out of your 'justifications'. For all I know, maybe he discusses these items a few pages earlier, anyways? Besides, unlike other simple solutions you criticize, he's not talking about an 'average' probability. He's talking about a strategy one may use in every play of the game, including the doors 1 and 3 pairing. You act like that's a bad thing. How utterly perverse! All this couldn't have been discussed first? What's the danger of leaving it? Who would be injured by leaving it? 16:18, 15 June 2010 (UTC)
- If there's anyone here who prefers Glkanter's version of this table, please speak up. To be absolutely clear, we're talking about Glkanter's version:
Door 1 | Original Probability | Host Has Opened Door 2 or Host Has Opened Door 3 | Remaining Door | Probability (unchanged) |
---|---|---|---|---|
Car | 1/3 | Goat | Goat | 1/3 |
Goat | 2/3 | Goat | Car | 2/3 |
- and this simpler version:
Player's door | Probability | Door the host opens | Remaining door |
---|---|---|---|
Car | 1/3 | Goat | Goat |
Goat | 2/3 | Goat | Car |
- If no one speaks up in support of Glkanter's version (other than Glkanter), I will change back to the simpler version again. -- Rick Block (talk) 14:12, 16 June 2010 (UTC)
- Rick, why do you continue to embarrass yourself this way? Either through ignorance or impunity, you are violating any number of Wikipedia Policies with this comment.
- Silence does not mean consent.
- Wikipedia doesn't care which table people 'like'. Wikipedia cares about the sources, and the readers.
- If you can build a case that the table in it's present form is not supported by the sources, then make the case and build a consensus. Sources, of course includes Whitaker's letter, vos Savant's column and solution, Carltion's solution, etc.
- If you can build a case that eliminating the door numbers from the column headings is better for the reader, then make the case and build a consensus.
- You're violating NPOV/Bias by trying to make this table look 'generic', and 'not really addressing Whitaker's question'.
- You're failing to show good faith with your actions as described above.
- Why not be honest and admit that Glkanter designed both of the above tables? I chose to add the 2nd occurance of probabilities to show that they don't change. Otherwise, all you've done is muddy the waters by changing some column headings.
- Glkanter (talk) 18:02, 16 June 2010 (UTC)
- Rick, why do you continue to embarrass yourself this way? Either through ignorance or impunity, you are violating any number of Wikipedia Policies with this comment.
- I'm sorry but personally I don't like either one of those tables. Considering how clear the verbal explanation that this table is supposed to be representing is I am sure we can come up with something better. I know that does not address the question as asked but that is just how I feel about it. Colincbn (talk) 17:12, 16 June 2010 (UTC)
- Thanks for your comment. I don't know how to make the table simpler. 3 columns show where the car and 2 goats are. 2 columns show the before and after probabilities. That's all there is. Please tell me how it fails to support the 'you always get the opposite' solution? Glkanter (talk) 18:02, 16 June 2010 (UTC)
I suggest we remove the table in the article. It is no more than OWN RESEARCH, and adds nothing. Nijdam (talk) 20:31, 16 June 2010 (UTC)
- If we can't agree on a table, removing it seems like a sensible idea to me. -- Rick Block (talk) 04:55, 17 June 2010 (UTC)
- It's interesting, but sad, that a so-called 'educator' is so anxious to suppress a simple table supported by reliably published sources rather than engage in honest dialog about it. Solely because it conflicts with his ideology that '1/3 <> 1/3'. It isn't surprising, though. Glkanter (talk) 20:53, 17 June 2010 (UTC)
- Rather than the two tables above, what is wrong with the one that is currently there? Seems fairly clear to me. I don't like the color-scheme that much, but I think it represents the intended sentence fairly well. It could be made easier to understand I suppose. Otherwise how about something like this:
Player's potential doors | result if switching | result if staying |
---|---|---|
Car | Goat | Car |
Goat | Car | Goat |
Goat | Car | Goat |
- The first table in this section is basically straight from vos Savant's solution published in Parade [1]. She presented two tables each with three lines varying the location of the car from door 1 to door 3, the first showing the result if you switch in each case and the second showing the result if you stay). Given that this source presents door numbers, I think this table should as well. -- Rick Block (talk) 13:21, 17 June 2010 (UTC)
Glkanter reported for edit warring
Another classic example of misinterpreting unambiguous text:
"== User:Glkanter reported by User:Rick Block (Result: No violation) =="
"* No violation I only count three reverts. Please feel free to open up another request if the user resumes reverting. --B (talk) 21:20, 15 June 2010 (UTC)"
— Preceding unsigned comment added by Glkanter (talk • contribs) Glkanter (talk) 00:20, 16 June 2010 (UTC)
- Yes, I reported you for edit warring. This is really not the appropriate forum for discussing this but you seem to want to insist. Fine. WP:AN3 is for reporting recent violations of the three-revert rule, and active edit warriors. The admin responding to this is saying you didn't violate wp:3rr (which you didn't, and I never claimed you did) - not that you weren't edit warring (which you were). -- Rick Block (talk) 00:15, 16 June 2010 (UTC)
- Just to keep the record straight here, Glkanter continued edit warring and was then blocked for a short time for violating wp:3rr. -- Rick Block (talk) 04:52, 17 June 2010 (UTC)
Rick, your long standing obstruction to various editors' good faith edits has to be addressed by Wikipedia at some point. I'll need to be able to present a concise argument with diffs to support my charges. The situation above is one perfect, simple example, as it demonstrates your violations of all 3 of these WP policies:
Glkanter (talk) 00:20, 16 June 2010 (UTC)
That bogus RfC/U on me that you conjured up with DickLyon is another perfect example. Now, that was Harassment, and Gaming the system, and... Glkanter (talk) 01:03, 16 June 2010 (UTC)
- We are still officially in mediation (although it's on hold pending the assignment of a different mediator). If mediation fails, Wikipedia:Arbitration would be the next step. If you want, feel free to pursue Wikipedia:Requests for comment/User conduct. -- Rick Block (talk) 14:34, 16 June 2010 (UTC)
- A stronger approach would be a community ban, per Wikipedia:Banning policy, requiring a consensus of users not involved in the dispute. In fact, come to think of it, why don't we have a straw poll. Would anyone watching this page who has not been involved in this dispute (I think this excludes at least Glkanter, me, Martin, Glopk, Nijdam, Kmhkmh), support a community imposed topic ban against either or both of myself or Glkanter which would ban this user from making edits to articles and talk pages relating to the Monty Hall problem? Please just sign in the appropriate list below. This is in no sense binding, the official way to do this is through a discussion at WP:ANI. If there seems to be significant support, the actual discussion would need to occur at WP:ANI - I'm suggesting a straw poll here just to get a sense for whether there might be some point in opening such a discussion. -- Rick Block (talk) 15:03, 16 June 2010 (UTC)
- I would be happy to sign for a ban against Glkanter, but am I not allowed?Nijdam (talk) 15:36, 16 June 2010 (UTC)
- In a discussion at WP:ANI about this, you would be welcome to comment, but since you're "involved" I think your "vote" would be more or less ignored. -- Rick Block (talk) 15:47, 16 June 2010 (UTC)
- I would be happy to sign for a ban against Glkanter, but am I not allowed?Nijdam (talk) 15:36, 16 June 2010 (UTC)
- Why support votes only? I don't know if I would be counted as being "involved" in this tempest in a teapot, but I'd oppose topic banning Rick. It would be most counterproductive - he's basically doing a good job, and I think he would even manage this article just fine all by his lonesome. On the other hand, Glkanter seems to be more disruptive than helpful (sorry to say). -- Coffee2theorems (talk) 15:44, 17 June 2010 (UTC)
- I also cannot vote below but agree with Coffee2theorems that this is a storm in a teacup. We have had a minor editing skirmish and the bandying around of words like 'harassement'. No animals got hurt and there is no need for anyone to be banned. Martin Hogbin (talk) 23:20, 17 June 2010 (UTC)
- I'd support a ban against anyone that's over-complicating this. A few moths ago when I last visited here, there were concise, easy to grasp explanations of what is generally perceived as the MHP. Now it's all muddled. Just like a real encyclopedia, or not. This is why Wikipedia has no real credibility anymore.Feddx (talk) 13:33, 26 June 2010 (UTC)
The following users would support a topic ban against Glkanter
- <1st sig here>
- <next sig here>
The following users would support a topic ban against Rick Block
- <1st sig here>
- <next sig here>
Is The Table That Supports Carlton's Solution OR?
I agree that he didn't include the table, but the 2 row 'choice-centric' design of the table is certainly consistent with his solution.
The column headings and the values in each cell come directly from:
- Whitaker's letter (which doors serve which purposes)
- Carlton's paper (the before-a-goat-is-revealed probabilities of 1/3 and 2/3),
- Simple math (the after-the-goat-is-revealed probabilities) as shown:
- (Probability of being a car * Probability of host revealing a goat (aka, a 100% condition) = Probability of being a car after revealing a goat)
- 1/3 * 100% = 1/3
- 2/3 * 100% = 2/3
I spent some time looking, and didn't find any WP policy prohibiting editor-created tables, as long as they are reliably sourced, NPOV, and verifiable. I think the table in it's current form meets these requirements. Glkanter (talk) 06:40, 18 June 2010 (UTC)
- I don't know, Rick. Maybe there are other editors who see merit in the table. You, yourself proposed 5 different tables for the exact same solution. You must have seen the need for a visual aide. Maybe you could address my issues, above, before you delete the table for reasons other than sources, verifiability, NPOV or non-benefit to the reader? What did Nijdam mean by his edit comment when he 'corrected' the column headings? That's not silence, and it wasn't a deletion. I think your deletion was pretty aggressive and unnecessary. Glkanter (talk) 14:20, 18 June 2010 (UTC)
- No one other than you is arguing in favor of this table. Not even Martin. It's time for you to accept that there is a consensus against this table. You are violating one of the basic guidelines of editing behavior, and are obviously aware of the consequences. -- Rick Block (talk) 02:50, 19 June 2010 (UTC)
One reason for much of the argument
Any statistician seriously considering the MHP is likely to be using the modern definition of probability theory in which we have consider a sample set of probability mass functions. This is a formal mathematical system to allow probabilities to be manipulated to solve probability problems. There is, however, one very important point that may be causing some argument here. To quote directly from probability theory article:
The function f(x), mapping a point in the sample space to the "probability" value is called a probability mass function abbreviated as pmf. The modern definition does not try to answer how probability mass functions are obtained; instead it builds a theory that assumes their existence.
In the MHP there are three undefined (in Whitaker's question) distributions, the producer's original placement of the car and the player's original choice of door, and the host's choice of door. Once these distributions have been determined, a formal calculation is carried out which produces an answer that does not depend on any person's state of knowledge or any real world assumptions. This is why some people have been arguing that that whether the host knows the host's door opening policy is not important. In probability theory, once the relevant distributions have been determined, the answer follows automatically from the mathematical formalism.
As is made clear in the relevant article, probability theory does not tell is how to obtain all the relevant probability mass functions. This is where the basis on which we choose to answer the question comes into play. The player's original choice of door is generally not contentious so I will ignore that here. That leaves two undefined distributions to consider, the initial car placement and the host goat door choice. In order decide these distributions we must use some logical basis. It is in the choice of these distributions that the vague question posed by Whitaker must be formulated into a probability problem.
Consider the producer's initial car placement. How should we decide the probability with which the car is placed behind each door? This may or may not have been done using some random process, but regardless of that, the host and the producer know where the car is. If we decide to answer the question from the point of view, or state of knowledge, of the host, even if the car was actually placed uniformly at random by some process (such as throwing a die) it is not randomly placed from the POV of the host. If we wished to set up the problem from the POV of the host we should not use a uniform distribution of initial car placement. If we had to answer the question from the Host's POV and were not told where the car was placed, our only option would be to parameterise the car placement and say the car was placed behind door 1 with probability C1 etc. This would provide a general answer where, when know, the actual values of the parameters used could be used in a probability calculation to give a numerical answer. The important point to note is that, it is in the setting up of the formal probability mass functions that the state of knowledge from which we wish to answer the question is decided. Even if the car was stated to have been initially placed uniformly at random, if we wanted to answer the question from the host's perspective, we would not use a uniform distribution.
The other distribution that we must consider is that of the host's choice of goat door. The way we must deal with this is in exactly way as we deal with the initial car placement. We have to decide from what state of knowledge we wish to address the problem. Suppose the host has a non-uniform policy of choosing which door to open, say he always chooses door 2 when possible. What distribution should we use in or formal calculation? Exactly as before, that depends on the basis on which we wish to answer the question.
One consistent basis or state of knowledge must be used to determine all unknown distributions in the MHP. Possibilities include:
- From the POV of the player, as strongly suggested by Whitaker's question.
- Strictly what is stated in the problem statement.
- From the POV of the audience
- From the POV of the host.
- From our general real world knowledge TV quiz shows.
Options 1, 2, and 3 give us no information whatever about the unknown distributions. In the case of 1 above note that it does matter whether or not the player knows the host's goat door policy. This gives us two further options, parameterise the probability mass functions and give an answer in parametric form into which the real values could be substituted when known which gives us a rather boring and unhelpful answer that probability of winning by switching is anything from 0 to 1 depending on where the car was initially placed and the host goat door policy. The only alternative is to apply the principle of indifference to the distributions and take them all as uniform (regardless of the real, but unknown, basis on which the car was initially placed and the host chooses a goat door). This gives the answer of exactly 2/3. This answer is now agreed by Morgan.
Option 4 is unlikely to be of interest to mist people as the host knows where the car is. Using our real word knowledge of TV shows it is most likely that both distributions would be uniform.
I think it would be more productive for editors to discuss these issues further to determine where exactly any disagreement lies, rather than propose editing blocks.
- The cases where the standard assumptions (which say i.a. that all the relevant distributions are uniform) do not apply are discussed in the variants section of the article, and do not seem to be the main point of contention at this time. So it would be more productive to stick to the standard assumptions for now and hash out any issues with the variants section once the stuff preceding that in the article has been dealt with (which has the projected timeframe of "sometime after hell freezes over", so let's not be overly ambitious here..). Accordingly, I will make the standard assumption in the sequel (as usual).
- It is of course possible to encode state of knowledge (SoK) into the unconditional probability distributions, and to some extent that is necessary in general, but that's generally how you deal with the fixed part of the SoK. The changing part of the SoK is what you condition on, either because that is the clearest way to proceed or because you want to compute the probabilities for different SoKs. In this case, the knowledge "host opens door 3" is not a part of the player's SoK before the host opens the door, but it is a part of her SoK after the fact, so it is natural (and usual!) to represent that by conditioning. It also allows us to formalize the idea that "the probability that the car is behind door 1 does not change by the opening of door 3" (i.o.w. P(car is behind door 1 | host opens door 3) = P(car is behind door 1)), which is something that is (arguably) important for understanding the MHP.
- You can also consider the "fixed" part of the SoK in this case to be a description of the actual behavior of the people involved, and maybe that is the clearest way. The uniform distributions would then mean that the people make their choices literally by throwing dice. Then it is clear that any SoK is not going to change that behavior by one iota, and the unconditional probability distributions must then reflect that reality (i.e. they are uniform). It is by conditioning, then, that you get answers in different kinds of situations.
- As an example, if you wanted to look at things from the POV of the host on the game show, then you'd condition on the knowledge that the car is behind door x (we don't know x, but the host does), and the resulting probability would, depending on x, be either 1 or 0 (the host knows everything worth knowing, so there is no uncertainty whatsoever). The resulting solution would be: "If the car is behind door 2, then switching wins with probability P(C=2 | H=3, S=1, C=2) = 1, and if it's not, then switching wins with probability P(C=2 | H=3, S=1, C=1) = 0. Or in other words, the probability is P(C=2 | H=3, S=1, C=x) = x-1, which the host can compute because he knows x." -- Coffee2theorems (talk) 14:22, 18 June 2010 (UTC)
- Per Carlton, the condition is not that 'door 3 is opened'. The condition is 'host reveals a goat'. Whitaker, 'say, door #3' uses this as an example. Opening door #3 is not a premise to the puzzle in any published formulation. How can it be a required condition? Glkanter (talk) 14:41, 18 June 2010 (UTC)
- Coffee2theorems, I know how to do treat the problem from the host's SoK but this is not what the problem is all about. The question is about how to formulate the problem from the player's SoK. I think it is generally taken that player does not know the host's policy on goat door opening. So it makes no difference what that policy is. When we formulate the problem from the players perspective we must take the initial car placement to be uniform at random by applying the principle of indifference (or else declare the answer to be indeterminate) and also take host's goat door choice as uniform at random and do the subsequent conditioning on that basis. As Morgan now say, if you consider the conditions implicit in the problem (that is to say we are to answer from perspective of the player, who does not know the host's door policy) '...the answer is 2/3, period'. Martin Hogbin (talk) 17:31, 18 June 2010 (UTC)
Another issue
Having established that, regardless of the actual host policy, on any reasonable basis, we must take his goat door choice distribution to be uniform at random (thus making the problem symmetrical) the question arises of whether a simple solution, in which the door opened by the host is not specified, is correct.
Regarding this point I think it is best for neither side to take a hard line. The simple solution gives the correct answer because of an obvious and intuitive symmetry in the problem. Whether it is formally correct could be argued forever. I suggest we use words like formal and informal to apply to the two types of solution (as suggested by someone earlier) to avoid any further conflict. We already give both types of solution in the article. Martin Hogbin (talk) 11:54, 18 June 2010 (UTC)
- Martin - if the article is "written from a neutral point of view, representing fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources" (per WP:NPOV), the questions you're raising here are moot. Any editor's opinion about what conditions "we must take" is completely irrelevant. Your lecture about probability theory at best belongs on the /Arguments page, not here. -- Rick Block (talk) 14:10, 18 June 2010 (UTC)
- Do you actually disagree with anything I say or are you just objecting on principle? It is all standard probability theory and is an attempt to explain the current heated argument about host bias. I am more than happy to discuss any of the issues involved with you on the /Arguments page if you wish. I think that would be far more productive than trying to get editors who disagree with you blocked.
- If you want to discuss what the sources actually say then I am happy to do that also on the /Sources page. Martin Hogbin (talk) 15:15, 18 June 2010 (UTC)
- Martin - we've had this discussion repeatedly. I know what you think. I suspect you know what I think. You say above that you think "it is best for neither side to take a hard line about whether a simple solution is correct". I completely agree with this and have been arguing (for months) that the article reflect this. It seems to me that it is you who are completely unwilling for the article NOT to take a hard line about this. If you really think it is best for neither side to take a hard line about this, then why do you keep arguing that the article must take the stance that the simple solution is absolutely correct and that the unconditional approach must be presented as the preferred approach? -- Rick Block (talk) 20:11, 18 June 2010 (UTC)
- I don't. Martin Hogbin (talk) 20:13, 18 June 2010 (UTC)
- Really? What do you think it means to insist that the simple solutions are presented first, without criticism, without even mentioning that many sources approach the problem conditionally (let alone that some sources explicitly criticize these solutions as not addressing the question), and then following up such solutions with an "aid to understanding" section that further elaborates the "insight" offered by these solutions - and only then in the weakest possible way saying that "some sources interpret the question differently"? I take this to mean the article is saying the simple solutions are absolutely correct and that they are the preferred approach. -- Rick Block (talk) 20:39, 18 June 2010 (UTC)
- I don't. Martin Hogbin (talk) 20:13, 18 June 2010 (UTC)
- Martin - we've had this discussion repeatedly. I know what you think. I suspect you know what I think. You say above that you think "it is best for neither side to take a hard line about whether a simple solution is correct". I completely agree with this and have been arguing (for months) that the article reflect this. It seems to me that it is you who are completely unwilling for the article NOT to take a hard line about this. If you really think it is best for neither side to take a hard line about this, then why do you keep arguing that the article must take the stance that the simple solution is absolutely correct and that the unconditional approach must be presented as the preferred approach? -- Rick Block (talk) 20:11, 18 June 2010 (UTC)
You don't really have a 'smoking gun', Rick. You're connecting the dots and making a conclusion. In this case, a flawed one. It's the same technique that Sarah Palin used to connect about 5 different aspects of Health Care Reform and conclude that DEATH PANELS are going to euthanize Grandma. Glkanter (talk) 20:53, 18 June 2010 (UTC)
Rick, I have explained my position several times before. I would like to see this article as useful as possible to as wide a range of readers as possible, from those with no interest in the subject who just want the answer to settle a bet for example, through elementary students of statistics, to experts in the subject, who still might get something useful from it.
I am not trying to hide anything or push any POV in the article but just suggesting that we organise the article to present all the information about the subject in a way that works best for our readers. It is logical to present a simple exposition of the subject first then proceed to a more detailed discussion. That is the order used on most text books and WP articles.
The general reader does not want to be distracted by arguments about conditional probability while they are trying to understand one of the world's hardest brain teasers. On the other hand, I doubt that the expert reader will object to skimming through a simplified solution to reach a proper mathematical discussion of the subject, where we can put all the stuff that you claim I am trying to hide.
This continued argument about conditional/unconditional solutions is something of an artifact of discussion here, started by over reliance on a paper by Morgan et al by some editors and still propagated here by some despite a somewhat softened appraoch by the paper's original authors. Martin Hogbin (talk) 10:43, 19 June 2010 (UTC)
- Whatever your motivations, are you disagreeing that the structure you keep insisting on has the effect of making the article say the simple solutions are absolutely correct and that they are the preferred approach? -- Rick Block (talk) 14:21, 19 June 2010 (UTC)
- Here you go, Rick. This is what a 'smoking gun' looks like:
- "Several authors point to the fact that this and some other simple solutions, are not complete, because they do not distinguish properly between the probabilities before and after the opening of a door by the host. (See the probability section for more details.)" Nijdam (talk | contribs) Revision as of 04:05, 18 June 2010 (edit) (undo)
- "Whatever your and Nijdam's motivations, Rick, are you disagreeing that the above unsourced commentary he keeps putting back in the Popular Solution section has the effect of making the article say the simple solutions are absolutely wrong and that they are the monority approach?" adapted from Rick's comments above Glkanter (talk) 08:26, 22 June 2010 (UTC)
- Here you go, Rick. This is what a 'smoking gun' looks like:
- @Rick, of course I disagree that the structure I keep insisting on has the effect of making the article say the simple solutions are absolutely correct and that they are the preferred approach - because the article says that this is not the case. Martin Hogbin (talk) 17:37, 22 June 2010 (UTC)
How's This? Act II
I. Very little mention of controversies (maybe just vos Savant's in Parade magazine, the others exist only in professional journals and text books) until after the solutions are provided.
II. One solution section: vos Savant, Carlton, Devlin, conditional. Yes, in that order.
III. Aids to Understanding
IV. The Controversies:
- A. vos Savant's 1/2 vs 2/3
- B. Door 3 must be revealed
- C. Morgan, including simple solutions are false, host bias, and that their's is the (only) 'correct resolution'
A boring litany of who said what, and why, only. No original research editorializing on who is right or who is wrong.
Maybe switch the order of III and IV...
V. Sources of Confusion
etc...
Before the solution section, I guess in the Problem statement, I would suggest including Selvin's comment from his 2nd letter (paraphrasing here): '...this solution relies on the host always offering the switch and choosing equally between...' and a mention that the importance of this will be addressed later in the article, with a link to that section.
And I would have a separate, later section for the other simple solutions and any other solutions people want to include. I would put this before the 'Variants' section.
So, anybody else see this as a NPOV way to get a better article and some closure? Glkanter (talk) 21:13, 18 June 2010 (UTC)
A rose by any other name
I think the article is looking a lot better. I object to the titles of the two main sections: "popular solution" and "probabilistic solution". There is nothing non-probabilistic about the popular solution. It is good probabilistic thinking. The so-called probabilistic solution is just "the" solution to a more refined and more complex problem. The popular solution is the proper probabilistic solution to a more simple and easy to solve problem. It is a matter of opinion which of the two problems, if either, is THE Monty Hall problem. There are also other options. Economists see THE Monty Hall problem as a problem of game theory or decision theory, not of probability at all. The problem is not "what is this, or that, probability?" The problem is "what should you decide to do?". Decision theory. Game/decision theory is something more complex than probability theory and requires ability to do probability, but goes beyond. Many people see the Monty Hall problem as a problem of psychology or behavioural science, and that is going even further since game theory and decision theory can be thought of as mathematiziations of parts of behavioural science (cf. economics).
I would prefer to refer to "simple problem, simple solution" and "refined problem, refined solution".
Gill110951 (talk) 05:54, 19 June 2010 (UTC)
Right, having said all that, I went ahead and did it. I cleaned up the mathematical solution to the conditional problem, added the game theoretic solution to the game theoretic version, cleaned the solution of the unconditional problem. I hope in a way which demonstrates a totally neutral POV. Still have to add lots of references... please help, if you don't disagree too strongly.
I am trying hard to keep my own Point of View out of this (ie what I think is the right formulation of the problem). I do believe there are three (or more) legitimate mathematizations out there, they can all be justified by recourse to various Authorities or to History or to Good Taste, or whatever, ... so what... They're out there in the wild and wikipedia should talk about them, neutrally, each on its own merits, in length at reasonable proportion to its importance, and go on to summarize the controversy.
Please lets separate maths from controversy. There is clean rigorous nice maths for every interesting variant of the problem. As a mathematician I am going to fight for clean rigorous nice maths. Gill110951 (talk) 07:07, 19 June 2010 (UTC)
- I agree with what you say but suggest that there is one case where a slight relaxation in mathematical rigour is justified in the interests of clarity. That is in the symmetrical conditional case, where the player chooses after a specific door has been opened by the host but the host is known to open a door uniformly at random, selected from all unchosen doors hiding a goat - the unambiguous formulation of K&W.
- In this case it might be argued that a conditional solution, in which the possible doors that might have been opened by the host are shown, is required. I suggest that an obvious and intuitive symmetry tells us that it is not necessary to distinguish between doors the host might have opened since the answer must be the same whichever door is actually chosen by the host. I believe use of the simple solution in this case is justified initially, although I have no objection to stating the potential problems with this approach later in the article. As Rosenthal says about the simple (unconditional) solution for this case, 'This solution is actually correct, but I consider it "shaky" because it fails for slight variants of the problem'. Martin Hogbin (talk) 11:32, 19 June 2010 (UTC)
- There's no problem with using the unconditional ("simple") solutions for solving the conditional problem, as long as it is mentioned that a symmetry argument or something similar is required. It is not enough to say on the talk page that "an obvious and intuitive symmetry tells us that it is not necessary to distinguish [...]" - a sentence like that must go into the article. It is not reasonable to assume that the reader will come to the talk page to read that sentence. The article should present complete solutions only; or some incomplete solutions, each with a caveat in the article, right next to that solution that the solution is not complete. This is not a question of rigor, but of honesty. Passing incomplete solutions off as complete solutions is dishonest.
- In an encyclopedia, there is no need to present any solutions at all to convince the reader. If we want the reader to know that the answer is 2/3, then it is enough to say so and to cite that claim. The citation is enough. The solutions are not there to convince the reader at all costs. They are there because a discussion of the MHP is not complete without presenting a solution, i.e. the reader should be told not only the "what" but also the "why". If we leave many readers with an impression that they understood the "why" when in fact they did not - as an incomplete solution without a caveat right next to it does - then we are doing them a disservice. The task of an encyclopedia is to inform, not to misinform. -- Coffee2theorems (talk) 15:01, 19 June 2010 (UTC)
- I would like to put a sentence along the lines that you suggest into the article but I do not know of a source that makes that specific statement (if you do please let me know). On the other hand there are many sources that just give the simple solution as a complete solution to the MHP. To some degree the true, and certainly most notable, MHP is the problem to which the simple solution is the solution. Martin Hogbin (talk) 00:42, 20 June 2010 (UTC)
- In an encyclopedia, there is no need to present any solutions at all to convince the reader. If we want the reader to know that the answer is 2/3, then it is enough to say so and to cite that claim. The citation is enough. The solutions are not there to convince the reader at all costs. They are there because a discussion of the MHP is not complete without presenting a solution, i.e. the reader should be told not only the "what" but also the "why". If we leave many readers with an impression that they understood the "why" when in fact they did not - as an incomplete solution without a caveat right next to it does - then we are doing them a disservice. The task of an encyclopedia is to inform, not to misinform. -- Coffee2theorems (talk) 15:01, 19 June 2010 (UTC)
Rose?
I really do not like the way Gill is intervening, acting like the elephant in the porcelain shop. I'm also very disappointed by him still defending the simple problem. A door (I do not care which one) has been opened before the player is given the offer to switch! Just the "simple" reader of the article will and should have this picture in mind. There is nothing to compromise. So, from now on, I suggest we start with the correct problem and the correct solution, be it in simple wording. Further on in the article we may mention some authors consider simplified versions of the problem! I'm through. Nijdam (talk) 08:07, 19 June 2010 (UTC)
May be someone can explain to me where from the problem formulation it may be deduced the player is given the offer to switch BEFORE the door with a goat is opened?Nijdam (talk) 08:09, 19 June 2010 (UTC)
Moreover, giving the right numerical answer on basis of wrong reasoning, is as bad as giving the 50-50 answer.Nijdam (talk) 10:55, 19 June 2010 (UTC)
- It is not entirely clear where any precise formulation of the MHP comes from. The most well known statement, Whitaker's question, is so vague that no method can solve it without some additional assumptions. The simple solutions fail if the host chooses a goat door non-uniformly, the conditional solution fails if the car is initially placed non-uniformly, and even the game theoretical solution fails if the TV station want to boost their viewing figures by engineering a win for the player.
- You seemingly care not to comment on the opened door. How vague the formulation may be, it's definitely not vague about the host opening a door and then offering the player to switch. Agree?
- No I do not agree. It is far from clear that Whitaker intended to specify a specific door or even that he intended to specify that the player must choose after the host has opened a door. I have tried to discuss this several times before but nobody seems interested. I am happy to talk about it again in a new section if you wish. Martin Hogbin (talk) 13:36, 19 June 2010 (UTC)
- You seemingly care not to comment on the opened door. How vague the formulation may be, it's definitely not vague about the host opening a door and then offering the player to switch. Agree?
- Oh please. From Whitaker's problem statement (emphasis added)
- You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?"
- If you do want to discuss the important subject of what exact question Whitaker actually wanted to know the answer to I suggest we do so in a new section. I have tried to raise this issue before, only to be hit with 'what do the sources say?'. They actually say very little on that subject. Martin Hogbin (talk) 18:38, 20 June 2010 (UTC)
- You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?"
- Seems abundantly clear that the player can see which door the host opens (the door the host opens is not obscured in any way, for example the player is not blindfolded) and the host is offering the chance to switch after he opens a door. The desired mental image is clearly that the player is standing in front of two specific doors (not in front of an "unknown" pair of doors in a Heisenberg uncertainty principle sense) and can see a specific third open door. If the problem is symmetrical the answer will be the same regardless of which specific pair of doors the player is standing in front of, but arguing that the player can't tell which pair of doors are involved is plainly ridiculous. As far as I know, there isn't any published solution (not even a single one - out of at least 50 that I've personally read) that asserts the player must decide before the host opens a door or asserts that the player can't tell which door the host opened. -- Rick Block (talk) 14:15, 19 June 2010 (UTC)
- Oh please. From Whitaker's problem statement (emphasis added)
- @Martin: why are you so reluctant to accept that the player is standing in front of two closed and one opened door showing a goat. Why? This is where the "paradox" stems from. Nijdam (talk) 16:37, 19 June 2010 (UTC)
- Because there may well be a difference between the question that Whitaker intended to ask and the actual words that he used. Any decent statistician would have tried to do what Seymann suggested (and vos Savant pretty well did) and answer the question that Whitaker probably meant to ask. It is extremely unhelpful for a professional to stick rigidly to the exact words used when an amateur asks a question. Martin Hogbin (talk) 17:44, 22 June 2010 (UTC)
- @Martin: why are you so reluctant to accept that the player is standing in front of two closed and one opened door showing a goat. Why? This is where the "paradox" stems from. Nijdam (talk) 16:37, 19 June 2010 (UTC)
- In the symmetrical case, the simple solutions necessarily give the correct answer because of an obvious and intuitive symmetry. Spotting symmetries and using them to simplify mathematical problems is a well used and respected mathematical technique. You have no right to demand that a problem can only be solved by a method of your choosing. Martin Hogbin (talk) 11:38, 19 June 2010 (UTC)
- Right. And if an unconditional solution actually says this, then it is no longer an unconditional solution but a conditional solution. The point you seem to keep ignoring is that YOU are adding this argument to the published simple solutions. Since they don't argue this themselves we can't argue this on their behalf - that would be WP:OR, even though is it perfectly true. -- Rick Block (talk) 14:28, 19 June 2010 (UTC)
- We need to cover all the angles in the article, in a way that us useful and informative to our readers. We should, and do, point out in the article the limitations of the simple solution. This should be the case for all solutions. Martin Hogbin (talk) 11:11, 19 June 2010 (UTC)
- Right, and we definitely should pass to any reader the warning of several authors about the flaw in the simple solutions.Nijdam (talk) 11:28, 19 June 2010 (UTC)
- Indeed, in the appropriate place. Martin Hogbin (talk) 11:38, 19 June 2010 (UTC)
- Right, and we definitely should pass to any reader the warning of several authors about the flaw in the simple solutions.Nijdam (talk) 11:28, 19 June 2010 (UTC)
Unambiguous?
The problem statement, even the supposedly unambiguous one, fails to state the fundamental rule of the game, which is that the player wins the car if he picks the car door. Or that he wins whatever is behind the chosen door and he prefers the car over goat. Ketorin (talk) 15:15, 19 June 2010 (UTC)
- The quoted problem statement from Parade includes an editorial note that the goats are booby prizes. This seems like a sufficient clarification. -- Rick Block (talk) 19:30, 19 June 2010 (UTC)
- Yes the "booby prize" clarification is within the conventional problem statement, and as such it would probably be tolerable for that statement alone (given the other discussed uncertainties as well). I am bothered with the so claimed "fully unambiguous" one that does not have this clarification at all.
- The point of the Krauss and Wang version is that it clarifies the ambiguities about the host's behavior and the nature of the question (in the standard way). Their description is "A formulation of the Monty Hall problem providing all of this missing information and avoiding possible ambiguities of the expression "say, number 3" would look like this (mathematically explicit version):", where the missing information they're referring to consists of the host's preference between two goats, the requirement that the host open a door to reveal a goat and make the offer the switch, and the initial distribution of the car behind the three doors. Would you prefer some different wording introducing this version? Perhaps by making this change:
- According to Krauss and Wang (2003:10), even if these assumptions are not explicitly stated, people generally assume them to be the case
. A fully unambiguous, mathematically explicit version of the standard problem is; their mathematically explicit version of the problem is:
- According to Krauss and Wang (2003:10), even if these assumptions are not explicitly stated, people generally assume them to be the case
- The understanding that the goats are undesirable booby prizes presumably carries forward from the earlier editorial note. -- Rick Block (talk) 21:07, 19 June 2010 (UTC)
- The point of the Krauss and Wang version is that it clarifies the ambiguities about the host's behavior and the nature of the question (in the standard way). Their description is "A formulation of the Monty Hall problem providing all of this missing information and avoiding possible ambiguities of the expression "say, number 3" would look like this (mathematically explicit version):", where the missing information they're referring to consists of the host's preference between two goats, the requirement that the host open a door to reveal a goat and make the offer the switch, and the initial distribution of the car behind the three doors. Would you prefer some different wording introducing this version? Perhaps by making this change:
- Does anyone see a problem with adding a sentence to the paragraph beginning 'Certain aspects...' making clear that the player receives the prize behind the finally-chosen door and that it is generally assumed that the objective of the game is to win the car'? There are sources which say something like that. This may be obvious to most people but this is a very pedantic subject and we should not leave any loopholes. Martin Hogbin (talk) 09:54, 20 June 2010 (UTC)
- The "Certain aspects" is specifically about the host's behavior - if we must add something about this I'd prefer it be separate from this (already long) paragraph. We could editorially annotate the K&W problem statement or perhaps add something at the very beginning of the section explaining this aspect of Let's Make a Deal. I'll try adding something. -- Rick Block (talk) 16:57, 20 June 2010 (UTC)
- I think it is a good idea to add something somewhere. This is a subject where obvious assumptions sometimes turn out to be wrong (although sometimes they do turn out to be correct). Best make everything crystal clear. Martin Hogbin (talk) 18:31, 20 June 2010 (UTC)
- Did anyone ever understood otherwise??Nijdam (talk) 19:38, 20 June 2010 (UTC)
- Maybe not, who knows, but this is a rather pedantic topic where unjustified assumptions are frowned upon. Martin Hogbin (talk) 17:35, 21 June 2010 (UTC)
- Did anyone ever understood otherwise??Nijdam (talk) 19:38, 20 June 2010 (UTC)
- Here's the diff of the change [2]. -- Rick Block (talk) 13:50, 21 June 2010 (UTC)
- I do not much like the idea of inserting text, even in square brackets, into a quotation. Also it is not clear that the statement refers to the finally-selected door. Could we add this sentence, 'Also it is generally assumed that the car is the desired prize and that the goats are booby prizes and that the player gets the prize behind the door that they finally select' before the sentence starting 'According to Krauss...'? Martin Hogbin (talk) 17:35, 21 June 2010 (UTC)
- This is really an entirely different topic than what the "Certain aspects ..." paragraph is about. Using square brackets is the standard way of annotating quotes. If we're not going to clarify this inline in the quote then I'd prefer saying nothing about this at all or clarifying this at the beginning of the section, i.e. start the section with "In the game show Let's Make a Deal, originally hosted by Monty Hall, contestants won prizes by picking doors or curtains hiding the prize, sometimes winning very valuable prizes like cars or sometimes winning undesirable booby prizes like goats. Steve Selvin described a problem loosely based on this show ...".
- @Ketorin - when reading this did you actually not understand that what is behind the player's choice of door is what the player wins and that the car is considered the preferred, or is this a hypothetical concern? I think anyone actually confused about the context would follow the link to Let's Make a Deal, which I think makes both of these points abundantly clear. This is frankly starting to sound like pedantry for pedantry's sake. -- Rick Block (talk) 18:18, 21 June 2010 (UTC)
Actually, this silly discussion exemplifies the way that some editors do not understand the true nature of the MHP. It is essentially a 'simple mathematical puzzle', in which it is customary to make all necessary assumptions to keep the problem simple. Once you start to consider real-world scenarios or the exact words that someone said, the problem looses it whole point.
In other words, we should assume whatever makes vos Savant's notable solution correct. The player wants the car, an unspecified door is opened by the host... You know the rest. Once you start to get stupidly pedantic about a simple puzzle you start a never ending spiral of pedantry, as is shown here. Martin Hogbin (talk) 17:29, 22 June 2010 (UTC)
- Alas, there is nothing to be assumed making Marilyn's so called solution correct. Also not you mentioning: ...an unspecified door is opened by the host.... Perhaps this form of the problem may be given as a variant. Face it: a door is opened by the host, so we (the player) knows which door. That's what counts. It need not be door 3, if you like door 2 better, please. Or the player may have chosen door 3, and the host opens door 1. It's fine with me.Nijdam (talk) 20:45, 22 June 2010 (UTC)
- The majority of sources that give the simple solution might be assumed to make different assumptions from those that give a conditional solution. Unfortunately, in common with most sources that give conditional solutions, they do not make their assumptions clear, however they do make their solutions clear. Why do we assume here that the majority of sources got the question wrong? Martin Hogbin (talk) 22:26, 22 June 2010 (UTC)
- Alas, there is nothing to be assumed making Marilyn's so called solution correct. Also not you mentioning: ...an unspecified door is opened by the host.... Perhaps this form of the problem may be given as a variant. Face it: a door is opened by the host, so we (the player) knows which door. That's what counts. It need not be door 3, if you like door 2 better, please. Or the player may have chosen door 3, and the host opens door 1. It's fine with me.Nijdam (talk) 20:45, 22 June 2010 (UTC)
- The clarification that the goats are unwanted has actually been requested several times and had recently been (until this edit) a separate sentence in the "Problem" section. I moved it to an editorial note in the quote of the Parade version to keep the clarification, but in a more compact form. I think that the player wins what is behind the chosen door is strongly implied by the wording (isn't this just the standard meaning of "choose"). How about if we keep the "[unwanted booby prizes]" clarification in the K&W quoted problem but ditch the clarification about winning what is behind the door you choose? -- Rick Block (talk) 18:09, 22 June 2010 (UTC)
- Whatever. That was not really my point. Martin Hogbin (talk) 18:23, 22 June 2010 (UTC)
- Yes, I know what your point is. I'm trying to keep this thread on the topic at hand. -- Rick Block (talk) 19:16, 22 June 2010 (UTC)
- Whatever. That was not really my point. Martin Hogbin (talk) 18:23, 22 June 2010 (UTC)
A Morgan Checklist
Selvin describes the problem with boxes. The contestant has chosen box 'B'. The host reveals nothing (a goat) inside of box 'A'.
Selvin first solved his problem with a table of 9 entries. The car in all 3 locations times the contestant choosing all 3 doors.
In this table, he showed each of the three 2-empty-boxes (2-goats) scenarios on a single line.
In his second letter he states that the host choosing randomly between goats is a premise of his puzzle.
He solves the puzzle conditionally using 1/2 to represent the host's probability of either door when faced with 2 goats.
Morgan's original paper focuses solely on Whitaker's letter to vos Savant, with the contestant selecting door 1, and the host revealing door 3.
In their response to the correction letter from Martin and Nijdam, they say:
- "Simply put, if the host must show a goat, the player should switch." - Morgan, et al
- "To wit, had we adopted conditions implicit in the problem, the answer is 2/3, period." - Morgan, et al
- The American Statistician, May 2010, Vol. 64, No. 2
- Morgan, J. P., Chaganty, N. R., Dahiya, R. C., and Doviak, M. J. (1991),
- “Let’s Make a Deal: The Player’s Dilemma,” The American Statistician, :45 (4), 284–287
- Comment by Hogbin and Nijdam and Response
I have three comments:
- Given that Morgan, et al only talk about Whitaker's letter in vos Savant's column in their original paper, they must be still talking about Whitaker's letter to vos Savant when they made the above 2 comments.
- Given that Selvin solves the problem he just originated unconditionally for all outcomes, it's illogical for Morgan and others to later argue that the MHP should be taken literally to mean doors 1 & 3.
- There's no way of knowing which portions of Morgan's original paper they still support. There is at least one math error, and two walk-backs.
The various sources are published, and per Wikipedia will be part of the article. But that doesn't mean the article has to slavishly follow this discredited POV. Glkanter (talk) 08:27, 23 June 2010 (UTC)
According To Morgan, There Is No Difference Between Prior and Posterior
"To wit, had we adopted conditions implicit in the problem, the answer is 2/3, period. Morgan, et al 1
"Simply put, if the host must show a goat, the player should switch." - Morgan, et al 1
That's "...the answer is 2/3, period."
and
That's not, "...after the host has shown a goat". It's "if the host must show a goat..."
Simply put, to wit, the Queen Bee of 'simple solutions are false' has recanted.
1 The American Statistician, May 2010, Vol. 64, No. 2
Morgan, J. P., Chaganty, N. R., Dahiya, R. C., and Doviak, M. J. (1991),
“Let’s Make a Deal: The Player’s Dilemma,” The American Statistician,
45 (4), 284–287: Comment by Hogbin and Nijdam and Response
Glkanter (talk) 13:59, 23 June 2010 (UTC); revised order of Morgan's comments - Glkanter (talk) 20:03, 23 June 2010 (UTC)
- It really doesn't matter how many times you repeat these non sequiturs about Morgan et al., they are not going to become true by the sole virtue of being repeated ad nauseam on this talk page. -- Coffee2theorems (talk) 19:29, 23 June 2010 (UTC)
- The Morgan paper criticizes the simple solutions for the arguments they use, not for the value (2/3) or the decision (the player should switch) that is obtained. Their criticism of the "simple" (i.e. unconditional) solutions is exemplified by statements like "F1's beauty as a false solution is that it is a true statement! It just does not solve the problem at hand." and "F5 is incorrect because it does not use the information in the number of the door shown." Clearly they say nothing that would constitute a retraction of these statements, so your conclusion that "the Queen Bee of 'simple solutions are false' has recanted" does not follow.
- Unfortunately, I do not have full access to the response by Morgan et al., only the first page of two at [3]. Their comment that "Simply put, if the host must show a goat, the player should switch." is not new, but is already contained in the original article. They are reiterating it just in case the reader has forgotten it ("In any case, it should not distract from the essential fact that [...]"). The response "To wit, had we adopted conditions implicit in the problem, the answer is 2/3, period." appears to be about an issue unrelated with the correction in the math, and the discussion gets cut short by the page break there (including the references to the letters where the issue was raised). My guess is that this is an admission that the standard assumptions (with the possible exception of the unbiasedness of the host) are implicit in the problem. That, however, again has nothing to do with the criticism of the "simple" (unconditional) solutions. -- Coffee2theorems (talk) 23:54, 24 June 2010 (UTC)
POV and disclaimers
There has been some edit warring over the addition of disclaimers in the various sections. Without doubt there are sources that give simple solutions and sources that say that the simple solutions are incomplete. There are also editors who hold both points of view.
It would seem to me least POV to not have a disclaimer in the simple section saying it is incomplete and also not have assertions in the Probabilistic section saying that some sources disagree. A full and sensible discussion of the reasons why some sources say the simple solutions are incomplete at the start of the Probabilistic section would seem the least POV to me. Martin Hogbin (talk) 15:44, 23 June 2010 (UTC)
- If no one can give a good reason why we should have POV disclaimers in the article, I will remove them. Martin Hogbin (talk) 20:06, 23 June 2010 (UTC)
Carlton's Decision Tree
Please offer comments, criticisms, and questions.
Glkanter (talk) 19:21, 23 June 2010 (UTC)
I'm fixing to add this to Carlton's solution in the simple section. Any comments, etc. before I do so? Glkanter (talk) 10:53, 26 June 2010 (UTC)
- This diagram does not match the diagram (Figure 5) in Carlton's paper. AFAICS this is original research. -- Coffee2theorems (talk) 14:16, 26 June 2010 (UTC)
- That's correct, this tree does NOT support that solution. Rather, this tree supports the 'intuitive solution' he gives at the very beginning of section five, which is included in the Wikipedia article's Simple solution section. It is the only solution without a visual aid. The branching on car/goat rather than door's is Carlton's. The initial 1/3 and 2/3 come from Cartlton. The calculations of 1/3 * 100% = 1/3, and 2/3 * 100% are simple math. Everything else in the tree comes from Whitaker's letter, which Carlton includes in full earlier in the paper.
- I don't agree, therefore, that there is any OR in this tree. Glkanter (talk) 14:26, 26 June 2010 (UTC)
- For the solutilon you're talking about, Carlton doesn't say anything about "initial probability" or "probability after revealing a goat". What you're doing is caled synthesis, which is a form of OR - please see WP:SYNTHESIS. -- Rick Block (talk) 15:58, 26 June 2010 (UTC)
Carlton repeats Whitaker's letter in it's entirety in his paper.
Combining the premises of a puzzle with the solution is not WP:Synthesis. It's called 'visual presentation'.
Carlton certainly mentions the starting 1/3 and 2/3 probabilities as part of his 'intuitive solution'.
Wikipedia does not regard simple math as OR.
I find your objection unsupported by Wikipedia policies. Glkanter
- Your diagram refers to a conditional probability ("Probability After Revealing a Goat"), even though Carlton's informal explanation, which the diagram is intended to illustrate, refers to no conditional probabilities at all. Further, the way this conditional probability is obtained is very odd. You said that it comes from your own calculation "2/3 * 100% = 2/3". Huh? The only conditional probability currently covered in the Wikipedia article is P(C = 2 | H = 3, S = 1), and this calculation is certainly not a justified way of computing that! I'm guessing that it is some other conditional probability instead. If so, it certainly should not be abruptly introduced in a diagram, but be covered in the article text before the diagram occurs. Of course you would also need a source mentioning that this new conditional probability is of interest in the context of the MHP. -- Coffee2theorems (talk) 22:55, 26 June 2010 (UTC)
Carlton's solution, like every other solution, relies on this, from the article: "The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it."
That's 100%. Glkanter (talk) 23:04, 26 June 2010 (UTC)
And it SURE DOESN'T say MUST BE DOOR #3, does it? Glkanter (talk) 23:05, 26 June 2010 (UTC)
Unreferenced Figure In The Conditional Solution section
Rick Block, it looks like you uploaded the various car/goat/door images in the 'expanded figure below', so I'll presume that you are the originator of that overall figure.
Neither the narrative or the figure are referenced. Can you please provide a source for this text and figure? Otherwise, I'm sure you would agree that it must be considered OR. Please advise.
"The conditional probability of winning by switching given which door the host opens can be determined referring to the expanded figure below, or to an equivalent decision tree..."
Glkanter (talk) 19:12, 26 June 2010 (UTC)
The first paragraph of the Solution section.
There's no references, etc., except a mention of Selvin. Otherwise is this paragraph from a source(s)? It reads as OR/editorial commentary. Glkanter (talk) 21:08, 23 June 2010 (UTC)
Besides, The Combined Doors solution does show door 3 being opened to reveal a goat. Nijdam's real complaint with that simple solution actually rests on his argument that a prior 1/3 probability is not the same as a posterior 1/3 probability. The paragraph adds no value whatsoever. Glkanter (talk) 21:14, 23 June 2010 (UTC)
Gill's changes
Gill has made some improvements to the readability and factual accuracy of this article as well as adding a section on a game theoretical solution. I hope editors will discuss whatever they feel might be wrong with these changes before blindly 'correcting' them. Martin Hogbin (talk) 15:04, 24 June 2010 (UTC)
I see that I was too late. Martin Hogbin (talk) 15:09, 24 June 2010 (UTC)
- The reason for the "correction" (it was actually revert), was that gill replaced old content without seeking consent, i.e. it was by all means a controversial edit without consent. The issue here in particular was not adding new material but removing the old formal solution, which in different was part of the article since it became featured. And the proper order for such a change is to seek consent first and not to make a controversial edits first and ask later.--Kmhkmh (talk) 15:16, 24 June 2010 (UTC)
- How about putting the formal solution back in a separate section after the 'Game theoretical' section. Very few people will be interested in it but, if it agreed as a rigorous formal solution, it should have a place. Martin Hogbin (talk) 15:23, 24 June 2010 (UTC)
- I have no objection against adding (rather than replacing) the odds-ratio approach and I told Gill explicitly so, nevertheless he insisted on removing it again.--Kmhkmh (talk) 15:32, 24 June 2010 (UTC)
- I think it should come later, following the general rule of simple stuff first. We might get something that everyone can agree on. Martin Hogbin (talk) 15:46, 24 June 2010 (UTC)
- How about putting the 'Game theoretical' in a separate article. Very few people will be interested in it but, if it agreed as a rigorous formal solution, it should have a place. glopk (talk) 15:37, 24 June 2010 (UTC)
- That is going too far. It is far more interesting that a rigorous formal solution, which is only there for completeness. Martin Hogbin (talk) 15:46, 24 June 2010 (UTC)
- Martin, who is "it"? To whom it is "far more interesting that a rigorous formal solution"? Are you representing a consensus larger than the number of neurons inside your skull? To me a rigorous formal solution is very interesting and useful in this article. It allows a mathematically conversant reader to cut through the cruft and get a satisfactory grasp for a correct solution.glopk (talk) 15:53, 24 June 2010 (UTC)
- Only a small fraction of our readers would understand a rigorous formal solution. They, like you would, no doubt find it interesting. I suspect that more people would be interested in knowing the result if the player and TV show adopt strategies to optimise their own position. But why not have both? Martin Hogbin (talk) 16:01, 24 June 2010 (UTC)
- Do you have hard numbers on how "small" a fraction of readers can read math? On the other hand, do you know how many readers would prefer a well-worked out solution to a soup of barely mentioned approaches? Again, please do not stand up on a box as the "advocate" of the "common reader" - you are just an editor like everyone else, pushing your view (one would hope in good faith). Lacking hard numbers, the best one can hope for here is informed editorial consensus. Lacking that, an informed majority. Clearly neither exist in support of Gill's (and now yours) bold changes. glopk (talk) 16:17, 24 June 2010 (UTC)
- No, I do not have figures, it is just my estimate. Let us just have both. Martin Hogbin (talk) 08:16, 25 June 2010 (UTC)
- Do you have hard numbers on how "small" a fraction of readers can read math? On the other hand, do you know how many readers would prefer a well-worked out solution to a soup of barely mentioned approaches? Again, please do not stand up on a box as the "advocate" of the "common reader" - you are just an editor like everyone else, pushing your view (one would hope in good faith). Lacking hard numbers, the best one can hope for here is informed editorial consensus. Lacking that, an informed majority. Clearly neither exist in support of Gill's (and now yours) bold changes. glopk (talk) 16:17, 24 June 2010 (UTC)
- Only a small fraction of our readers would understand a rigorous formal solution. They, like you would, no doubt find it interesting. I suspect that more people would be interested in knowing the result if the player and TV show adopt strategies to optimise their own position. But why not have both? Martin Hogbin (talk) 16:01, 24 June 2010 (UTC)
- Martin, who is "it"? To whom it is "far more interesting that a rigorous formal solution"? Are you representing a consensus larger than the number of neurons inside your skull? To me a rigorous formal solution is very interesting and useful in this article. It allows a mathematically conversant reader to cut through the cruft and get a satisfactory grasp for a correct solution.glopk (talk) 15:53, 24 June 2010 (UTC)
- That is going too far. It is far more interesting that a rigorous formal solution, which is only there for completeness. Martin Hogbin (talk) 15:46, 24 June 2010 (UTC)
- How about putting the 'Game theoretical' in a separate article. Very few people will be interested in it but, if it agreed as a rigorous formal solution, it should have a place. glopk (talk) 15:37, 24 June 2010 (UTC)
- The game theoretic approach belongs in the variant sectioni. It is not a solution to the standard problem but a variant where the host has more flexibility. Rick Block (talk) 00:57, 25 June 2010 (UTC)
- It is a good solution for vague problem statements such as Whitaker's. There re still three doors, the standard rules still apply, host always offers the swap, and always opens an unchosen door to reveal a goat. Martin Hogbin (talk) 08:16, 25 June 2010 (UTC)
- I think it should come later, following the general rule of simple stuff first. We might get something that everyone can agree on. Martin Hogbin (talk) 15:46, 24 June 2010 (UTC)
- I have no objection against adding (rather than replacing) the odds-ratio approach and I told Gill explicitly so, nevertheless he insisted on removing it again.--Kmhkmh (talk) 15:32, 24 June 2010 (UTC)
- How about putting the formal solution back in a separate section after the 'Game theoretical' section. Very few people will be interested in it but, if it agreed as a rigorous formal solution, it should have a place. Martin Hogbin (talk) 15:23, 24 June 2010 (UTC)
Nice discussion.
In my opinion the odds form of Bayes rule is the simplest rule and the rule which nowadays you will find many authorities using.
The long formal calculation which comes down to proving Bayes rule by hand is superfluous since it exists on another wikipedia page.
The game theoretic version of the story goes back to Nalebuff hence to pre-vos Savant days and "everyone in economics and game theory" knows it, they just are too busy getting Nobel prizes or fat consulting jobs (using game theory to advice governments and multinationals) to waste their time writing up a five second "in your mind" exercise during Game Theory 101. Sorry, that's how they are.
The proof by symmetry, the proof by odds, the proof of the unconditional version by easy logical deduction are all equally mathematically rigorous. The mathematical correctness of a proof does not depend on the number of lines of formula manipulation which the writer feels necessary to write out at length. Mathematicians prefer proofs by using exciting and important concepts. Gill110951 (talk) 09:11, 25 June 2010 (UTC)
My aim by the way is to restore simplicity and unity and clarity to the Month Hall page, and to restore a neutral point of view. An ambiguous sentence like vos Savants' which made the problem famous and is the reason we have this page at all, and the reason these discussions go on endlessly, is open to multiple scientific and in particular mathematical formulations. Anyone who says that their formulation is the only true formulation is a joker or an ignoramus or a terrorist. I suggest all editors of these pages actually read all the literature and also make some effort to understand different points of view, even if they are different from a point of view which they have held strongly for a long time. A true scientist is prepared to change their mind on seeing new evidence. That is what Bayes theorem tells us. A not true scientist makes up their mind and interprets all new information through their prejudiced point of view.
I have changed my mind on Monty Hall many times! I am a mathematician and all the mathematicians I know, care about the unconditional problem. But I know there are lots of folks out there who love the conditional one. It is beautiful, I think, to bring these approaches closer together and show their relations. Finally the whole is more than the sum of the parts. —Preceding unsigned comment added by Gill110951 (talk • contribs) 09:16, 25 June 2010 (UTC)
By the way, the simple verbal solution to the simple unconditional problem and the many alternative and simple verbal solutions to the more complex conditional probability question can all be converted "word for word" into equally short formal algebraic computations, if that is what some of you mean by a rigorous mathematical proof. Real mathematicians like proofs which they can tell their friends while making a walk in the forest. They like to replace calculations by ideas, so that you know why the answer is correct without any calculation. Algebraic manipulations are boring and better left to computers or accountants. There is no way anyone can say that the present tedious lines of formulas are better mathematics than a couple of sentences containing real logical deductions from hard ideas. Just try to digest the ideas in the sentences. Now try to digest the ideas in the formula manipulation. Which meal ends up making you feel you have learnt something, that you understand the problem better now, at a deeper level?
By the way the suggestion some of you made earlier that I am here to push my Own Research is such an incredible joke. Being a senior professional mathematician my professional standing is harmed, not improved, by spending my time discussing elementary probability on wikipedia and writing up educational/amusement maths articles for obscure journals about what I have learnt from all you great guys here. Seriously, I love a debate, I love a fight. That is how science progresses. I am putting in an acknowledgement to all wikipedia editors for the fantastic ideas and fantastic stimulus which comes from this thing. I love it. Gill110951 (talk) 13:22, 25 June 2010 (UTC)
Bayes vs. Gill's argument
IMO, the mathematical formulation should be the classic Bayes expansion (as it was before Gill's changes). This expansion appears in dozens of sources and should be included. The simpler conditional argument (equivalent to Rosenthal's) could be included as well but it is distinctly rare (in sources). -- Rick Block (talk) 15:12, 24 June 2010 (UTC)
- I agree I see zero reason to remove it and zero reason to assume any consent in that matter. Hence I reverted it originally, however to avoid triggering an edit I refrained from further reverts after Gill insisted on making the change without consent.--Kmhkmh (talk) 15:19, 24 June 2010 (UTC)
- How about a separate section again? We then have several solutions, simplest first moving to the more complex. Martin Hogbin (talk) 15:27, 24 June 2010 (UTC)
- Essentially that would be fine with me at least as long as no other editors have additional issues regarding the new material.--Kmhkmh (talk) 15:38, 24 June 2010 (UTC)
- I have issues both general and specific. General, in regard to the fact that the page is already very long. Specifically, in regard to Gill's verbose and basically unreadable formal treatment - really, I thought the argument on whether math notation is more informative than verbiage had been resolved two centuries ago.
- As for the "Game theoretic" section added by Gill, I object to its being little more than an expanded list of references, short on explanation for why it is a genuinely relevant approach. Anyone worth their mathematical salt can come up with a new reformulation of the MHP and call it "original" and "novel". The job of the editors of an encyclopedic article is to both report AND select sources for relevance. glopk (talk) 15:49, 24 June 2010 (UTC)
- Maybe the section could be improved but it does represent an alternative approach based different assumptions. Martin Hogbin (talk) 15:53, 24 June 2010 (UTC)
- Regarding the article length, the variants section would be the best thing to move if the article is considered too long. Martin Hogbin (talk) 15:56, 24 June 2010 (UTC)
- Essentially that would be fine with me at least as long as no other editors have additional issues regarding the new material.--Kmhkmh (talk) 15:38, 24 June 2010 (UTC)
- How about a separate section again? We then have several solutions, simplest first moving to the more complex. Martin Hogbin (talk) 15:27, 24 June 2010 (UTC)
Gill's argument is Bayes argument in a short verbal form which requires less computation and is moreover highly illuminating for understanding the problem.
- Translation into English: "It's my own proof, I like it better, everybody else must like it too!"
The long mathematical proof is a sequence of formula manipulation steps, suitable for computers, not for humans. To understand it, requires hard work and expansion.
- The "long" mathematical proof consists of 1 definition and 6 equations, all of them commented at lenght. As far as symoblic proofs go it's very very short. IMO, A reader who cannot do the "hard work" of reading is obviously not mathematically conversant enough to read any math, so won't likely be interested in any formal proof, either in symbols or in words.
Moreover the formal Bayes computation is contained elsewhere on wikipedia so need not be duplicated here. However to give all you sceptics a chance, I have simply added the "odds form" proof as an alternative proof, and added two more short alternative proofs. I am astounded that half a page of formal formula manipulations is thought to be more illuminating and useful for the general reader than a couple of sentences based on "posterior odds equals prior odds times Bayes factor".
- You must be new to editing WP articles (or interacting with any online community). There are many many thing you'll find astounding along the way, but worry not, you'll learn to be humble and listen rather than just speak,
I agree, you have to start thinking about odds instead of probabilities. That is difficult for non Anglo-Saxons but still it is useful to get used to the idea.
- For example, one thing you will quickly learn (under penalty of ridicule or ban) is that en.wikipedia.org is read and edited by many many English-fluent people who are not Anglo-Saxons, and that Anglo-Saxon people do not have any special status therein. Statements like yours above are seen as condescending at best, offensive at worst. Being seen as condescending or offensive does not add to your reliability as a wikipedia editor.
Of course my writing may not be optimal so please go ahead and improve, cut verbiage, clean and smooth... but first read and try to understand. Then we can discuss again.
- That works well in general. However, when a WP article has been peer reviewed enough and deemed of high enough quality to be "Featured", prudence must be exercised in editing. Exactly because the content of WP is not OR, it is rare that anything new is dicovered, that makes a featured article suddenly obsolete. Therefore radical edits to a featured article tend to be seen with skepticism, and the onus is on the editor (YOU) to show to the community that a radical re-edit is worthwhile.
I repeat that the long section with the formal manipulation is totally out of place on this page and moreover superfluous since you can find it elsewhere on wikipedia.
- Believe me, I can read, and understand that this is YOUR opinion. Now, open your eyes and read: there are other editors (your peers) that think you are wrong. This is not black and white, it is not a matter of truth of falsity of a theorem. It's an editorial decision, a matter of opinion that intelligent and informed people can compromise upon. Your diktats don't work here.
I am stunned that anyone objects to the short proof based on symmetry. It is beautiful and entirely correct.
- I am stunned that you cannot engage in a constructive conversation about your edits. You have so far used a steamroller approach. I do believe there is some merit in some of the material you have offered, as far as the MHP article. However, your way of offering them has so far only managed to get you a few reverts. Don't you think there is a more constructive way?
I am stunned that noone appreciates learning that Monty Hall is not the exclusive ownership of statisticians. Read the literature, learn Game Theory 101; it can be as important for your intellectual development as Probability 101. Gill110951 (talk) 05:30, 25 June 2010 (UTC)
- You get stunned easily, perhaps WP editing is not for you after all. glopk (talk) 22:08, 25 June 2010 (UTC)
BTW the recent and very nice Rosenthal book and Rosenthal article expound the odds version of Bayes in clarifying Monty Hall conditional versus unconditional. It makes Monty Hall so much better understandable. No formulaic mumbo jumbo. Just a simple intuitively appealing and extremely power correct principle which everyone would benefit from knowing. Gill110951 (talk) 13:38, 25 June 2010 (UTC)
Unreferenced OR
Most of Gill's recent changes are completely unreferenced and therrfore appear to be OR. Also they're not written in the appropriate style (see MOS:MATH. I have very limited connectivity at the moment but would suggest we move the new sections here to work on rather than have them in the article in their current state. Rick Block (talk) 16:59, 25 June 2010 (UTC)
- Agree, moving sections below. glopk (talk) 22:31, 25 June 2010 (UTC)
- I trust we can all see this move as a temporary step whilst supporting references are added. The additions certainly improve the article and will move it forward towards becoming a modern and comprehensive exposition of the subject rater than a shrine to a narrow, not very notable, and somewhat contrived treatment. Martin Hogbin (talk) 09:17, 26 June 2010 (UTC)
- The odds-approach can be found in Rosenhouse's book or Rosenthal's paper if I'm not mistaken. For the other parts I'm not sure. I think it might be a good idea to keep those sections here until we have them properly sourced and maybe optimized style/language a bit as well. Once that is done in sufficient manner, they should be reintegrated into the article.--Kmhkmh (talk) 10:08, 26 June 2010 (UTC)-
Alternative Proof: odds form of Bayes law
It is illuminating to use Bayes' theorem in the memorable form, see [4]:
posterior probability is proportional to Bayes prior times likelihood
or equivalently
posterior odds equals prior odds times likelihood ratio, also known as the Bayes factor.
Odds are the ratio of two probabilities. In this case we are look at the ratio of the two probabilities that the car is behind Door 2 or Door 1, first prior to observing a Goat behind door 3, and then posterior to this event. The likelihood ratio, also known as the Bayes factor, is the ratio of the probabilities of what is observed, Goat behind Door 3, given each of the two competing hypotheses (Car behind Door 2;Car behind Door 1). Suppose the Car is initially equally likely to be behind any of the three doors, and suppose the Player has chosen Door 1. We then observed the Host opening Door 3, revealing a Goat. Throughout, we take the Player's initial choice, Door 1, as fixed. Initially the two hypotheses of interest, Car behind Door 2 and Car behind Door 1, were equally likely, hence their prior odds are 1 to 1. We want to know how the odds change by acquiring the information that there is a Goat behind Door 3. The likelihood ratio is by definition the ratio of the probabilities of this information - Host opened Door 3 revealing Goat - under the two competing hypotheses: Car behind Door 2, and Car behind Door 1. If the Car was behind Door 2, the Host was forced to open Door 3, so the probability thereof was 1. If the Car was behind Door 1, the Host had a choice, and if he chooses uniformly at random, the chance that he actually opened Door 3 was 1/2. Hence the likelihood ratio is 1 to 1/2 or 2:1 for Door 2 versus Door 1. Odds of 2 to 1 corresponds to posterior probabilities 2/3 and 1/3 since these are the only two, mututally exclusive, possibilities.
More generally the Host could use any probability between 0 and 1 with which to decide to open Door 3 when he has a choice between Doors 2 and 3. Thus the odds for Door 2 to Door 1 can lie anywhere between 1 to 0 and 1 to 1; i.e., the conditional probability given the initial choice door 1 and the opened door 3 lies anywhere between 1 (odds of 1 to 0) and 50% (odds of 1 to 1). Whichever it is, it is never unfavourable to switch doors (see Morgan et al., 2010). Even if it sometimes makes no difference to switch, there always are situations where it is favourable to switch, since the average probability that switching gives success is 2/3. —Preceding unsigned comment added by Glopk (talk • contribs) 22:33, 25 June 2010 (UTC)
Proof by symmetry
In the case that the Host makes his choice uniformly at random when he has a choice, we can argue using mathematical invariance very rapidly that the conditional probability must be 2/3, since it is, by the law of total probability equal to the average of the conditional probabilities. But in the situation without host bias, the problem remains the same under relabelling of the doors. So all the conditional probabilities (given choices of player and host respectively) are equal, and all must therefore be equal to their average, 2/3. —Preceding unsigned comment added by Glopk (talk • contribs) 22:33, 25 June 2010 (UTC)
Forcing symmetry
According to Marilyn vos Savant the door numbers mentioned in her restatement of Whitaker's question were just added to aid the readers' imagination, and should not play any role in the solution. There is a neat way to mathematically enforce the requirement that the solution should not depend on arbitrary door numberings, as follows. Let the door numbers mentioned in the problem have been introduced by the player for his own personal purposes (semantics: "say Door 1" could mean "I will call this, my first choice, Door 1"), and let him do this (uniformly) at random. First he chooses his own door at random and names that himself "Door 1". He next selects one of the remaining doors at random and names it "Door 2". The third door he names of course, "Door 3". He observes the Host open Door 3 revealing a Goat. By the player's symmetrization of the problem, i.e., by randomization of door names, the host is equally likely to open either of the doors which he himself named Door 2, Door 3, when the car is actually behind the player's initial choice. His conditional probability of finding the car by switching to the door which he named Door 2, is therefore 2/3. —Preceding unsigned comment added by Glopk (talk • contribs) 22:35, 25 June 2010 (UTC)
Symmetry by ignorance
For many people, perhaps starting with Laplace, probability is a measure of subjective degree of belief. From this point of view, it is reasonable to suppose that the player has no idea about whether or not the host has any bias, and that his beliefs about the possibly bias are completely symmetric. For such a player, the host when allowed to choose, chooses with some probability p which lies somewhere between 0 and 1, and about which the player knows nothing. If the player is totally ignorant about this possible bias he will be equally likely e.g. to believe it is below 1/4 as above 3/4. His belief is symmetric. Thus the player expects 0 bias, or in other words, the player's expectation of the host's p is 1/2. Morgan et al., as corrected by wikipedia editors Hogbin and Nijdam (2010), and acknowledged by Morgan et al. (2010), come out with a posterior subjective probability of 2/3 that switching gives the car, in this scenario. The computations using Bayes theorem can be avoided by remarking that the symmetry of the Bayes hyperprior on p makes the problem symmetric in the door numbers, and thereby forces the conditional probabilities to be 2/3 again.
Game theoretic version
According to economists and game theorists, for instance the famous Nalebuff (1987) and Rinnoy Kan (1990), the Monty Hall problem is a problem of game theory or decision theory. According to this formulation, we see this final phase of the quiz show as a game between the Host, who has hidden the Car behind a Door, and later opens a Door to reveal a Goat, and the Player, who first chooses a Door, and later has the opportunity to revise his initial choice. The Host wants to keep the Car, the Player wants to get it. This is a finite two-person zero-sum game so there exists a solution, equilibrium, or saddle-point by the famous minimax theorem of John von Neumann (1926), the founder of game theory and hence of much of mathematical economics. At least, there exists a solution, as long as we allow both persons (Host and Player) to use randomization (i.e., toss coins or dice to determine their choices). In this context, a solution is a strategy for the Host, called his minimax strategy, a strategy for the Player (his minimax strategy), and a number p, such that if the Player uses his minimax strategy he is guaranteed to win the Car with at least probability p, whatever strategy is used by the Host, and if the Host uses his minimax strategy, he is guaranteed to lose the Car with at most probability p, whatever strategy is used by the Player. Now consider the following pairs of strategies: Host: hide Car uniformly at random (with equal probabilities over all possibilities); later open a door uniformly at random if he has a choice; Player: choose Door uniformly at random; later always Switch. With the Player's minimax strategy the Player wins the Car with at least probability 2/3 (in fact, with exactly probability 2/3) whatever strategy is used by the Host. Conversely, with the Hosts' minimax strategy, the Player can't do better, since his initial choice is irrelevant if the car is hidden uniformly at random, and switching gives him 2/3 probability to get the car, not switching only 1/3 probabilitiy, so all he can get (by randomization of these two deterministic strategies) are probabilities between 1/3 and 2/3.
This comes from the Conditional Solution section.
..."The simple solutions correctly show that the probability of winning for a player who always switches is 2/3, but without further assumptions this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens...
I don't think that's a true statement. What further assumptions are required, and what other results will a simple solution return? Glkanter (talk) 14:57, 26 June 2010 (UTC)
With Selvin's stated premises: "Host always offers switch, host chooses randomly when faced with 2 goats, and the K & W version, I think most of this paragraph from the Conditional solution section is a bunch of hot air. Further, Morgan's later comments contradict a lot of what's written.
Also, 2 of the simple solutions *do* show door 3 being opened: The combined Doors and (with the tree) Carlton's.
Selvin DOES NOT say the problem requires a conditional solution. He simply offered a conditional proof, and made no other comment on his original solution of a table of all outcomes. His citation should be removed from "...That probability is a conditional probability (Selvin 1975b;..."
I have twice offered a proposed format (above) of the article where the differences from the sources are addressed only after the solutions have been given. The sole exception would be vos Savant's Parade controversy would come before the solutions.
"Conditional probability solution (as currently in the article)
The simple solutions show in various ways that a contestant who is going to switch will win the car with probability 2/3, and hence that switching is a winning strategy. Some sources, however, state that although the simple solutions give a correct numerical answer, they are incomplete or solve the wrong problem. These sources consider the question: given that the contestant has chosen Door 1 and given that the host has opened Door 3, revealing a goat, what is now the probability that the car is behind Door 2?
In particular, Morgan et al. (1991) state that many popular solutions are incomplete because they do not explicitly address their interpretation of Whitaker's original question (Seymann). The popular solutions correctly show that the probability of winning for a player who always switches is 2/3, but without further assumptions this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens. That probability is a conditional probability (Selvin 1975b; Morgan et al. 1991; Gillman 1992; Grinstead and Snell 2006:137; Gill 2009b). The difference is whether the analysis is of the average probability over all possible combinations of initial player choice and door the host opens, or of only one specific case—to be specific, the case where the player picks Door 1 and the host opens Door 3. Another way to express the difference is 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); either way, the player is interested in the probability of winning at the time they make their decision. Although the conditional and unconditional probabilities are both 2/3 for the problem statement with all details completely specified - in particular a completely random choice by the host of which door to open when he has a choice - the conditional probability may differ from the overall probability and the latter is not determined without a complete specification of the problem (Gill 2009b). However as long as the initial choice has probability 1/3 of being correct, it is never to the contestants' disadvantage to switch, as the conditional probability of winning by switching is always at least 1/2."
Glkanter (talk) 15:22, 26 June 2010 (UTC)
The Mathematical Contrivance
In between his 2 letters to The American Statistician, it looks like Selvin heard from a few of his peers. That's likely why he clarified things with:
- The host always offers the switch
- The host will choose randomly when faced with 2 goats
- He offered the more formal conditional solution
Well, as we see from Carlton's solution, it's not the random selection premise that's needed at all. Selvin included it, because his buddies convinced him that he had to state that it was 50/50. Because you can't do the customary doors-based conditional solution without it.
Carlton's car/goat based 'conditional' solution (if you consider 100% 'conditional) shows we simply don't care how he chooses between doors. Because, whatever he does, and however he gets to it, the host is revealing a goat.
Because it's the contestant's State of Knowledge about how the host chooses the doors that would matter. And there is NOTHING in the puzzle about that. You would need a new premise. I don't know what it would be, and I don't care. Because that would be a different puzzle.
All those conditional tree solutions with 50/50 when the host has 2 goats? They're BS. The puzzle doesn't require it. But that popular, flawed, overly-complex conditional solution sure does. It doesn't work without it. Everyone that uses it is wrong. Or maybe just lazy. Selvin was wrong, because he believed them (his original simple solution just lumped the 2 goat doors on a single row of his table, with no % split). It could be any split on the face of the Earth, even 0/100, it doesn't matter. Without a contestant's SoK premise to this affect it makes no difference, and Carlton's solution shows this. And as an added bonus, Carlton's solution proves Nijdam's never-ending '1/3 prior probability <> 1/3 posterior probability' filibuster is BS, too. Any number multiplied by 100% equals the original number. duh.
But, they're reliably published (except Nijdam's 'point', whatever it was), so in the article they go. But the heavy handed editorializing that the Combined Doors and Carlton's simple solutions are flawed should stop right now. Glkanter (talk) 22:16, 26 June 2010 (UTC)
Oh, yeah. And that gang of idiots, Morgan, et al, wrote a entire peer-reviewed article in a professional journal about the host having to choose randomly. And beat up that nice lady for no reason. When it's nothing but a lazy mathematical contrivance. Glkanter (talk) 22:26, 26 June 2010 (UTC)