Rick Block (talk | contribs) →So, What Are The Significant Events, And Why, Of The Monty Hall Problem Paradox: so what are you suggesting? |
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:::::Another Straw Man, aka Aunt Sallie. Don't you have anything better to do with your time? [[User:Glkanter|Glkanter]] ([[User talk:Glkanter|talk]]) 16:08, 13 January 2010 (UTC) |
:::::Another Straw Man, aka Aunt Sallie. Don't you have anything better to do with your time? [[User:Glkanter|Glkanter]] ([[User talk:Glkanter|talk]]) 16:08, 13 January 2010 (UTC) |
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::::I'm just trying to understand what you're talking about. You say the article is rife with a pro-Morgan POV and at least imply you think everything after Solutions might as well be deleted. I asked if this is indeed what you are suggesting. Your reply says you're not talking about deleting everything but that you'll be part of a consensus that makes changes, making it sound like some "consensus of editors" will have carte blanche to make whatever changes they collectively want. What I'm saying is that this is not how it works and you're at least implying you know that. OK. So please say what specific changes you are suggesting. Take any one (or more) section I've listed above. Say "I, Glkanter, would like the pro-Morgan POV in section blah blah blah to be eliminated by changing <something> to <something else>". Thank you. -- [[user:Rick Block|Rick Block]] <small>([[user talk:Rick Block|talk]])</small> 18:08, 13 January 2010 (UTC) |
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== Door numbers matter == |
== Door numbers matter == |
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Is The Contestant Aware?
Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?
It seems to me that this is a (unstated) premise of the problem, as both vos Savant (Whitaker) and Krauss and Wang begin the problem statement with: 'Suppose you're on a game show'. I read this as clearly stating it is only the contestant's point of view we are concerned about. And, being a game show, the host is prohibited from divulging to the contestant either where the car is, or where the car is not.
Is there agreement on this, or is this in dispute? Glkanter (talk) 11:28, 29 November 2009 (UTC)
- As far as I know, this is the only known instance where a contestant at home was able to determine a game show's strategy. It was an aberration, an unexpected outcome, and steps were immediately taken to prevent it from happening again.
- In my humble opinion, a particular contestant gaining usable information from the hosts actions, which would make that contestant's SoK something different than the 'average' contestant's SoK of '2/3 likelihood of a car if I switch' is inconsistent with the published problem statement: 'Suppose you're on a game show'. The problem statement would become: 'Suppose you are not on a game show'. This is not merely a 'variant', or a new or changed premise. It is a completely different problem. And why this completely different problem should be referenced as often and prominently as it is in the Wikipedia article does not make sense to me.
- There are countless reliable published sources which use the combining doors solution to derive the '2/3 likelihood of a car if I switch'. The contestant uses this method to determine the 2/3, then properly says to himself, 'Monty's actions haven't given me any new information, so I'll go with the 2/3 for the average contestant'. Glkanter (talk) 14:49, 29 November 2009 (UTC)
- Is there a specific change you're suggesting? If not, I'd suggest moving this thread to the /Arguments page. -- Rick Block (talk) 01:15, 30 November 2009 (UTC)
- Rick, this is the central element in my criticism of the over-reliance on Morgan's paper in the article. In order to develop a consensus that includes you, I need to know your position on it, so I would greatly appreciate your response to the original question:
- "Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?."
- I see this as an 'editing' question more than a 'mathematics' question, so I'd prefer to leave it here. Thank you. Glkanter (talk) 13:51, 30 November 2009 (UTC)
- Rick, this is the central element in my criticism of the over-reliance on Morgan's paper in the article. In order to develop a consensus that includes you, I need to know your position on it, so I would greatly appreciate your response to the original question:
- Is there a published source that takes the stance that "Suppose you're on a game show" means what you're suggesting? If not, then what you're talking about is WP:OR which makes it a moot point as far as editing is concerned. I'm not saying it's a bad or invalid argument, just that if it's not published it's worthless for editing purposes. -- Rick Block (talk) 14:52, 30 November 2009 (UTC)
- Yes, I have previously linked to the Wikipedia articles showing the only 2 instances where individual contestants had a different State of Knowledge than would the average contestant. The 1950s Quiz Show Scandal and Whammy/Press Your Luck. Both were considered as extraordinary events.
- It wouldn't matter, even if it was OR. I'm not going to put this critical mistake of Morgan's in the article (unless other editors want to build a consensus for it). I'm only using it to decide, using facts rather than my personal opinion, how much emphasis Morgan should get in the article.
Rick, I have directly asked you this question many times, and have never seen a direct 'yes or no' answer from you. As this is a crucial element of the consensus that has been built, it is essential that we understand your reasons if you do not agree with the paragraph above:
- "It seems to me that this is a (unstated) premise of the problem, as both vos Savant (Whitaker) and Krauss and Wang begin the problem statement with: 'Suppose you're on a game show'. I read this as clearly stating it is only the contestant's point of view we are concerned about. And, being a game show, the host is prohibited from divulging to the contestant either where the car is, or where the car is not."
I look forward to your response. Glkanter (talk) 23:02, 1 December 2009 (UTC)
- Rick has responded under his comments section. For clarity and closure, I'll copy them into this discussion.
- "Glkanter asks why I haven't responded about his "Is The Contestant Aware?" question. Why should I? Glkanter has repeatedly demonstrated a complete lack of comprehension of nearly everything I've ever said. It's like trying to explain something to a cat. At some point you just have to give up. However, I'll give it another go. Meow, meeeow, meow, meowww. I'm not sure I have that quite right since I don't speak cat, but it's probably about as comprehensible to him as anything else I could say."
- Nijdam, your response to the question at the beginning of this section is also of great interest to the other editors. In the MHP that begins with "Suppose you are on a game show", is the contestant aware of Monty's door choice behaviour when there are 2 goats remaining? Thank you. Glkanter (talk) 15:14, 4 December 2009 (UTC)
- Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method? Yes, but that doesn't mean the contestant is prevented from wondering what affect this might have on her chances of winning by switching.
- Rick, how am I to parse 'yes, but' in the above paragraph? Is that "yes, I agree that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?" Or, "but..." Let the contestant wonder, it's a free country. But that doesn't affect the game play. Glkanter (talk) 18:51, 5 December 2009 (UTC)
- And, being a game show, the host is prohibited from divulging to the contestant either where the car is, or where the car is not. Is there agreement on this, or is this in dispute? The problem specifies that a game show is the setting. How much information the host divulges to the contestant depends on how the problem is specifically phrased, not on what "must" be true of game shows.
- These are both questions that are more appropriate for the /Arguments page than here, since as editors what we believe to be the TRUTH is ultimately of no importance. What is important is what reliable sources say. Our task as editors is to accurately represent what these sources say regardless of whether we individually agree. This one of the three fundamental policies of Wikipedia, see WP:NPOV. -- Rick Block (talk) 17:44, 5 December 2009 (UTC)
Changes suggested by JeffJor, Martin Hogbin, and Glkanter
If you're here because you've been invited to comment, there are ,two,. three (related) suggestions.
- #Glkanter's suggestion: Eliminate all 'host behaviour, etc' influenced discussion, save for the Wikipedia minimum necessary references to Morgan and his ilk, as the 'conditional' problem is the converse of "Suppose you are on a game show."
- If nobody minds, I'd like to revise my proposal to make it more reflective of the literature: 3 Sections to the article: The unconditional MHP, A brief discussion on why Morgan and the 'conditional variants' are not the MHP, and 'diversions' - which includes 'variants', etc.
- #JeffJor's suggestion: The so-called conditional problem needs to be a separate article, with "conditional" in its title.
- #Martin Hogbin's suggestion: This article should concentrate on the unconditional solution with the Morgan's conditional solution in a variations section.
Please indicate in subsections below whether you favor or oppose each of these suggested changes.
The intent is to try to determine whether there is community consensus for any of these changes. I would suggest one subsection per user who is commenting, and to avoid endless arguments, restricting your comments to your own section (this is modeled after the process used at Wikipedia:Arbitration Committee). I've precreated sections for everyone I've explicitly invited to comment. -- Rick Block (talk) 15:31, 2 December 2009 (UTC)
Discussion about setting up this section and inviting folks to comment
In this section please summarize the changes you're suggesting. I'll be asking the set of folks I mentioned to Glkanter above to come here and offer their opinions, so please keep it as brief as possible. Please let me know when you think this section is ready for others to comment on. -- Rick Block (talk) 01:07, 1 December 2009 (UTC)
- Rick, have you invited Boris Tsirel, William Connolley, or C S to contribute their opinions? What sort of time frame do you have in mind before 'In essence, silence implies consent' as per Wikipedia policy? Glkanter (talk) 15:47, 2 December 2009 (UTC)
- I haven't invited anyone yet. The list of folks is the set of users I added below, plus I'll post something at Wikipedia:WikiProject Mathematics. I can specifically invite Boris, William, and C S if you'd like. As far as the timeframe, I was thinking maybe something like a week or two. -- Rick Block (talk) 15:53, 2 December 2009 (UTC)
- I was going to, but had an edit conflict with Martin as he added his new section below. We need to straighten this out first. -- Rick Block (talk) 16:06, 2 December 2009 (UTC)
- I've notified all (including Wikipedia:WikiProject Mathematics) using template:please see referring them to this section. -- Rick Block (talk) 04:20, 3 December 2009 (UTC)
Glkanter's suggestion
- This is from the section above.
- Each instance in the article (and the various FAQs) of a host behaviour, or host bias, or host prejudice is indicative of a reliance on Morgan's paper. Regardless of what is being illustrated, this topic only exists among Morgan and a few others.
- Since the problem statement of both vos Savant (Whitaker) and Krauss & Wang begins with: "Suppose you are on a game show", we know that this host behaviour will not be shared with the contestant, whose State of Knowledge is the only one asked for in The Monty Hall problem.
- So, while Morgan is published, his argument is irretrievably flawed. The moment the problem is restated to rely on a host behaviour, it's no longer the Monty Hall problem. The problem statement becomes: 'Suppose you are not on a game show'. Which is the exact opposite of how both Monty Hall problem statements in the article begin: "Suppose you are on a game show". Morgan's criticism and his solutions are not relevant to the Monty Hall game show problem, which is the subject of this article.
Only because it's been published, Morgan should be referenced, but with such an obviously erroneous argument, it hardly deserves the great emphasis it currently enjoys. All other references to host behavior, etc., 'conditional vs unconditional', 'variants', and the Popular solutions being in any way inadequate should be removed from the article.
- A second section would explain why Morgan and 'conditional variants' are not the Monty Hall problem
- A final section on 'diversions' would include 'variants' and whatever else.
JeffJor's suggestion
- Rick, I've changed my mind on one thing. The so-called conditional problem needs to be a separate article, with "conditional" in its title. It can be linked to the MHP, but it is not the MHP. For justification, see (and cite in the article) [url=http://www.jstor.org/stable/187880] Maya Bar-Hillel's article "How to Solve Probability Teasers," Philosophy of Science, Vol. 56, No. 2 (Jun., 1989), pp. 348-358. That addresses several points critical to the problem, that are quite specific to all of the arguments we have had here, incuding documented evidence. Specifically: (1) It's just a puzzle. It isn't supposed to present a rigorously-defined mathematical problem, (2) The simple assumptions implied by the informal problem statement are intended, and almost universally accepted by anyone who isn't expecting such a rigorously-defined mathematical problem, (3) Even when presented with alternate wordings that explicitly include elements of host strategy, the general audience does not take that strategy into account in their solutions, and (4) the clear majority of respondents get the wrong answer (1/2) that is based on naive intuition rather than a formal solution.
- By separating the articles this way, Wikipedia can clearly present both problems in a fair and uncluttered manner, allowing any reader who wants to depend on the more formal approach to do so, and allowing those who do not see that formalism as necessary to limit themselves to the information that is of interest to them. JeffJor (talk) 15:03, 1 December 2009 (UTC)
Martin Hogbin's suggestion
We should take the current K & W statement as our starting definition of the MHP.
- I suggest that we give the Whitaker statement first then say that the K & W statement is how this is generally interpreted. The K & W paper itself supports this view.
The primary solution and explanation should not use conditional probability
- Although it can be argued that, even in the case where the host is defined to choose a legal door randomly, conditional probability should still be used because the action of choosing a particular door reduces the sample set and thus the opening of a specific door represents a conditioning of the sample set, it is clear that this is a trivial condition that it is not necessary to consider. This is quite evident either from the symmetry of the problem or from the fact that the revealing of random information tells us nothing. I am sure that we can find reliable sources to support this view.
The Morgan paper clearly does not answer the question as stated in the article and thus should not be regarded as our ultimate reliable source.
- The Morgan paper introduces a parameter q for something that is defined by the article problem statement to have only the value of 1/2. The Morgan paper thus answers a different problem (I suggest that we call it the Morgan scenario) from that posed in the article. In the Morgan scenario it is known that the host might have some preference for one of the legal goat doors.
The Morgan solution should be introduced in a later section of the article that deals with variations of the problem.
- There are many variations of this problem and the Morgan Scenario is just one of many.
Colincbn's comments
(referring to JeffJor's suggestion)
- Hear, hear!! Colincbn (talk) 15:09, 1 December 2009 (UTC)
(referring to Glkanter's suggestion)
I really don't know jack about probability and whatnot, but I still tend to agree with Glkanter's points. I came to this article through looking up various paradoxes and this was a really neat one that I got to try out in the real world (see simulation question above). As I understand it the "Monty Hall problem" states that the host chooses randomly, so any other discussion about host behavior should be limited to the "Variants" section under "Other host behaviors". Just my 2 cents, Colincbn (talk) 02:41, 1 December 2009 (UTC)
- Just to clarify, I think a mention of the Host behaviour/Conditional problem should be made in a subsection of this article, such as the Variants section, with a "main article" link (ie: {{main|MHP Conditional solution}} ) to a separate article that goes into Morgan's conditional problem in detail. I figure this will give the casual reader all the info he/she is looking for with an easy way to delve into the mathematics more deeply if they want. (also thanks to Rick for maintaining this section!) Colincbn (talk) 01:07, 3 December 2009 (UTC)
Martin Hogbin's comments
- I agree that this article should concentrate on the simple and notable interpretation of the MHP, namely the version in which a conditional solution is an unnecessary complication. Morgan's academic problem could be a section of this article or could form a new one. Martin Hogbin (talk) 22:02, 1 December 2009 (UTC)
Glkanter's comments
By my count, that's 4 in favor of the proposed changes, and 0 against. I've been championing these changes since October, 2008, Martin prior to that, and countless other editors for about 5 years. When can we declare an end to the pointless filibustering, acknowledge a consensus, and move on? Rick, will you be offering your comments? Have you contacted the others? Glkanter (talk) 22:29, 1 December 2009 (UTC)
About Martin Hogbin's suggestion - :I agree 100% with your proposed changes. I would like to add my 2 cents to the rationale, however. Morgan is criticizing and solving something other than the Monty Hall game show problem in the article. The introduction of the contestant being aware of any 'host behaviour' when selecting from 2 remaining goats changes the Problem Statement of both vos Savant/Whitaker and Karauss & Wang from "Suppose you are on a game show" to the converse, "Suppose you are not on a game show". Individual contestants on game shows are never provided more information than the 'average' contestant will have. There can be no 'condition'. It's illogical. Glkanter (talk) 15:33, 2 December 2009 (UTC)
JeffJor's comments
[Repeated in part from comments below]
The point of separating the articles is not to eliminate any POVs. It is to emphasize them. To not let one facet of the MHP (simple solution w nonintuitive result) become overpowered by the other (good teaching tool for conditional probabilites). If we don't physically separate them, we need to more clearly divide the article. The first part should be about the classic (unconditional) MHP, as stated by MvS (not K&W), and listing the set of assumptions she has said (and 99.9% of readers agree) are implied: interchangable doors, and any kind bias becomes irrelevant because of interchangeable doors. Then a section about game protocals (part of what some call host stratgies) such as always opening a door or revealing a goat, WITHOUT mention of bias or conditional problems. This mostly exists. Finally, you can cite Gillman (not Morgan) as a reference that introduces the possibility that the conditional problem is intended, but matters only if there is a bias. Use the K&W statement here, not Gillman's misquote. Gillman is better than Morgan because it is clearer, includes placement bias, and does not launch into possibilities that we are never told how to use. I think this is pretty consistent with Martin's suggestion. JeffJor (talk) 17:44, 4 December 2009 (UTC)
- Rick, no paper that uses q<>1/2 is addressing the K&W problem. They allow for it as a very specific variant of what they are addressing. But make no mistake: they are treating the problem statement we are supposed to be working with as the variant. That is wrong. There is nothing wrong with addressing their solutions as the variant, because it is a (more general) variation of what the article is supposed to be about. It isn't even a variant that is supposed to be used: no references use it, they just present it and say you don't need to use it. And I feel you have been just as much as stone wall on points relating to this as you accuse others of being. Meow. JeffJor (talk) 17:56, 4 December 2009 (UTC)
- Rick, you keep treating the Morgan POV as though it is involiate. It is not. Morgan misquotes the MvS problem statement, and so their claim that "the conditional problem is intended" cannot be taken as a reliable interpretation of the MHP. It is just a possible interpretation. Any reference that derives from Morgan is similarly suspect. Gillman misquotes, too, but in different ways. Bar-Hillel's survey proves that few (she found none) readers think of the conditional problem. More references exist that ignore it completely, than that address it. Krauss and Wang admit what the mis-quoters do not - I'll repeat it since when I said it before, it was apparently in cat language before - "Semantically, Door 3 in the standard version is named merely as an example." Grinstead and Snell separate the problem in the exact same two ways I suggested (and in fact, were a model for the suggestion). In short, it is a very minor POV that the conditional problem is meant, and it is based on citable misquotation and misinterpretation. JeffJor (talk) 18:24, 4 December 2009 (UTC)
Rick Block's comments
As a matter of fundamental Wikipedia policy, articles MUST be written from a neutral point of view. What the proponents of these changes are essentially suggesting is that this article take the POV that the interpretation of the problem described by a significant number of reliable sources (the Morgan et al. reference and others) is invalid. Even if this were a stance taken by reliable sources (which, as far as I know, is not the case), by relegating the "Morgan" interpretation to a "variant" subsection or splitting it into a POV fork this article would then be taking the "anti-Morgan" POV. I've made this point to these editors numerous times before, but yet they keep tendentiously arguing that the "Morgan" POV is wrong, or the Morgan et al. reference has errors, or (most recently) that the Morgan POV is NOT about the "real" Monty Hall problem (as if by convincing me that their POV is "correct" I would then agree with the changes they're suggesting).
I sincerely hope the "consensus" from this process is against making these changes, because even if there is a consensus for these changes they cannot be implemented - doing so would violate Wikipedia policy. -- Rick Block (talk) 04:01, 3 December 2009 (UTC)
You all realize Martin's proposal implies the article will not even mention conditional probability except in a "variant" section, don't you? How anyone can think this is not a blatant POV issue escapes me. -- Rick Block (talk) 20:19, 3 December 2009 (UTC)
- Yes Rick, all three proposals are consistent that way. It's based on this very recent and brief section of this talk page (following long and lengthy discussions on various 'talk' and 'argument' pages), 'Is The Contestant Aware?':
- I started this section at 11:28, 29 November 2009 (UTC). You responded with 2 vague, filibuster-style questions, and at 23:02, 1 December 2009 (UTC) I wrote this:
- "Rick, I have directly asked you this question many times, and have never seen a direct 'yes or no' answer from you. As this is a crucial element of the consensus that has been built, it is essential that we understand your reasons if you do not agree with the paragraph above:
- "Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?". [This question was ommitted when I asked Rick the 3rd time. I include it here for clarity]
- ""It seems to me that this is a (unstated) premise of the problem, as both vos Savant (Whitaker) and Krauss and Wang begin the problem statement with: 'Suppose you're on a game show'. I read this as clearly stating it is only the contestant's point of view we are concerned about. And, being a game show, the host is prohibited from divulging to the contestant either where the car is, or where the car is not.""
- To date, at 20:36, 3 December 2009 (UTC), you have still not responded directly to this question. Glkanter (talk) 20:36, 3 December 2009 (UTC)
I have to state the opposite view, which is that you have taken a ridiculously pro-Morgan POV. There are many reliable sources that relate to the MHP and not all of them have a host door choice parameter. Those that do generally quote Morgan as the source for this.
The article already takes a problem statement from a reliable source (K & W) and that same source confirms that this is how most people view the problem. In that statement, the host is defined to choose a legal goat door randomly. It is thus a simple matter of fact that the Morgan paper does not address that problem in so far as it allows a door choice parameter where none is permitted by the problem statement.
The Morgan paper clearly addresses a scenario where where the player is somehow aware of the host's policy for choosing a legal goat door. This rather bizarre scenario is not the one described by our problem statement and thus it should be viewed as a variant of the MHP as it is most commonly understood. Martin Hogbin (talk) 21:30, 3 December 2009 (UTC)
I thought I made it clear we were to use arbcom style rules here, which are that you only comment in your own section (it really does help keep the threads from getting absurdly long). However, since you've been rude enough to post here I'll respond to each of you, BUT please do not continue this as a thread here.
Glkanter asks why so dramatic? The argument has shifted from "present an unconditional analysis first (and don't criticize it)" to "exclude the conditional analysis completely (except as a variant)". This is a huge difference.
Glkanter asks why I haven't responded about his "Is The Contestant Aware?" question. Why should I? Glkanter has repeatedly demonstrated a complete lack of comprehension of nearly everything I've ever said. It's like trying to explain something to a cat. At some point you just have to give up. However, I'll give it another go. Meow, meeeow, meow, meowww. I'm not sure I have that quite right since I don't speak cat, but it's probably about as comprehensible to him as anything else I could say.
Martin (incorrectly) claims again that the Morgan et al. paper does not address the K&W version of the problem. Quote from the paper: "Incidentally, Pr(Ws | D3) = 2/3 iff p = q = 1/2". This is the solution to the K&R version of the problem statement. The Morgan et al. paper (and the Gillman paper and many, many others who approach the problem conditionally) absolutely address the K&R version. Because they also address other versions doesn't mean they don't address the K&R version.
- In the K & W statement q=1/2 by definition thus any problem in which q might not be equal to 1/2 must be a different problem. It is that simple. Martin Hogbin (talk) 19:04, 4 December 2009 (UTC)
Martin and Glkanter are both apparently completely incapable of understanding the main point of the Morgan et al. paper (and the Gillman paper, and what Grinstead and Snell have to say) which is that the MHP is fundamentally a conditional probability problem and that there's a difference between an unconditional and conditional solution. What these sources are saying is that a conditional solution clearly addresses the MHP (as they view the problem), but an unconditional solution doesn't unless it's accompanied by some argument for why it applies to the conditional case as well (and there are many valid arguments, but no argument at all which is what is generally provided with most unconditional solutions is not one of them). The fact that the problem can be (and typically is meant to be) defined in such a way that unconditional and conditional solutions have the same numeric answer in no way invalidates what these sources say. To have the article take the stance that the conditional solution is invalid (which would be truly absurd), or that the criticism these sources make of unconditional solutions is incorrect, or that a conditional solution applies only to a "variant" is making the article take a POV. This would be a direct violation of a FUNDAMENTAL Wikipedia policy. -- Rick Block (talk) 01:53, 4 December 2009 (UTC)
Antaeus Feldspar's comments
Glopk's comments
Delayed response (am not a very active editor at all these days), but here it is. Statement
I support Rick Block's statement as expressed above, and am in favor of keeping the article more or less in the state in which it passed the last FA review, with minor edits where needed. I am in strong disagreement with all three suggestions above (JeffJor, Glkanter, Martin Hogbin). In particular, I am in strong support of keeping the language that differentiates between the conditional (Bayesian) interpretation of the problem and the unconditional (elementary) one.
Motivation. The purpose of an encyclopedia is to present a "best" selection from the body of knowledge about each topic, being POV neutral as well as reader-neutral. --glopk (talk) 18:53, 29 December 2009 (UTC)
Father Goose's comments
Chardish's comments
Thanks for the invitation to comment. In my opinion, Martin Hogbin's suggestion seems the post prudent. The Monty Hall problem as popularly explained doesn't rely on conditional probability, and the Whitman explanation seems sufficient for anyone who is not a mathematician. Wikipedia is a general-purpose encyclopedia, and as such main articles should focus on explaining topics as they are popularly understood, with specific scientific analysis relegated to separate articles.
And, to be honest, the article as it stands is much harder to read and understand (as a layperson) than it was several years ago. NPOV isn't "pleasing everyone equally"; don't let efforts towards neutrality wind up hurting the article. - Chardish (talk) 02:53, 6 December 2009 (UTC)
Michael Hardy's comments
PMAnderson's comments
Melchoir's comments
Just from reading the present Wikipedia article, I agree with Martin Hogbin's suggestion, because I don't see why allowing the host to prefer one goat over the other is a more relevant generalization than allowing the host other behaviors. Melchoir (talk) 06:47, 3 December 2009 (UTC)
jbmurray's comments
Nijdam's comments
I fully support Rick's view. Nijdam (talk) 10:34, 3 December 2009 (UTC)
To make my position crystal clear: there is no such as an unconditional solution. There are different problems: an unconditional problem and a conditional one. The latter generally being called the MHP. Nijdam (talk) 22:24, 3 December 2009 (UTC)
- Please read my proposal. I do not claim that the MHP is an unconditional problem. What I say is that in the problem definition given in the article the host is taken to choose a legal goat door randomly. Morgan address the case where this choice is non-random,thus they do not address the problem as defined in this article. Martin Hogbin (talk) 22:39, 3 December 2009 (UTC)
- I'm in the audience looking at the stage. I see three doors and a player pointing to one of them. From the two remaining doors one is opened and shows a goat. That's what I call the MHP. (And I know of the random placement of the car and the random choice of the host.) Nijdam (talk) 22:50, 3 December 2009 (UTC)
- Quite, and that is not the problem that the Morgan paper addresses. The Morgan paper addresses the case where the host door choice is not random. Thus the Morgan paper addresses a variation on what we all agree is the MHP. Martin Hogbin (talk) 23:28, 3 December 2009 (UTC)
- The problem that you and K & W describe, which is the problem addressed by this article, is one in which q=1/2 by definition. Therefore, any problem in which there is a possibility that q might not equal 1/2 must be a different problem. Morgan clearly consider a problem in which it is possible for q to have a value other than 1/2. The problem they consider therefore must be different from that in which q is defined to be 1/2. Morgan do indeed address a (bizarrely) more general problem than the one we are considering but it is, for sure, a different problem. Martin Hogbin (talk) 19:35, 4 December 2009 (UTC)
Dicklyon's comments
I haven't been watching this article for a while; glad to see the K&W treatment up front; that looks like the most sensible article I've seen on it. As for the Morgan conditional approach, I think it's an unnecessary distraction, but it's out there in mainstream reliable sources about the topic, so we ought to cover it in the article. I think Martin Hogbin's proposal sounds best. Dicklyon (talk) 05:01, 3 December 2009 (UTC)
I agree with Rick Block that the other two proposals essentially violate WP:NPOV; but I disagree that moving the conditional stuff to a more minor position is a problem; his heavy promotion of the conditional approach violates WP:UNDUE in my opinion. Dicklyon (talk) 16:29, 3 December 2009 (UTC)
Henning Makholm's comments
I have long since given up on following these discussions, and am not even a very active editor these days. However, since somebody went to the length of creating a heading for me, here are my general recommendations -- for whatever they are worth:
- The article absolutely should discuss assumptions about the host's behavior. It is impossible to derive a valid answer without making some assumptions, and differences in which assumptions are implicit are one of the main reasons why smart people can disagree on the solution when the problem is stated sloppily. It would be a sorry encyclopedia that purported to treat the Monty Hall problem without explicitly pointing out this kind of confusion.
- The analysis that involves conditional probabilities and the one that considers whole-game expectations under different player strategies are both valid ways of approaching the problem, each with its own advantages and disadvantages. The article should present both, and must not suggest that one of them is inherently better or more correct than the other. (For this reason I would oppose splitting one of the analyses into a separate article, suggesting that it solves a fundamentally different problem, rather than being an alternative way of approaching the same problem).
- There has been far too much microlinguistic analysis about precise wordings of the problem in this source or that one, trying to argue that this analysis or that one is the one that most directly addresses the question being asked (implying that the other is a detour via a different but non-canonical presentation of the problem). Which analysis one chooses depends depends far more on which properties (besides being valid) one wants of it. For example, raw convincing power for a lay audience would favor the whole-game analysis, whereas a more in-depth discussion of the effect of different assumptions of the host's behavior is most easily done using conditional probabilities.
- Editors should keep in mind that Wikipedia is an encyclopedia, not a textbook, an question-and-answer database, or a Court of Public Opinion. The goal of an encyclopedia article is not to answer one particular question but to present a body of knowledge. Therefore the amount of energy spent on negotiating "the" question that this article should be about answering is fundamentally misspent. The body of knowledge the article ought to present encompasses several different but related questions (some of which are sometimes mistaken for each other), and several different way of approaching some of them. An approach that restricts ourselves to discussing just one of them would fail to cover the topic encyclopedically.
- I have no strong opinion about which analysis should be first in the article, as long as it is not being touted as inherently superior or inferior by virtue of its position. However, the general principle of progressing from the "quick and easily understood" to the "more complex but also more general and (possibly) enlightening" would seem to suggest starting with the whole-game analysis.
–Henning Makholm (talk) 07:13, 3 December 2009 (UTC)
- Most people, including Rick, think that the problems should be addresses from the player's point of view (state of knowledge). As has been pointed out by many people, it is extremely unlikely that the player would have any knowledge of the host's door opening policy, thus from the player's point of view the host policy must be taken as random (within the rules).
- I have no objection to the Morgan scenario (in which the payer is assumed to know the host's policy) as well as the more simple case being presented here provided that it is made clear exactly what case this applies to.
- What you call, 'microlinguistic analysis about precise wordings of the problem' was started by Morgan et al. who added a pointless layer of obfuscation to a simple puzzle that most people get wrong.
- The point is that the simple/symmetrical/non-conditional problem is the notable one and therefore it should come first. More complex versions should come later for the few that are interested in such complications. Martin Hogbin (talk) 22:36, 4 December 2009 (UTC)
Boris Tsirelson's comments
I summarize my position in two points:
- 1. The symmetric case is more important for an encyclopedia than the general case. (Likewise, a circle is more important for an encyclopedia than an arbitrary curve.)
- 2. The coexistence of the conditional and the unconditional can be more peaceful. (Not just "numeric coincidence" in the symmetric case; see #Not just words and #Formulas, not words.)
Boris Tsirelson (talk) 06:44, 9 December 2009 (UTC)
Being invited by Glkanter, I quote here some paragraphs of a discussion that happened on my talk page on February 2009. As far as I understand, my position is close to that of JeffJor. Boris Tsirelson (talk) 17:20, 2 December 2009 (UTC)
Why split? Because of different importance. The "conditional" article will be, say, of middle importance, while the "unconditional" article – of high importance. We surely have our point of view about importance (rather than content). Boris Tsirelson (talk) 05:54, 4 December 2009 (UTC)
The quotes follow.
Each time giving the course "Introduction to probability" for our first-year students (math+stat+cs) I spend 20-30 min on the Monty Hall paradox. I compare two cases: (a) the given case: the host knows what's behind the doors, and (b) the alternative case: he does not know, and it is his good luck that he opens a door which has a goat. Im addition I treat the case of 100 (rather than 3) doors (just like Monty Hall problem#Increasing the number of doors). And, I believe, students understand it.
I have no idea, why some people spend much more time on the Monty Hall paradox (and even publish papers). (Boris Tsirelson)
This simple little problem is deeper than it might appear, and likely well worth more than 20-30 mins of lecture time. Perhaps even worth revisiting once or twice during a term to explore its more subtle aspects. (Rick Block)
Deeper than it might appear? OK, why not; but still, for now I am not enthusiastic to deep into it. Tastes differ. I find it more instructive, to restrict myself to the simpler, symmetric case, and compare the two cases mentioned above.
If an article leaves many readers puzzled, why it is unnecessarily complicated, it is a drawback. (Boris Tsirelson)
If a problem that appears so simple to me, like the Monty Hall problem, is not sufficiently solved using my unconditional proof, in what circumstances is the unconditional proof appropriate? Thank you. (Glkanter)
The unconditional argument shows that "always switch" is better than "never switch". This is what it can do. Let me add: if you (that is, the player) are not informed about possible asymmetry then you cannot do better than these two strategies, either "always switch" or "never switch". (Boris Tsirelson)
- Well, I gotta ask. Do you still prefer JeffJor's proposal among the 3 proposals put forth? Glkanter (talk) 07:19, 9 December 2009 (UTC)
- Really, I have nothing to add to the two points that summarize my position (above). Any move toward them is good for me. My resolution power, and my acquaintance with the literature, are too low for choosing between different proposals; I leave this matter to more informed (and less lazy) editors. Boris Tsirelson (talk) 08:22, 9 December 2009 (UTC)
William Connolley's comments
C S's comments
kmhkmh's comments
I'll start with a clear statement and give some more detailed information afterwards:
- I strongly disagree with any of the 3 suggestions (JeffJor, Glkanter, Martin Hogbin) and aside from minor difference fully support Rick Block's approach
If one surveys the available literature literature/publications on the topic, you pretty much get an relatively obvious outline for the article: original problem (in vos savant's column), unconditional solution (basically vos savant and/or various math sources), conditional solution (Morgan and almost in any math source), various problem variation and caveats, history of the problem, application of the problem outside the math domain. Which is essentially for the most part, what we already had and what Rick managed to maintain. In that context I fully agree with Henning Makholm's comments above, who puts it fairly well. The article wouldn't have such problems if all participants would follow that rationale.
The fuzz over quality or minor mistakes in Morgan's paper is a somewhat ridiculous distraction, since Morgan's paper is not needed to argue the conditional solution or caveats to the unconditional solution at all. There is plenty of other math literature dealing with the problem in more or less the same manner.
My personal advice would be to pass the article for final thorough review and modification to the math or a science portal. During that review neither of the 4 disagreeing authors (JeffJor, Glkanter, Martin Hogbin, Rick Block) are allowed to participate/edit. After that review the article should be fully protected for good.
I've seen what happened to the German version, that had similar problems (without a Rick Block around to constantly remain some standard). So we had a lot of people with a somewhat fanatic approach constantly pushing for their favoured explanation and constantly ignoring wiki standards, common sense and more important the available literature on the subject. As result mathematicians and scientists basically dumped the article and gave up on improving it.An effect this article has partially seen as well.--Kmhkmh (talk) 16:45, 4 December 2009 (UTC)
- Kmhkmh, no one is proposing a reduction in the quality of this article but you miss some essential points out in your outline. We should have: 'original problem (in vos savant's column), unambiguous problem definition (K&W), solution to the unambiguous problem (which is trivially conditional but need not be treated so, basically vos savant and/or various math sources), the Morgan scenario (in which the player knows host door choice policy) the conditional solution (Morgan). Martin Hogbin (talk) 19:46, 4 December 2009 (UTC)
- I don't quite see how that is "missing" in my outline above nor do I see any particular reason to give (K&W) a preferred treatment, such an approach does not reflect the publications on the topic.--Kmhkmh (talk) 20:11, 4 December 2009 (UTC)
- I am not proposing that we give K&W and preference but must have a clear and unambiguous problem statement before we (or anyone else) can attempt to answer the problem. Morgan do not have such a statement in their paper so we must use one from another reliable source, in this case another published paper. Note that the lack of clear problem statement in the Morgan paper is not just my opinion, that same point is made clear in the comment by Prof Seymann published in the same journal immediately after the Morgan paper. Martin Hogbin (talk) 20:20, 4 December 2009 (UTC)
- I'm really not interested in repeating now here the discussion that you're pushing for almost over a year now and which frankly from my perspective is entirely pointless and misguided. The original problem in vos Savant's column was ambiguous and hence various articles on the topic and its variations provide their own specifications. As pointed out above already Morgan doesn't really matter in that regard. What the Wikipedia article has to do, is to describe the all various specification and not arbitrarily picking one like K&W as the "right" one. I'd recommend you to reread Henning Makholm's comments carefully. Or to put it rather bluntly - you asked for my comment here it is: Leave the article alone.--Kmhkmh (talk) 21:46, 4 December 2009 (UTC)
- Actually I did not ask for your comment here and I certainly did not ask for, and do not need, your permission to edit Wikipedia. Neither did I pick K&W as the 'right one' as you put it, somebody else put it in the article as a clear and unambiguous description of the problem. As it happens I agree with whoever did this as K & W is the only published paper to seriously address the question of how most people interpret the MHP. It is therefore an excellent place to start the article. Martin Hogbin (talk) 22:25, 4 December 2009 (UTC)
- I'm really not interested in repeating now here the discussion that you're pushing for almost over a year now and which frankly from my perspective is entirely pointless and misguided. The original problem in vos Savant's column was ambiguous and hence various articles on the topic and its variations provide their own specifications. As pointed out above already Morgan doesn't really matter in that regard. What the Wikipedia article has to do, is to describe the all various specification and not arbitrarily picking one like K&W as the "right" one. I'd recommend you to reread Henning Makholm's comments carefully. Or to put it rather bluntly - you asked for my comment here it is: Leave the article alone.--Kmhkmh (talk) 21:46, 4 December 2009 (UTC)
- I am not proposing that we give K&W and preference but must have a clear and unambiguous problem statement before we (or anyone else) can attempt to answer the problem. Morgan do not have such a statement in their paper so we must use one from another reliable source, in this case another published paper. Note that the lack of clear problem statement in the Morgan paper is not just my opinion, that same point is made clear in the comment by Prof Seymann published in the same journal immediately after the Morgan paper. Martin Hogbin (talk) 20:20, 4 December 2009 (UTC)
- I don't quite see how that is "missing" in my outline above nor do I see any particular reason to give (K&W) a preferred treatment, such an approach does not reflect the publications on the topic.--Kmhkmh (talk) 20:11, 4 December 2009 (UTC)
Gill110951's comments
No comment right now. But a lot of Christmas break reading to do here, to catch up. Happy Wikipedia Christmas, everyone! Gill110951 (talk) 13:27, 20 December 2009 (UTC)
Friday's comments
Summary of opinions
I have added names to the sections below based on comments above. If I have got it wrong please move yourself.
Please do not make comments in this section.
Editors are invited to sign against their names to confirm that they are in the right section. Martin Hogbin (talk) 11:29, 5 December 2009 (UTC)
- Martin, What exactly do you mean by "for change" and "against change"? Dicklyon and JeffJor's opinions (for example) seem very different to me. By categorizing them both as "for change" I think you may be misrepresenting the situation. It would be better to be more specific about what change you're talking about, i.e. Glkanter's suggestion (
remove any mention of conditional probability and any host behavior variants)Eliminate all 'host behaviour, etc' influenced discussion, save for the Wikipedia minimum necessary references to Morgan and his ilk, JeffJor's suggestion (separate articles - basically Glkanter's suggestion plus create a new article for the "conditional" treatment), your suggestion (I'm not exactly sure precisely how to summarize yours). In addition, rather than "against change" the other alternative should probably be described in terms of what it is for, which I think could be described as "present both unconditional and conditional solutions without taking a POV about the validity of either one". And, I'll note that for the article to say that Morgan et al. criticize the unconditional solutions is not the same as taking that POV. You do understand this difference, don't you? -- Rick Block (talk) 16:52, 5
- Although there may be some discussion over the details it is fairly clear that several people would like to see the simple/unconditional solution/problem given more prominence here. This is the change that I am referring to. 'Against change' is fairly self explanatory Martin Hogbin (talk) 16:58, 5 December 2009 (UTC)
For change
Colincbn
Martin Hogbin Martin Hogbin (talk) 11:29, 5 December 2009 (UTC)
Glkanter Glkanter (talk) 12:16, 5 December 2009 (UTC)
JeffJor
Melchoir
Dicklyon
Boris Tsirelson Boris Tsirelson (talk) 15:27, 5 December 2009 (UTC)
Gill110951 (talk) 13:28, 20 December 2009 (UTC)
Against change
Rick Block
Nijdam
kmhkmh
Glopk
Unable to classify
Please move your name to the correct section if appropriate. Martin Hogbin (talk) 11:24, 5 December 2009 (UTC)
Henning Makholm
Chardish (I object to summary classification of my comments. - Chardish (talk) 00:59, 11 December 2009 (UTC))
Words, words, words
Because in words one often doesn't distinquish between probability and conditional probability, a lot of the misunderstanding arises. As I tried before, I ask anyone of the discussiant to formulate the problem and the solution in rigorous mathematical notation. I'll do the kick-off:
- Three doors with numbers 1,2 and 3
- C is the random variable indicating the number of the door with the car
- X is the random variable indicating the number of the door chosen by the player
- C and X are independent
- H is the random variable indicating the number of the door opened by the host
As C and X are independent we may without loss of generality consider the case "X=1" and without explicit mentioning condition on this event. Probabilities are:
Did I leave something out? Is something wrong? Please comment! In my opinion the whole point of the MHP is the calculation of P(C=2|H=3). I'm looking forward to other opinions. Nijdam (talk) 12:40, 5 December 2009 (UTC)
- No, your calculation is fine and as I have made clear before I accept that the MHP problem (as defined by the K & W statement in which specific doors are mentioned) is a problem of conditional probability. The opening of a door by the host reduces the sample set (clearly the player can no longer change to that door the host has opened) thus the sample set has been conditioned and the problem is one of conditional probability.
But, even with this problem formulation we can ask the question as to whether we need to use conditional probability to answer the question. We can immediately note that there is an obvious symmetry in the question. Namely that:
and therefore that: P(C=2|H=3)=P(C=3|H=2) Thus, as Boris has ,said the conditional solution, whichever door the host chooses, must equal the unconditional solution. There is nothing wrong with this approach, many intractable mathematical problems have been solved by noticing some symmetry in the problem.
>>>Even so, as you say, we have to calculate the conditional probability. Nijdam (talk) 15:57, 5 December 2009 (UTC)
Alternatively we might say that, because the hosts choice is random (when he has a choice) it gives no information about the initial placement of the car and thus the original
must hold good after the host has opened a door.
>>>??It doesn't:
- You are right, of course. I should have just said that P(C=1)=1/3; Martin Hogbin (talk) 16:39, 5 December 2009 (UTC)
So to sum up, with the K & W formulation, the problem may be strictly conditional but it is clear that a simple unconditional solution will give the correct answer.
However there is more to the issue than this. The MHP is essentially a mathematical paradox and thus it is logical to formulate the problem so that the solution is as simple as possible. This is what was done before the Morgan paper was published. The door numbers were considered irrelevant. I have answered your question, now perhaps you will answer mine. Do you really believe that what Whitaker wanted to know was what the probability of winning was given only that specific doors had been chosen and opened? I would very much like to hear your answer to that question.
>>>Whitaker definitely wanted to know what for a chosen door and an opened one the (conditional) probability was for the car to be behind the remaining unopened door. Nijdam (talk) 15:57, 5 December 2009 (UTC)
I would say that the question that he actually wanted the answer to is: 'Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, and the host, who knows what's behind the doors, opens another door, which has a goat. He then says to you, "Do you want to pick the remaining door?" Is it generally to your advantage to switch your choice?'
He obviously wants to know what is the best strategy in playing the game: to switch or not. This a simple question with a simple answer that most people get wrong.
>>>Always given the situation the player is in. Nijdam (talk) 15:57, 5 December 2009 (UTC)
I can only repeat, yet again, the words of the wise Prof Seymann (with my emphasis) in the hopes that you will heed them: ' Without a clear understanding of the precise intent of the questioner, there can be no single correct solution to any problem. Thus, with respect to the three door problem, the answer is dependent on the assumptions one makes about the intent of the one who initially posed the question '. In my opinion Morgan et al. have completely misrepresented the original point of the question, resulting in all this conditional nonsense which only serves to obfuscate an interesting problem. Martin Hogbin (talk) 14:02, 5 December 2009 (UTC)
>>>There you have a point. The original problem may be interpreted in different ways. Also in the way of Morgan et al. BTW. But my main concern is that even in the precise definition (like the K&W formulation) many people, seemingly also mathematicians, probabalists and statisticians, give the wrong analysis. Nijdam (talk) 15:57, 5 December 2009 (UTC)
- I don't completely disagree with your description. However making claims about Whittaker's intentions is clearly WP:OR. Either his intentions are published somewhere, then they are known and we can incorporate them or they are not. In the latter case personal speculations of WP editors do not belong in the article. As far possible or real misinterpretation are concerned, WP has to describe or summarize how Whitaker was interpreted in various publication - period.--Kmhkmh (talk) 14:12, 5 December 2009 (UTC)
- I understand your point about sources; that is why I have not edited the article but continued to argue here. On the other hand may editors feel passionately that the current format of the article is wrong. There are many sources on this problem and it is, to some degree, up to us which ones we use and how we use them. See my suggestion below on mediation. Martin Hogbin (talk) 14:33, 5 December 2009 (UTC)
- I agree completely with the analysis done above by Nijdam and Martin Hogbin. Of course, the symmetry assumption should be formulated more accurately in the "official" text; not only C and X are independent, but also C is uniform on the 3-point set, and H is conditionally uniform on the rest (be it 1-point or 2-point set). Boris Tsirelson (talk) 15:04, 5 December 2009 (UTC)
- Actually Martin's description leaves me a bit confused since in my perception it is not quite in line with what he was pushing for earlier. However if we all essentially agree now, I don't quite get what the actual dissent is now. Are we just arguing about the order of various sections but agree about the general content (see also the outline under my comment)? I can live with more or less any order in the article as long as the (sourced) content stays (in whatever section). Or to be specific regarding the much debated Morgan, his approach (or that in similar publications) hast to be mentioned and his caveat to the vos savant's approach/simple solution as well, in which section of the article this happens however doesn't really matter.--Kmhkmh (talk) 15:22, 5 December 2009 (UTC)
- As for me, I only say that the symmetric case is more important for an encyclopedia than the general case. (Likewise, a circle is more important for an encyclopedia than an arbitrary curve.) Boris Tsirelson (talk) 15:32, 5 December 2009 (UTC)
- Actually Martin's description leaves me a bit confused since in my perception it is not quite in line with what he was pushing for earlier. However if we all essentially agree now, I don't quite get what the actual dissent is now. Are we just arguing about the order of various sections but agree about the general content (see also the outline under my comment)? I can live with more or less any order in the article as long as the (sourced) content stays (in whatever section). Or to be specific regarding the much debated Morgan, his approach (or that in similar publications) hast to be mentioned and his caveat to the vos savant's approach/simple solution as well, in which section of the article this happens however doesn't really matter.--Kmhkmh (talk) 15:22, 5 December 2009 (UTC)
- I agree completely with the analysis done above by Nijdam and Martin Hogbin. Of course, the symmetry assumption should be formulated more accurately in the "official" text; not only C and X are independent, but also C is uniform on the 3-point set, and H is conditionally uniform on the rest (be it 1-point or 2-point set). Boris Tsirelson (talk) 15:04, 5 December 2009 (UTC)
- Indeed; but I'm afraid, Rick Block will say that a featured article must be perfectly sourced... Boris Tsirelson (talk) 16:15, 5 December 2009 (UTC)
- I meant, please respond to my call. Formulate here your ideas about the symmetric case and its solution in the above terminology!! Nijdam (talk) 16:21, 5 December 2009 (UTC)
- OK, wait a bit. Boris Tsirelson (talk) 16:25, 5 December 2009 (UTC)
- I meant, please respond to my call. Formulate here your ideas about the symmetric case and its solution in the above terminology!! Nijdam (talk) 16:21, 5 December 2009 (UTC)
- Indeed; but I'm afraid, Rick Block will say that a featured article must be perfectly sourced... Boris Tsirelson (talk) 16:15, 5 December 2009 (UTC)
Not just words
Notation:
- Three doors with numbers 1,2 and 3
- C is the random variable indicating the number of the door with the car
- X is the random variable indicating the number of the door chosen by the player
- H is the random variable indicating the number of the door opened by the host
Assumptions:
- C is uniform on the set {1,2,3}
- C and X are independent
- H is conditionally uniform on the complement of the set {C,X} in the set {1,2,3} (be the complement a 1-point or 2-point set)
The "unconditional" solution:
A pure strategy of the player is a function that maps every possible pair (X,H) to either X or the "third" element of {1,2,3} (different from X and H).
A mixed strategy of the player is a function that maps every possible pair (X,H) to a probability measure on {1,2,3} that vanishes on {H}.
The winning probability is a function of a strategy. It is invariant under the rearrangement group of {1,2,3} (by the symmetry, of course). Therefore it is sufficient to consider only invariant mixed strategies. Such a strategy is nothing but a probability of switching. It is sufficient to consider only extremal values (0 and 1) of the probability. Thus, the question boils down to: to switch or not to switch. The simple unconditional calculation completes the analysis.
The "semi-conditional" solution:
The conditional winning probability without switching, P(X=C|X,H), is a function of X and H. It is invariant under the rearrangement group of {1,2,3} (by the symmetry, of course). Therefore it is the constant function. Taking into account that its expectation is equal to the unconditional probability (the total probability formula) we see that P(X=C|X,H) = P(X=C); that is, the conditional probability is equal to the unconditional probability. The simple unconditional calculation (the same as above) completes the analysis.
(You see, the conditional probability is used; but is reduced to the unconditional probability.)
There is also the "all-conditional" solution, not mentioning the unconditional probability; but probably I do not need to write it here.
What do you think about it? Boris Tsirelson (talk) 17:08, 5 December 2009 (UTC)
Maybe some words here sound a frightening math for some people; but they are just a formalization of the symmetry argument intuitively clear to most people. So much clear that they probably do not feel any need in formalization... Boris Tsirelson (talk) 17:27, 5 December 2009 (UTC)
- As I said; words, words, words and more words. Give me the correct formulation of this decision theoretical approach in formulas. BTW: I hope you don't tell this story to your first grade students in your introductory course. What I am actually interested in is what you tell in this course and then in the above formalism, with no more words than needed. Think you can manage? Nijdam (talk) 10:03, 6 December 2009 (UTC)
- Sorry, now I do not understand what do you mean by "words". Every mathematical paper consists of words (and formulas inside). The formulation above is correct. Yes, this is not what I teach. The reason is that I do not teach MHP; I teach probability, especially conditioning, and at that moment I use MHP as an instructive example. The goal of Wikipedia article is probably quite different. Boris Tsirelson (talk) 13:26, 6 December 2009 (UTC)
- I am not used to explain math in a sarcastic environment. If someone will ask me a reasonably specific question in a reasonably polite manner, I'll be reasonably helpful.
- I claim that the above texts are correct proofs (well, somewhat sketchy). I am not an anon; I am a professional mathematician.
- If you want to say that this is anyway too complicated for the article, just say so. In fact, I never proposed these solutions for the article. And I did not object against conditional probability. I only object against the general case treated as no less important (for Wikipedia, not for science) than the symmetric case. Boris Tsirelson (talk) 14:48, 6 December 2009 (UTC)
- I think the argument for this extensive treatment of the general case is, that is somewhat reflects the academic publications/treatment of that topic. I have hardly any objection against Martin's reaction to Nijdam's comment. Featuring the special case or unconditional solution and its benefits prominently in the first part of the article is perfectly fine. However if you review some of Martin's other or earlier comments and even more so some comments of JeffJor or Glkanter, then you can see they are pushing for things which partially factually false or unsourced and definitely not in line with bulk of reputable literature on the subject. Among these are glkanter "legal arguments", JeffJor's attempt to define the conditional approach as a "non MHP"-problem. There also seems to be an attempt to remove reputable sources from the article (via badmouthing) and ignoring that other sources more or less state same conclusions anyway. In particular there is one thing the WP-article cannot do, that is treating the MHP as a unconditional problem only, while the bulk of the literature treats it at least as a conditional problem as well. Another thing is, that the WP article cannot redefine the MHP to remove its ambiguity, since the original problem is ambiguous and much of the literature explicitly deals with the ambiguity. WP needs to report the definitions of others and not set its own.--Kmhkmh (talk) 15:38, 6 December 2009 (UTC)
- I agree. Boris Tsirelson (talk) 16:02, 6 December 2009 (UTC)
- I think the argument for this extensive treatment of the general case is, that is somewhat reflects the academic publications/treatment of that topic. I have hardly any objection against Martin's reaction to Nijdam's comment. Featuring the special case or unconditional solution and its benefits prominently in the first part of the article is perfectly fine. However if you review some of Martin's other or earlier comments and even more so some comments of JeffJor or Glkanter, then you can see they are pushing for things which partially factually false or unsourced and definitely not in line with bulk of reputable literature on the subject. Among these are glkanter "legal arguments", JeffJor's attempt to define the conditional approach as a "non MHP"-problem. There also seems to be an attempt to remove reputable sources from the article (via badmouthing) and ignoring that other sources more or less state same conclusions anyway. In particular there is one thing the WP-article cannot do, that is treating the MHP as a unconditional problem only, while the bulk of the literature treats it at least as a conditional problem as well. Another thing is, that the WP article cannot redefine the MHP to remove its ambiguity, since the original problem is ambiguous and much of the literature explicitly deals with the ambiguity. WP needs to report the definitions of others and not set its own.--Kmhkmh (talk) 15:38, 6 December 2009 (UTC)
- If you want to say that this is anyway too complicated for the article, just say so. In fact, I never proposed these solutions for the article. And I did not object against conditional probability. I only object against the general case treated as no less important (for Wikipedia, not for science) than the symmetric case. Boris Tsirelson (talk) 14:48, 6 December 2009 (UTC)
- I'm confused. If you agree then why are you listed here as supporting changing the article (to attempt to redefine the MHP to remove its ambiguity and to remove, badmouth, and ignore reputable sources). I suspect we don't have a common understanding of what changes are suggested. -- Rick Block (talk) 17:19, 6 December 2009 (UTC)
- Here is my position (once again): I only say that the symmetric case is more important for an encyclopedia than the general case. (Likewise, a circle is more important for an encyclopedia than an arbitrary curve.) Boris Tsirelson (talk) 18:43, 6 December 2009 (UTC)
- Then, I would suggest you change where your name appears in Martin's "summary" of opinions. I believe what Martin, and Glkanter, and JeffJor are actually arguing for is to banish any mention of conditional probability to a "variant" section. And they think you agree with this. -- Rick Block (talk) 20:20, 6 December 2009 (UTC)
- But I do support some change. First, prepare a list with a different formulation. Boris Tsirelson (talk) 20:42, 6 December 2009 (UTC)
- Rick, you are overstating my request. I do not want to banish any mention of conditional probability to a variant section but I do want to state that it is only really important if the host chooses non-randomly - the Morgan scenario. Martin Hogbin (talk) 21:55, 6 December 2009 (UTC)
Formulas, not words
Here is the "semi-conditional" solution rewritten for these that hate words and like formulas:
P(X=C|X=1,H=2) = P(X=C|X=1,H=3) = P(X=C|X=2,H=1) = P(X=C|X=2,H=3) = P(X=C|X=3,H=1) = P(X=C|X=3,H=2);
P(X=C) = P(X=C|X=1,H=2) P(X=1,H=2) + P(X=C|X=1,H=3) P(X=1,H=3) + P(X=C|X=2,H=1) P(X=2,H=1) + P(X=C|X=2,H=3) P(X=2,H=3) + P(X=C|X=3,H=1) P(X=3,H=1) + P(X=C|X=3,H=2) P(X=3,H=2);
P(X=1,H=2) + P(X=1,H=3) + P(X=2,H=1) + P(X=2,H=3) + P(X=3,H=1) + P(X=3,H=2) = 1;
therefore
P(X=C|X=1,H=2) = P(X=C|X=1,H=3) = P(X=C|X=2,H=1) = P(X=C|X=2,H=3) = P(X=C|X=3,H=1) = P(X=C|X=3,H=2) = P(X=C).
That is, the conditional probability is equal to the unconditional probability (in the symmetric case, of course). Boris Tsirelson (talk) 20:21, 7 December 2009 (UTC)
- Well, I don't hate words, but I do hate them when they hide an underlying problem. And, sorry, I do not understand what "semi-conditional" means. I gave before extensive solutions using Bayes and symmetry arguments. In both ways it is necessary to calculate a conditional probability. I would very much like anyone who claims there is a solution without such a calculation, demonstrate it, in proper terminology. Till now, no-one tried. Yes, in words, words, words. That's why. BTW: The above analysis, without words, also calculates a conditional probability. Nijdam (talk) 13:39, 8 December 2009 (UTC)
- "also calculates a conditional probability"? Ultimately, yes, of course. However, this way the fact that it equals to the unconditional probability becomes clear BEFORE we get the values of this or that probability. Not "numeric coincidence" but a logical necessity. Does it matter for you? Boris Tsirelson (talk) 16:53, 8 December 2009 (UTC)
- As another example, imagine you have to calculate the perimeter of an ellipse with major axis j and minor axis n. You are told that j = n. Is it better to do the calculation for an ellipse and then set j = n or should you observe that the ellipse is, in fact, a circle and do the calculation on that basis? Martin Hogbin (talk) 12:20, 13 December 2009 (UTC)
Theorem ( Boris Tsirelson, Dec 2009, published in the first time, all rights reserved :-) ) Let an even E and a random variable X be such that the conditional probability P(E|X) is a constant function of X. Then the conditional probability is equal to the unconditional probability: P(E|X) = P(E).
Eureka! Boris Tsirelson (talk) 17:38, 8 December 2009 (UTC)
- You puzzle me. What can I say more? Everything is said. What kind of point do you expect? Well, I repeat, but surely you will not be satisfied (and I do not understand why).
- First, I find the symmetric case more interesting. Second, I am glad to see that in this case it is rather easy to see (before specific calculations, just from simple general arguments) that the conditional probability (you know, of which event) is equal to the unconditional probability. I believe that this is a (or rather, the) mathematical formalization of a fact understood by many non-mathematicians by intuition and/or arguments about "no other useful information available" and all that. Boris Tsirelson (talk) 21:27, 8 December 2009 (UTC)
- I summarize my position in two points:
- 1. The symmetric case is more important for an encyclopedia than the general case. (Likewise, a circle is more important for an encyclopedia than an arbitrary curve.)
- 2. The coexistence of the conditional and the unconditional can be more peaceful. (Not just "numeric coincidence" in the symmetric case.)
- Boris Tsirelson (talk) 06:38, 9 December 2009 (UTC)
- I agree with you, and indeed is the conditional probability of the considered event not just coincidentally equal to the unconditional. But in my opinion it is necessary to mention the conditional probability as the probability of interest. That's why I asked you to formalize your reasoning. Nijdam (talk) 11:17, 9 December 2009 (UTC)
- For a better understanding of my position, I may add, that a lot of people, amonst them students, teachers etc., reason as follows: the car is with probability 1/3 behind the chosen door. Hence the probability to find it behind one of the two remaining doors is 2/3. After the opening of one of them this (!) probability now applies to the remaining unopened door. I consider this as insufficient. One needs to add here that the opening of one of the door "does not influence" the probability of the car being behind the chosen door. And what is the meaning of "does not influence"? Well that the conditional probability and the unconditional one are the same. Agree? Nijdam (talk) 11:26, 9 December 2009 (UTC)
- Agree! Boris Tsirelson (talk) 16:08, 9 December 2009 (UTC)
- I did not object against conditional probability.
- "After the opening of one of them this (!) probability now applies to the remaining unopened door." — For me this is the central (!) point of MHP. The only (!) point making it instructive to my students (in my opinion). A quote from #Boris Tsirelson's comments: "I spend 20-30 min on the Monty Hall paradox. I compare two cases: (a) the given case: the host knows what's behind the doors, and (b) the alternative case: he does not know, and it is his good luck that he opens a door which has a goat." I do so exactly for explaining why in (a) the 2/3 jump to the remaining door, while in (b) it scatters to both doors. Boris Tsirelson (talk) 16:28, 9 December 2009 (UTC)
- What makes things cumbersome? First of all, the lack of symmetry. And second, – to a much less extent, – conditioning. Boris Tsirelson (talk) 16:34, 9 December 2009 (UTC)
- I don't follow you any more. To put things straight: the probability 2/3 of the event that the car is behind one of the not chosen doors, differs (in type, not in value) from the conditional probability the car is behind the remaining unopened door given the other door opened. It concerns different probability measures. That's why I asked for formulas. Nijdam (talk) 11:51, 10 December 2009 (UTC)
- Nijdam, I completely agree with your words above. Why do you feel that you don't follow me any more? Boris Tsirelson (talk) 14:45, 10 December 2009 (UTC)
- Where you write: >>"After the opening of one of them this (!) probability now applies to the remaining unopened door." — For me this is the central (!) point of MHP. The only (!) point making it instructive to my students (in my opinion). << Because this is not in line with my explanation above. It is not this probability, but a different probability with the same value as the former. Nijdam (talk) 22:05, 10 December 2009 (UTC)
- (unindent) Ah, now I see; I was missing your point. Yes, of course. But on the other hand: in many cases people feel intuitively that some conditioning does not change some probability; and they are right often (probably, not always). There is a big difference between the approach of a mathematician (unless he speaks informally) and that of others. And I think, we should not be too insistent. I'd recommend a tone like that: ...thus the probability is 2/3. This probability is called unconditional, since ... In contrast, by the conditional probability one means ... It is worth to ask what is the conditional probability. It seems plausible that it is the same (2/3) since ...; and it is really the case. In general, conditional probability differs from unconditional probability, it is either bigger or smaller, depending on the condition; but on the average it is equal. In particular, in our case the conditional probability does not depend on the condition by symmetry. Thus it cannot be bigger or smaller, but only equal to the unconditional probability. However, without symmetry it is different; see Section "Asymmetric generalizations" below. Boris Tsirelson (talk) 07:11, 11 December 2009 (UTC)
- This has been Rick's and my opinion all along. We won't put much emphasize on it. But anyhow it should be mentioned that after opening of the door by the host a new situation has risen in which becuase of the symmetry the probability for the car being behind the chosen door is the same (has the same value) as before. This is not clear to many of the discussiants and some even fight this fact. Like in what is called the "combined door solution". I hope we may find you on "our" side? Nijdam (talk) 09:39, 11 December 2009 (UTC)
- It is a bit risky to say "yes" (or "no"), since we sometime misunderstand each other. Another example: let two fair coins be tossed; two independent events A,B of probability 0.5 appear. Now someone could say: P(B|A) = 0.5 = p(B), but this is a numeric coincidence only; P(.|A) is another probability measure different from P(.); never say "this 0.5", say "another copy of the number 0.5", etc. As for me, this is much too formal; definitely inappropriate when talking to non-mathematicians. It seems to me, you told this way. But maybe I again misinterpret your position. If so, then the better. Boris Tsirelson (talk) 11:39, 11 December 2009 (UTC)
- Nor Rick, me or Kmkmh have the intention to be formal (except of course in our minds). What we want is to be sure the distinction between the probability before the opening of a door by the host and the probability thereafter is somehow mentioned. I.e. we want to make clear or at least mention they are not the same. Why? Well, because some people, as I wrote before, reason as follows, or in a similar way: the probability of the car behind the chosen door 1 is 1/3. After opening of door 3 the probability for the car to be there is 0, hence (???) the probability for the other unopened door is 2/3. This "simple" explanation (solution?) is wrong. It turns up in many forms, all omitting the difference between the unconditional and conditional probabilities. This way of reasoning is copied by teachers, pupils, students and alas also by some mathematicians. It is the reasoning in the "combined door solution". It also turns up in many of the simulations on the internet. I suggested a very modest phrasing month ago: After the player has chosen a door, the probability it hides the car is 1/3. This probability is not influenced by the opening of a door with a goat by Monty, hence after Monty has opened a door with a goat, the probability the original chosen door hides the car is also 1/3. Because clearly the open door does not show the car, the remaining closed door must hide the car with probability 2/3. Hence switching increases the probability of winning the car from 1/3 to 2/3, which is formally correct. In a next, more formal section we could go into some more detail about the probability not being influenced. Of course this all is not Wikipedia:OR. It is found in many texts about the MHP, amongst them Morgan et al's. Nijdam (talk) 13:52, 11 December 2009 (UTC)
- Please pardon the intrusion. Nijdam, you wrote this above: "I consider this as insufficient." It seems to me that your opinion contrasts with many reliably published sources. I think Wikipedia's policy is pretty clear on this, and I imagine Rick Block would agree. Are you suggesting, insisting actually, that the article be edited to your opinion, rather than the reliably published sources? And although you did not ask me, I think your argument in the above paragraph is a rather weak insistence on the necessity of the so-called 'conditional' problem statement. How can a 'conditional' problem statement be a Monty Hall problem statement, when it doesn't even follow the 'words' of the actual problem statement from the published sources? Glkanter (talk) 11:52, 9 December 2009 (UTC)
- Boris, Nijdam - As Boris has suggested, the concept of a condition is to some degree arbitrary. What is the chance I will get a head if I toss fair coin given that I clap my hands before tossing it. Strictly speaking we could call that a conditional problem but nearly everyone would intuitively say that clapping your hands will not make any difference to the outcome so it can be ignored, and who can disagree? There has to come a time where it is permissible to say that a condition is unimportant and can be ignored. It is intuitive to most people that if a random legal goat door is opened it will not matter which one, this is also a true fact. It seems right to me, therefore, that we can ignore this detail in our initial presentation of the problem and its solution. Martin Hogbin (talk) 12:18, 11 December 2009 (UTC)
- (1) I agree with Martin (as usual).
- (2) About finding me on this or that side... Asked (by Nijdam) what is my point I have formulated it (in two concise items, you know). If each of us will do so, then it will become clear what are the possible coalitions (and the old "yes-no" list will become obsolete). I would be especially glad to reach a common position of Rick, Nijdam, Martin and myself. Boris Tsirelson (talk) 13:55, 11 December 2009 (UTC)
- Sorry Martin and Boris (I discoverd you reacted in the mean time). We have discussed these ideas over and over a long time ago. I'd rather see you coming to some insight. I write the conditioning completely out; chosen is door 1, door chosen by the host to be opened in boldface:
door* 1 door 2 door 3 probability car goat goat 1/6 car goat goat 1/6 goat car goat 1/3 goat goat car 1/3
- The host opens door 3 showing a goat:
door* 1 door 2 door 3 conditonal probability car goat goat 1/3 goat car goat 2/3
- I hope you admit there is factually conditioning and not some handclapping or whatever. Nijdam (talk) 14:25, 11 December 2009 (UTC)
- There is some conditioning in so far as a door has been opened, I think this is obvious to everyone. But we should allow the case that either door 2 or door 3 is opened to reveal a goat (even though in Whitaker's question he gives door 3 as an example), thus it is not important which door has been opened. Martin Hogbin (talk) 17:40, 11 December 2009 (UTC)
- I hope you admit there is factually conditioning and not some handclapping or whatever. Nijdam (talk) 14:25, 11 December 2009 (UTC)
- Okay, door 2 might be opened too. Here it comes:
- The host opens door 2 showing a goat:
door* 1 door 2 door 3 conditional probability car goat goat 1/3 goat goat car 2/3
- Again factual conditioning. You permanently "forget" that the player is on stage and sees which door is open. In fact the player might have chosen door 2 or 3 as her initial choice. Any combination of choice and opened door leads to the same analysis. That's why, the given combination serves as an example for the complete problem. Nijdam (talk) 09:20, 12 December 2009 (UTC)
Comment Boris
(I added a section break for easy editing)
- At the moment when X is already known (but H and C are not) we can say the following. If C=X then H is distributed uniformly on {1,2,3}-{X}.
>>>Okay, let us start from X given; then P(H=h|C=X)=1/2 for h!=X Nijdam (talk) 22:44, 14 December 2009 (UTC)
>>>>Yes; this is exactly what I wrote. To be uniform on a two-point set means just probabilities 1/2 at each of these two points. What is the problem here? Boris Tsirelson (talk) 05:59, 15 December 2009 (UTC) >>>>>No problem hereNijdam (talk) 14:22, 15 December 2009 (UTC) Otherwise, if C differs from X, then still, H is distributed uniformly on {1,2,3}-{X}. ("Given C != X", not "given C"!
>>>P(H=h|C!=X)=1/2 for h!=X ???? In my opinion: P(H=h|C!=X)=0 for h=C,XNijdam (talk) 22:44, 14 December 2009 (UTC)
>>>>No! Again: "Given C != X", not "given C"! It cannot be 0 for h=C simply because C is not given! This conditional probability does not depend on C. Boris Tsirelson (talk) 05:59, 15 December 2009 (UTC)
>>>>>Okay,I see what you mean.Nijdam (talk) 14:22, 15 December 2009 (UTC)
>>>>Use the force, do formulas not words! Words are sometimes ambiguous; formulas are not. The formula P(H=h|C!=X) means P(H=h|A) where A is the event C!=X. This event includes both values of C. In words: the condition "C != X" does not disclosure the value of C; it gives only a partial knowledge about C.
>>>>To be honest, these are not the true formulas, because one premise remains words: "At the moment when X is already known (but H and C are not)". But it is easy to get rid of the words completely, inserting the condition on X everywhere. That is, we consider P ( H = h | X, A ) where A is the event C != X. Or, if you prefer longer style formulas, P ( H = h | X=x, A ). In any form, it is important that the given condition leaves to C two possible values. Boris Tsirelson (talk) 06:54, 15 December 2009 (UTC)
The (3!) symmetry is already broken, but a (2!) symmetry persists.) That is, the (conditional) distribution of H does not depend on the event A = {C=X}. In other words, A and H are independent.
>>>>Before it was A={C!=X}?Nijdam (talk) 14:22, 15 December 2009 (UTC)
>>>I don't understand Nijdam (talk) 22:44, 14 December 2009 (UTC)
>>>>See above. And think again. Boris Tsirelson (talk) 06:38, 15 December 2009 (UTC)
>>>>It's clear now what you mean. Nijdam (talk) 14:22, 15 December 2009 (UTC)
>>>>You may also think about the following (rather standard) exercise. A fair coin is tossed 10 times. What is the (conditional) probability of "head" in the first trial given that there are exactly 7 "heads" in the total? Boris Tsirelson (talk) 07:10, 15 December 2009 (UTC)
Thus, the distinction between P(A) and P(A|H) is rather academical (like the distinction between (100-1)+25 and 100+(-1+25) in my example elsewhere); I think so. On the other hand, Rick says that somehow this distinction helps to many people to avoid errors. Maybe; this is beyond my expertise. Boris Tsirelson (talk) 12:40, 13 December 2009 (UTC)
>>>I did show in the above tables the factual conditioning. Where am I wrong? Nijdam (talk) 22:44, 14 December 2009 (UTC)
>>>>I never told there is no conditioning. I always agree that P(.|B) is not the same probability measure as P(.). I only say that the conditioning is ineffective on the considered event, due to independence. And the independence is not a numerical coincidence, but a consequence of the symmetry. Boris Tsirelson (talk) 06:43, 15 December 2009 (UTC)
>>>>>Okay, but I doubt whether these are the arguments the (I wrote adversaries, but meant) advocates of the simple solution have in mind or whether this makes it better understandable for the interested readers. I also cannot imagine you use this reasoning in your introductory course. Nijdam (talk) 14:22, 15 December 2009 (UTC)
- Well, we now agree on the mathematics (modulo a miserable problem of denoting by A different things on different days of discussion), and I feel my mission finished. About my introductory course: I had no reason to use this argument, but I could, and it would not make more troubles than other topics; however, this is hardly relevant to WP. About adversaries I do not know; I only feel that arguments of symmetry are quite easily guessed by many non-mathematicians (they do not vast effort to the trouble of formalization...). Boris Tsirelson (talk) 15:23, 15 December 2009 (UTC)
[outdent]I do not forget that fact. You seem to forget that we are addressing the formulation in which the host chooses randomly between legal goat doors. There is therefore a strict conditioning, just as there is in my urn problem, but it is quite obvious that this conditioning can be ignored. If the host chooses at random, it cannot matter which door he opens. It is just like my hand clap, is does physically occur but it is obviously irrelevant.
- It is relevant. Look at the probability the car is behind the door to be opened by the host. Before opening this equals 1/3, after opening 0. They are different probabilities. The same with de door left closed: before 1/3, after 2/3. How do we calculate these? By looking at the chosen door: before 1/3, after (also) 1/3, but although the same value, different probabilities! Before we have the (unconditional) distribution: 1/3, 1/3, 1/3; after (conditional): 1/3, 2/3, 0. Different distributions. If you're not convinced, please give a proper analysis, in formulas (with words to explain), not just in words. Nijdam (talk) 10:33, 12 December 2009 (UTC)
- It is not relevant to calculating the probability of interest, which is that the car is behind door 1. Martin Hogbin (talk) 10:51, 12 December 2009 (UTC)
Terminology and choice of sample set
The above heading is essentially for ease of editing - this section follows on from the thread above.
I do see your point, Nijdam, it is not easy to produce neat diagram (or formula) that proves my point, but I will have a go. To some degree the choice of initial sample set is arbitrary and this choice defines how the calculation proceeds. It is the choice of initial sample set that has to be defended with words. It may be that with your choice, my point is not easy to make. The problem is that other choices (say based on goat doors and car doors) may be difficult to justify. Let me think about it. Martin Hogbin (talk) 11:23, 13 December 2009 (UTC)
- But I took the challenge to answer to Nijdam in his own terms, see above. Boris Tsirelson (talk) 13:00, 13 December 2009 (UTC)
What about this? Not the terminology that you choose but it implicitly includes the symmetry of the situation in that the action of the host makes no difference to the answer.
1/3 | 1/3 | 1/3 | |||
Goat | Goat | Car | |||
The host opens door 3 to reveal a goat | |||||
Stick | Swap | Stick | Swap | Stick | Swap |
Goat | Car | Goat | Car | Car | Goat |
Martin Hogbin (talk) 12:14, 13 December 2009 (UTC)
- Hey, Nijdam, why don't you quit ignoring my edits and questions? It's pretty rude. If Huckleberry had used your method instead of Devlin's method, what would have been different in the game play that went on infinitely? Would he win a 'different' 2/3 of the time, or the 'same' 2/3 of the time? Wouldn't you agree it made no difference whatsoever to Huckleberry which door Monty opened? Isn't this known as 'indifference' in Mathematics? And, there is no Wikipedia policy stating story problem paradoxes must be solved using formal notation. Wouldn't be such a popular paradox in the real world with that requirement, would it? Glkanter (talk) 13:08, 12 December 2009 (UTC)
- There is also a words, not formulas explanation - which I think would be accessible to a layperson - at Talk:Monty Hall problem/FAQ. And, to address a point Glkanter attempts to interject above - this argument is firmly based on wp:reliable sources, not wp:or. Nijdam is reiterating what Morgan et al. and Gillman, and Grinstead and Snell, and most elementary probability textbooks (as claimed by Kmhkmh) say. Glkanter definitely, and Martin to a lesser extent, misinterpret the main point of the Morgan et al. paper. I believe Glkanter genuinely does not understand that there is a difference between asking about the probability of winning by switching (in general, for all players) and asking about the probability of winning given knowledge of which door the host opens, i.e. is having trouble with the basic concepts of conditional probability (see, for example, this edit). I think Martin understands the difference but insists that a complete explanation of the "notable" MHP can be made without mentioning this difference and it should therefore remain unmentioned until (essentially) a variant section. JeffJor understands this difference and insists that the MHP is explicitly asking about the former rather than the latter and therefore any source that says the MHP is asking about the latter is WRONG and should be excluded from THIS article.
- You told me not to speculate on your likely actions regarding this consensus. How then is it appropriate for you to speculate on my personal knowledge of the subject matter, given that we have had no communications whatsoever other than on the various Wikipedia pages? Glkanter (talk) 16:44, 11 December 2009 (UTC)
- I agree with both of Boris's "position points" above, but would add for the 2nd point that co-existence CANNOT mean excluding one in favor of the other (per FUNDAMENTAL Wikipedia policy, i.e. WP:NPOV). I've been referencing this version of the Solution on other threads as well, which I think is far closer to adhering to both of these points than what Glkanter, Martin, and Jeffjor are suggesting. -- Rick Block (talk) 16:24, 11 December 2009 (UTC)
- I guess I do agree with you, Rick, that I do not understand the main point of the Morgan paper, but that is because it is so badly written that it is impossible do divine what the main point is. I can assure you, however, that I do understand the subject and the issues involved. As you will see from my discussions with Nijdam, I accept that, strictly speaking and in some formulations, the problem is conditional, but in the symmetric case (host opens a legal goat door randomly) the condition (which door the host opens) is clearly irrelevant to calculating the probability of interest, thus it can be ignored. I also agree with Jeff, that the MHP can be reasonably interpreted to ask an unconditional question.
- Because of the both the facts presented above, I believe that the article should start with a complete section in which the problem is treated unconditionally (as in many reliable sources). After that (or even at the end of that section) I am happy to point out that even if the host chooses randomly the problem might be treated conditionally, with a reference to Morgan. This could then lead on to a variations section which would include the Morgan scenario (we know the door opening policy of the host). Martin Hogbin (talk) 11:09, 12 December 2009 (UTC)
- (unindent) back from weekend Here is a proof that 99+25=124:
- 99+25=100-1+25=124.
- Is it a correct, complete proof? Or should I show that I understand the conceptual difference between (100-1)+25 and 100+(-1+25) and use the associative law in order to overcome the difficulty? Boris Tsirelson (talk) 15:33, 12 December 2009 (UTC)
- People never write complete proofs. Yes, absolutely never. If in doubt, look at Mizar system; there you can find source texts and programs that allow to generate some complete proofs (but probably you'll never have enough paper in order to print such a proof).
- The more so, we should not insist on complete proofs in the encyclopedic context. Symmetry arguments are usually treated by non-mathematicians as too evident for being proved. Boris Tsirelson (talk) 15:40, 12 December 2009 (UTC)
- Thus, we'll never be able to convince the majority of our readers that the conditioning is really relevant in the symmetric case. Likewise, we could not convince children that 99+25=124 cannot be accepted if the associative law is not involved.
- And still, I think, it is not bad if conditioning will be mentioned in the article, as far as we'll not be insistent about its relevance in the symmetric case. Boris Tsirelson (talk) 15:48, 12 December 2009 (UTC)
- Isn't the "host forgets' case also symmetric? If we're using "symmetry" to say the host's action does not affect the player's initial chance of selecting the car, why is it that this same argument does not apply in this case? I know the reason and you know the reason but I'm willing to bet many people who "understand" the popular solutions to the MHP do not understand why these solutions DO NOT apply in this case. See, for example, this exchange on the arguments subpage.
- Rick, we've argued this many, many times. The problems are greatly different. Just one example, there are times the forgetful host doesn't offer the switch. Because he revealed the car. Two stated premises of the MHP are that he always reveals a goat, and always offers the switch. I'd hate to see any editor invest time by responding to this oldy, moldy filibuster. Glkanter (talk) 18:36, 12 December 2009 (UTC)
- (I've basically said this before - in this section, above) In a fairly recent column, vos Savant addresses the "host forgets" variant. In her analysis of this version she laments [1] [unfortunately now a dead link, and not available on the Internet Archive ] "Back in 1990, everyone was convinced that it didn’t help to switch, whether the host opened a losing door on purpose or not. ... Now everyone is convinced that it always helps to switch, regardless of what the host knows. But this is just as incorrect!" This is absolutely true. And, IMO, it's precisely because the popular sources do NOT address the "classic" MHP using conditional probability. By avoiding addressing the problem in this way, the popular sources have simply replaced one incorrect notion (two unopened doors always means each has equal probability) with another (whatever the host does he can never change the 1/3 probability of the player's initially selected door). As a featured article on Wikipedia, IMO this article must not make the same mistake. -- Rick Block (talk) 18:17, 12 December 2009 (UTC)
- Really? Well, if the conditioning helps to not make errors then of course it is useful. Then we only have to explain to the reader why it is important (namely, demonstrate him typical errors) before bothering him by conditioning; then hopefully he will not be disturbed. Boris Tsirelson (talk) 18:58, 12 December 2009 (UTC)
- Especially if we deal with the problem in more detail later on in the article for those who are interested. Martin Hogbin (talk) 13:25, 13 December 2009 (UTC)
This whole discussion could have reached an amicable end long ago
If the Morgan supporters would please read and respond to the existing section "Is The Contestant Aware?", and it's question:
- "Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?
- "It seems to me that this is a (unstated) premise of the problem, as both vos Savant (Whitaker) and Krauss and Wang begin the problem statement with: 'Suppose you're on a game show'. I read this as clearly stating it is only the contestant's point of view we are concerned about. And, being a game show, the host is prohibited from divulging to the contestant either where the car is, or where the car is not.
- "Is there agreement on this, or is this in dispute? Glkanter (talk) 11:28, 29 November 2009 (UTC)"
- http://en.wikipedia.org/wiki/Talk:Monty_Hall_problem#Is_The_Contestant_Aware.3F —Preceding unsigned comment added by Glkanter (talk • contribs) 16:15, 5 December 2009 (UTC)
- Yes this is in dispute. The view(s) that have to be described in the article, are the views taken in publication on the Monty Hall problem and there treatmeants which assume different perspectives. What we probably can be agreed on, that those different perspectives should be handled in a variations or generalization section and that they don't belong into the article lead. Also it is probably not appropriate to assume (specific und possibly) legal gameshow regulations. Since most of our readers, much of the original audience, potentially even whitaker himself and some of the people that have published on it, are possibly not aware of specific regulation and legal restrictions and definitely do not mention them. This means while the article can and should discuss how actual regulations (or the real monty's behaviour) might influence the analysis of the problem, but that definitely doesn't belong in the lead section nor can any such regulation simply be considered as an obvious "given" for analysing the problem.--Kmhkmh (talk) 17:01, 5 December 2009 (UTC)
- But it is indisputable. I don't care whether the problem starts with "Suppose you are on a game show," or what you think that implies. In every version of the MHP I have ever seen, the question is "Should the contestant switch?" It is not "What is the probability of winning the car if the contestant switches?" Or "Describe a parametric formula that allows the contestant, once given the proper data, to decide whether or not to switch." It is a yes/no question in the purest sense: only two answers address the question. "Yes, she should switch" and "No, she should not switch." If you are required to make assumptions to reduce your answer completely to one of those, there are acceptable means to do so. The point about the contestant being aware is indistutable because we have to make whatever assumptions are necessary to reduce the answer to "'yes' in all possibilities considered" or "'no' in all possibilities considered," which means that we have to apply the same knowledge we assume the contestant will use. When they address the actual MHP question, the sources that address the conditional problem make such assumptions; just not that specific assumption about how the host chooses between two goats. They do make the neutral contestant-knowledge assumptions for what I call "game protocol" (e.g., always revealing a goat) and placement bias.
- Morgan's point is NEVER that the answer to the MHP is 1/(1+q), or whatever. It is that one specific assumption is not really necessary when the problem is taken literally in conditional form (which itself is disputable: the sources that claim it is conditioanl misquote their sources in such a way that makes it conditional). You don't need to make the assumption that reduces 1/(1+q) to 2/3, since 1/(1+q) is always greater than 1/2. But that conditional form loses the real beauty of the MHP, that the seemingly paradoxical answer is correct. Far more sources discuss this aspect of the problem than address the "conditional solution," which is where some people's POV is interfering with their ability to approach the article objectively. Yes, the conditional problem is addressed by some sources. It is a minority, and neutrality means it should be treated as a variant. JeffJor (talk) 13:31, 7 December 2009 (UTC)
- I imagine Selvin explained his version of the game show this way: The contestant makes a random choice of doors. The probability she did NOT choose the car is 2/3. Monty reveals a goat, giving her no new information about either remaining door. She is indifferent to which door Monty opened. Being a sentient being, she says to herself, my selection's 2/3 chance of not being the car has not changed, therefore, on average, the remaining door has a 2/3 chance of being the car. I don't think I can do better than that. I'll take the switch! Glkanter (talk) 13:59, 8 December 2009 (UTC)
- These are the only 2 known English language Game Show situations (Whammy and the 1950s Scandal) in which the particular contestant had more 'information' than an 'average' contestant would have. Both situations were considered highly unexpected aberrations, and many steps were taken to prevent either from being repeated.
- So, yes, there is a commonly understood expectation of all game show watchers that each contestant has no information otherwise not available to other contestants, certainly not coming from the host (or the shows producers). And there's tons of laws and lawyers watching for this in the US.
- This is not ambiguous: 'Suppose you're on a game show' Glkanter (talk) 17:33, 5 December 2009 (UTC)
- You miss the point you are making assumptions based on your knowledge and perception of game shows. There is no evidence whatsoever that most of the readers have the same legal background as you have nor do most publications on subject deal with legal restrictions or "gameshow regulations". Basically this boils down to stick to the sources and no speculations by WP-Editors.--Kmhkmh (talk) 17:52, 5 December 2009 (UTC)
- Please see my response above. -- Rick Block (talk) 17:50, 5 December 2009 (UTC)
It is so pervasive, nobody bothered to mention it explicitly as an assumption. Everyone in the US understands this rule of game shows. The idea that one of the contestants on the screen has 'inside' knowledge? Folly. It's in the fine print that runs at the end of each and every episode. And I documented the only known times it improperly happened. This is not my opinion or interpretation. It's a defining characteristic of a game show, without which, it is no longer a game show. Your twisted interpretation more resembles a street hustler with a card table and a deck of cards. Or three shells and a pea, perhaps. Glkanter (talk) 18:11, 5 December 2009 (UTC)
- You make a bold claim (It is so pervasive, nobody bothered to mention it explicitly as an assumption.) without providing any evidence and moreover you miss the point again. The question was not posed to actual game show contestants (being most likely aware of all regulations) but to readers of weekly column and later through various (international) publications to a broader audience. Even if your most likely false claim was true for all readers of Marylin's column, it is certainly not true for the international audiences reading the various publications. I've sampled quite a lot of material on the subject in English and in German and on top of my head i cannot recall any of them mentioning legal restrictions and national game show regulations. In particular the German articles on the subject with a readership definitely being unaware of any such regulations do not mention any legal conditions whatsoever. The 2 cases you've posted above are not in question, however they do not prove your claim. What's in question here is your (false) line of reasoning.--Kmhkmh (talk) 18:35, 5 December 2009 (UTC)
- The MHP is really, as this article confirms, a probability puzzle. In such puzzles it is usual to make certain assumptions, which in the MHP would be that the cars are initially placed randomly, the player chooses randomly, and that the host chooses a legal goat door randomly. The assumptions are precisely those Glkanter claims above.
- If in the other hand, you want to claim that the puzzle is based on the question by Whitaker then I suggest that you first read the real Whitaker statement rather than the inacurate version misquoted by Morgan. In Whitaker's original question it is clear that the door number given for that initially chosen by the player and the door number opened by the host are intended to be examples rather than specified doors. I might add that, as no information is given in the question, we should still take all the human choices to be random by the principle of indifference. Martin Hogbin (talk) 11:27, 12 December 2009 (UTC)
Informal mediation requested
I have filed a request with the mediation cabal, see Wikipedia:Mediation Cabal/Cases/2009-12-06/Monty Hall problem. -- Rick Block (talk) 18:20, 6 December 2009 (UTC)
- What is the procedure now? Should we prepare a statement of what we definitely all do agree on? Martin Hogbin (talk) 18:56, 6 December 2009 (UTC)
- This mediation is completely informal and there's no guarantee anyone will accept this case (trying this first is generally a requirement before proceeding to formal mediation handled by the Mediation Committee). For now, I think we continue as best we can. -- Rick Block (talk) 19:17, 6 December 2009 (UTC)
The informal mediation request has been closed, with a recommendation for formal mediation.
Martin or Glkanter - would one of you like to file the request for formal mediation? See Wikipedia:Requests for mediation/Guide to filing a case. -- Rick Block (talk) 19:36, 17 December 2009 (UTC)
The Mathematics Rule I Am Properly Applying
Way back in junior high, we did some proofs or problems or something to do with absolute values. That's all I can remember.
But the thing I do remember is that after you 'solved' the problem, you had to go back and check each of the results to make sure it didn't violate the original problem statement in some way.
That's all I'm saying about Morgan and the rest. When you check your work with some 'host behaviour' variant, it no longer meets the original problem statement, "Suppose you're on a game show..." Go ahead and argue. Better you should save your breath. Hosts don't tell contestants where the car is.
So, as an encyclopedia, Wikipedia will properly refer to reliably published sources like Morgan. And Devlin. No problem.
But, as a self-appointed 'explainer' of all things MHP, I think the article improperly gives the conditional solutions way too much emphasis. Because it doesn't match the original problem statement any longer. Glkanter (talk) 21:47, 6 December 2009 (UTC)
- Often one solves a problem by saying something like "let x be the distance travelled". One converts the problem to algebra, and finds a solution "x=-2 or x=3". Lazy students quit there. But good students think. We go back and remember that we wanted a distance and it had to be positive, so the real answer is x=3. This is a fine problem solving strategy. One solves a relaxation of the problem, that is to say one solves the problem while forgetting about some of the constraints, finds some answers, and then looks to see if any satisfy the original problem. But I wouldn't call this strategy a "mathematics rule". I'm not sure it applies in this case, where the original problem is somewhat ambiguous. It turns out that there are a number of interesting MHPs. Gill110951 (talk) 05:19, 22 December 2009 (UTC)
- What I'm saying is that the so-called 'variants' do not satisfy the original problem. As I see it, the contestant being aware of any host bias is mutually exclusive with 'Suppose you're on a game show'. That's why the question the 'opposed' editors refuse to answer, 'Is the Contestant Aware...' is so pivotal to these discussions. A game-breaker, really. Glkanter (talk) 05:32, 22 December 2009 (UTC)
What "the conditional problem" and "the unconditional problem" mean
I wasn't quite sure where to respond, so a started a section.
- There are multiple ways to address any probability problem. If I ask "What is the probability a die rolled a 3, if we know it rolled odd?" we can solve it conditionally or unconditionally. We can say there are six equiprobable outcomes, and the conditional probability is found by P(3|odd)=P(3 and odd)/(P(1)+P(3)+P(5)) = (1/6)/(1/6+1/6+1/6)=1/3, or that there are three equiprobable outcomes that are odd, so P(3)=1/3. The unconditional problem is isomorphic with the conditional one.
- The MHP IS a conditional problem, but depending on how you address it, you can solve an isomorphic unconditional problem in its place. Just like my trivial example. That is not what we want for the so-called "unconditional problem" that the main part of the article should address.
- What we want, is for the contestant to base her decision on P(switch to car|she choose a door and host opened another door) as opposed to P(switch to car|she choose door #1 and host opened door #3). And I worded that carefully, because the first option is EXACTLY what the Whitiker problem statement says, and what K&W admit is the sematic meaning of the statement. We call it "the unconditional problem" because it is not conditioned on knowing how the host treats the doors differently. The doors cannot distinguished from each other in terms how probability applies. That does not mean that they can't be distinguished, it just means it can't be used.
- The question in the classic MHP is "should the contestant switch." We want "the unconditional problem" because we are not told how the host treats the different doors in any statement of the problem that I am aware of. So there is no known source that uses the unique aspects of "the conditional problem" to DIRECTLY answer the MHP. None.
- The two sources that seem to be recognized as starting the "game show" variation of this problem (as opposed to the related and much older "Three Prisoners" problem and Bertrand's Box Paradox), that is to say Steve Selvin's 1975 The American Statistician article and Marilyn vos Savant's Parade article, both used names for the boxes/doors as examples. But neither used those names to treat them differently in the solution. So they are both addressing the unconditional problem. Marilyn vos Savant, at least, has clarified that she intended the the "unconditional" problem.
- There are only two aspects to "the conditional problem." (There are more that are what I call "game protocol," like whether the host always offers a switch. I disregard those, because we have to assume the game protocol is well represented in the problem statement.) There are two random chioces, not one, that are part of the uncertainty in the game and that require assumptions. Not all of the "conditional problem" sources use both (Morgan does not), but those that do universally make the assumptions about the car placement that they eschew for the host's choice. Tat is, they treat it like "the unconditional problem." No conclusions can be drawn, nor HAVE been drawn, about the answer to the full "unconditioanl problem."
- There are, in general, three types of sources for the MHP: (A) Those that address problem from a common-knowledge standpoint, and that universally use "the unconditional problem" WHETHER OR NOT THEY USE CONDITIONAL PROBABILITY in their solution method. (B) Those that address the cognitive issues that cause confusion in people when first presented with the problem, and that emphasize in non-intuitive aaspect of switching to improve probability. This has nothing to do with the "condititional problem." (C) Those that explore it from a more rigorous mathematical viewpoint. Some emphasize either the unconditional, or the conditional. Call them CU and CC. Now, I haven't read all the references, but I did a quick run-down of the ones listed in the article. Of those that I could find quickly, 14 were in category A, 5 in category B, 4 in category CU, and 3 in CC. That shows that most people view the MHP as unconditional problem.
- Nobody is saying to disregard "the conditional problem." It is just a variation. JeffJor (talk) 20:22, 7 December 2009 (UTC)
Well said! Are you a new editor or did you forget to log in? I removed the linefeeds to fix your numbering. Martin Hogbin (talk) 18:40, 7 December 2009 (UTC)
No, I do log in - but somehow in switching windows I keep ending up in a non-logged-in window, and I don't notice that when the edit comes up. JeffJor (talk) 20:22, 7 December 2009 (UTC)
- See also #Formulas, not words above. Boris Tsirelson (talk) 20:29, 7 December 2009 (UTC)
@128.244.9.7:
- I completely agree
- I agree for the most part, however the bold line is a bit iffy. Who is exactly is "we"?
- This is bit iffy. K&W don't admit but assess what the real meaning is according to them, other publications differ on that. Your reading of Whitakers intent is fine, but alas it is only valid option to read the problem. You can also see it as P(switch to car|she choose a door and host opened a particalur door). If you use particular instead of another you end up exactly where Morgan & bulk of mathematical treatments go and you do use the particular door, i.e. doors can be distinguished and used. Note that both version treat door 1 and door 3 as examples.
- Not knowing the host behaviour per se does not mean we can't model it nor that we cannot draw conclusion from it. In fact Morgan & Co argue that no matter what the host behaviour is, you are never worse off by switching.
- Your descripion is not quite correct here. Selvin the mathematician/statistician who coined the problem 15 years before the parade buzz provides a unconditional and conditional solution to it (see Rosenhouse). This btw also defeats a point Martin keeps rising for more than half a year now, i.e. that the problem is essential not a math problem, but the mathematical perspective is just a minor sideshow. If the original problem was posed, solved and named by mathematician in math journal and this is not supposed to to be a math problem, then frankly i don't what is. That vos Savant solved the problem as an unconditional one is fine, however since she did not pose the problem, this tells us nothing of the "real" intent/perspective of the question.
- Here i admit, i'm not quite sure what point you're trying to make. What exactly is the "full unconditional" and to what literature/publications areyou refering for having drawn no conclusion. I assume you are aware that by now almost any recent primer into probability (in various language) contains a conditional treatment of MHP.
- I disagree with your assessment of (B). One thing about conditional probilities is that they are considered "unintuitive", i.e. unintuitive aspects and in particular the handling a posteriori/additional knowledge for reasoning are all about conditional probabilities. As far as (C) goes I don't quite buy your category count at first glance but even if that turns out to be accurate it does match for overall treatment in literature to my experience. As i pointed out already almost any modern probability primer has a conditional approach to the problem.
- yes and no, that depends on reading between the lines and carefully reviewing all the statement made here over time. Also see the recent suggestion by Jeffor and Glkanter, which to some degree you can consider as an attempt to remove the conditional approach from this article. Another thing is here.how you read/use the term variation. In light what else has been said here recently, I'd rather argue if you consider 2 problems to be isomorphic they are the "same".
- One additional reminder: In our attempts to understand/explain to "what was really meant", we easily overlook that as far as WP is concerned (in particular for contentious issues), we are supposed to describe what various reputable sources say and not what we think is right.
--Kmhkmh (talk) 20:32, 7 December 2009 (UTC)
2. "We," I believe, are those who think the "conditional" problem is a variant. Sorry if it sounded to general.
3. K&W say that the doors numbers are examples only; that means it isn't intended literally. But AFAICT their treatment does not allow their participants to consider that the host is biased. Both are needed to make it "the conditional solution."
4. While you can model it, you can't use it to directly answer the question "Should the contestant switch?" because that question has to be answered from the contestant's SoK. It is just a coincidence that, for that one bias, it always goes up. For example, if the the host tells you the placement is biased 50%:25%:25% for D1:D2:D3, and that his opening bias (afte you choose Door #1) is either 25%:75% or 75%:25 for D2:D3, but not which? Then the probability of winning goes up in one case and down in the other, BUT IT IS STILL ADVANTAGEOUS TO SWITCH BECAUSE THE AVERAGE PROBABILITY GOES UP. The point is that it doesn't help the contestant IN GENERAL to say what the probability is based on hidden knowledge, if that knowledge is not also given.
5. Yes, Selvin did. See #2 above. Did you miss the part where Selvin said that the key to his solution was that he assumed the host chose randomly between two doors, making his solution a conditional approach to what we call the unconditional problem? Martin's point is entirely correct, and was the point of Selvin's and Savant's questions.
6. If door numbers are important, the solution has to treat the possibility of placement bias the same way it treats opening bias. If you do that, the question "Should she switch?" is unanswerable. THE CONDITIONAL PROBLEM CANNOT BE ANSWERED. So all the treatments eventually get around to assuming car placement is 1/3 to each door. The question they actually answer is halfway in between being the conditional one and the unconditional one.
7. Show me a survey where respondants took host bias into account.
8. Both approaches are to be included, even if separated into parts or even different articles. But any problem that does not address (1) The formal statement for the article or (2) The problem intended by either of the two articles that started the controversy, does not belong on an equal footing.
9. Agreed. In what way is the suggestion to keep them separate not? JeffJor (talk) 22:16, 7 December 2009 (UTC)
- Jeff - You've quoted the K&W statement about the semantic meaning of the problem before. I must assume you've read the rest of the paragraph. If not I've quoted the entire paragraph above (this edit). What they're actually saying is that although Door 3 is semantically used as an example most people (97% of their sample) don't treat it this way. You (and they) are taking this to mean that the semantically precise interpretation is the "correct" one and thus that most people are more or less tricked by this problem statement into trying to solve the problem conditionally (and when they do this they come up with the 1/2 answer). Given this source says most (nearly all) people interpret the problem conditionally (meaning they're thinking about the specific case where Door 1 is the initial pick and Door 3 is the door the host has opened), and that this conditional problem is completely isomorphic to what K&W claim is the actual problem, and that other sources say that this conditional problem is what is meant, wouldn't it be rather helpful to our readers to address this issue head-on by providing a correct conditional solution? This is totally aside from the issue of whether addressing the problem only unconditionally satisfies NPOV. I don't know if anyone exactly puts it like this (I have most of the sources cited in the article - I'll look at some point), but I very strongly suspect that MOST of the refusal to believe the unconditional 2/3 answers comes from people using an incorrect conditional solution (not from failing to understand that the problem is "supposed" to be interpreted unconditionally).
- As I put it above, the fully unconditional solution is essentially saying "Ha ha, the problem tricked you into looking at the problem wrong - what is really being asked isn't the conditional case you're trying to solve but the general chance of winning by switching. You can easily see this must be 2/3 like this: <your favorite unconditional solution here>." What the conditional solution says is "Your approach was fine, but it's not quite right because you forgot that the host opens Door 2 sometimes when the car is behind Door 1. You see the host open Door 3 all the time if the car is behind Door 2, but only 1/2 of the time if the car is behind Door 1. So, if you see the host open Door 3 the car is twice as likely to be behind Door 2." The segue into the "half conditional" problem happens quite naturally because of the "1/2" in this solution. Where exactly does it come from and what does it mean? It is of course the much maligned host preference. I would be fine deferring consideration of this "1/2" to a variant, but insisting that this "conditional approach" be treated as a variant or as addressing a "different question" seems absurd.
- If you reply to this can you please try to separate your opinions from what you think the preponderance of reliable sources might say? For example, do you agree with Kmhkmh's assessment that "almost any modern probability primer has a conditional approach to the problem" (I'm taking this to mean either only a conditional approach or both conditional and unconditional approaches)? If you agree with this, I think NPOV says we MUST treat the conditional and unconditional approaches equally without favoring either one. -- Rick Block (talk) 03:46, 8 December 2009 (UTC)
- 1. I do not agree. The conditional solution calculates the conditional probability in the unconditional probability space, but the so-called unconditional solution calculates an unconditional probability in a conditioned probability space. There always is a condition. Nijdam (talk) 14:09, 8 December 2009 (UTC)
Yes, Rick, I've read the whole paragraph. The only point I ever made from that quote is that it is valid (and in fact proper) to interpret the Whitaker/MvS problem statement as not requiring specific door numbers; that they were examples. I didn't try to address whether K&W were using the conditional or unconditional problem, since they solve the full conditional one later (more below), and addressing it could be seen as POV. The "clarification" you provided, which you interpreted according to your POV, only shows that K&W recognized that respondants treated the door numbers the same way MvS did; as examples. Not as implying any dependency on the actual door numbers. And it was, at least in part, because the non-required use of examples was reinforced by K&W (not Whitaker/MvS) in the form their next question took: "Do you want to switch to Door #2?" That can't be answered without using the examples. So K&W followed in Morgan's and Gillman's footsteps, in rewording the MvS problem statement to make it become the conditional problem. But that paragraph, and my excerpt of it, had nothing to do with how the problem is addressed anywhere.
Once again, since I am apparently still speaking in cat when I say this (or your POV gets in the way of comprehension): The MHP is a conditional problem, but depending on what you see in it, it can be solved unconditioanlly. It is valid to do this: see G&S, p138, where they choose not to make an assumption which they say could be made without loss of generality. Their Figure 4.3 illustrates what Nijdam described using his own POV, as "an unconditional probability in a conditioned probability space" (emphasis added). Assuming it is a conditioned space means you are assuming there is a bias in either car placement and/or host choice. But that treatment is not what I call the unconditional problem. The "unconditional solution" is represented on p139 in Figure 4.4, and is a conditional solution that is isomorphic with the fully unconditional one (the isomorphism comes from the generalization they declined).
The point we (well, at least I) am trying to make here is that this pair of solutions represents what most people see in the MHP. K&W bear this out. People can a take what appears to be an unconditional approach (Figure 4.3 in G&S), but that is more complicated and provides no more generality than the alternative. With the generlization that a single door used in the solution represents the uninformed (to the contestant) distribution of all possibilities, and so its probability distribution must be the average of those possibilities, they can use conditional probability to get the same result. (It is possible that biases exist in this system: I am not assuming them away. But they all get eliminated in either solution. And I have no reference for this, but it is the approach most people, who can answer simple probability questions, actually take. I've seen it refered to as "symmetry." The paradox in the MHP is that they overextend this symmetry, not realizing how the revealed information makes things assymetric.) What we want the article to separate out, is the solution G&S add after Figure 4.4, where suddenly a bias is assumed without valid reason. That is what I have called "the conditional problem," because it requires the use of conditional probability, and ignores the symmetry which K&W always apply to their formulation of the probability. They don't use the conditional problem - they allow for it but always put in parametric values representative of symmerty, and thus the unconditional solution. They even provide arguments for why people assume symmetry. And I can't speak for "most primers," but I would think that what Kmhkmh has noticed is a conditional probability not based on biases; or if it is, wheere biases get eliminated by symmetry. 128.244.9.7 (talk) 16:34, 8 December 2009 (UTC)
- As far as most primers are concerned, you get some impression via Google Books as well (or alternatively sample recent lectures on university websites or the classic libraries/bookstore approach). Also reading the beginning of your comment, there seems to be some misunderstanding. I'm not aware of anybody here (aside Nijdam maybe) claiming the unconditional would not be valid - certainly not me. As often in science there a various valid perspectives on a particular problem and various ways to model it accurately. There is also imho no issue with featuring the unconditional solution first and separate from the conditional one. There is however an issue with (subtle) attempts to redefine the problem as such in a manner, which is not in line with the bulk of the sources. This concerns in particular formulations insinuating, that the conditional approach is not solving the "real" MHP problem, that the MHP is not a math problem (but a layman's creation), that an "question-answering-expert" knows best what the problem "really" is and what the intent of the question being posed to her was, or that psychological study on how average people react or think of the problem, tells us what the problem really is. Because these various points are at best if not outright wrong the reflection of an individual publication and not the bulk of literature on subject. Hence the WP article cannot assume such a position. I'm not opposed to rearrangements/modification of the article as long as they don't create the critical points I've mentioned above. From my personal perspective the article could be arranged rather differently (starting with the ambiguous Parade formulation and the simple unconditional solution, after that separate chapters for mathematical analysis (conditional,conditional,criticism, subtle changes of perspective,various modelling approaches and the requirements, problem variants, etc.), psychological/cognitive analysis (K&M,Mueser, Granberg and others), occurrences of MHP in other areas/sciences and a historical overview/timeline (from Gardner until now). Having said that however in general i prefer a rather conservative editing policy for featured articles, because every major change requires a reevaluation of the features status. Also in my experience in contentious cases good articles can easily deteriorate with a lot disagreeing people editing around, that's actually one of the main reasons why I won't support a change until I see a feasible editor compromise emerge first, that a) provides a meaningful improvement rather than marginal changes only and b) is in line with WP guidelines.--Kmhkmh (talk) 17:13, 9 December 2009 (UTC)
- Jeff - can you be more specific about the exact changes you're talking about? Is it:
- Delete the entire section "Probabilistic solution"
- Delete the 4th paragraph of "Sources of confusion"
- Delete "Other host behaviors" under "Variants" (or only some)
- Delete "Bayesian Analysis"
- Or, are you suggesting something else?
- From your comments above, it sounds like you have no fundamental issue with a conditional analysis so long as it assumes no host bias - i.e. you're only insisting that it be taken by definition that the host open one of two goat doors with equal probability. This makes me think you should be OK with the "Bayesian analysis" section, and even the "Probabilistic solution" section with fairly minor changes (like deleting the last paragraph in this section about the "host preference" variant or moving this paragraph to a "variant" section) - or even a single "Solution" section more like what was there following the last FARC (i.e. this version) which made no mention of a "host preference" variant. I think how we got to where we are is contention over the transition paragraphs in this solution section (the paragraph starting with "The reasoning above ..." and the following one). Would you be OK with reverting to this version and working on these paragraphs? -- Rick Block (talk) 20:34, 9 December 2009 (UTC)
The unconditional problem
Let us formulate the unconditional problem: A player will be offered the choice of one out of three doors, one of them hiding a car. What is the probability she wins the car? Nijdam (talk) 14:13, 8 December 2009 (UTC)
- More OR? Ho hum. Let's focus on sources, and the commonly understood meaning of a game show. This is Wikipedia, and the MHP from Selvin (who may have originally called the puzzle 'Let's Make a Deal problem', the name of Monty Hall's show) through K & W begins 'Suppose you are on a game show...' Hosts don't tell contestant where the car is on a game show. Many reliable sources use logical notation, and/or words, to solve the problem. Is there a problem with that technique? Glkanter (talk) 14:27, 8 December 2009 (UTC)
- That is a very interesting question and it shows why it is not just mathematics that is required to answer probability problems. For anyone to answer they need to know exactly what you mean. Do you want the question answered from the point of view (state of knowledge) of the player. If so, are we to assume that she has no knowledge of where the car was placed?
- Perhaps you would prefer a more modern approach. In that case I would want to know the initial distribution of the car behind the doors and the distribution of the players door choice.
- The point is that without stating exactly what it is that you want to know your question cannot be answered. Martin Hogbin (talk) 15:45, 8 December 2009 (UTC)
A problem for you in return
There is an urn containing 9 balls numbered 1-9. You have to pick ball 9 to win. We assume that your picks are at random. What is the probability that you will win in one pick? I think we can agree that it is 1/9.
Now suppose that you have two picks and the ball from the first pick is not replaced before your second pick. You do not win on your first pick. What is the probability you will win on your second pick and how would you calculate this? Martin Hogbin (talk) 16:31, 8 December 2009 (UTC)
- I assume with pick you mean random pick.
- Yes that is why I said, 'We assume that your picks are at random'. This is, of course, generally assumed anyway in an urn problem.
- I have to calculate: with the outcome of the first and second pick respectively. The calculation may be with the use of the apropriate laws, but you may arrive at the same answer by using a symmetry argument, stating that the second pick is from a conditioned urn, with 8 balls with number 9 amongst them. Note, that you always calculate the conditional probability. Nijdam (talk) 21:01, 8 December 2009 (UTC)
- You can obviously see what I am getting at. Nobody would suggest that the correct calculation is to take all the possible conditions after the removal of the first ball (That is to say ball 1 missing, ball 2 missing...ball 8 missing) calculate the probability of getting a 9 separately in each case then combine these probabilities weighted according to the (equal) probability of each condition. We can just observe that, provided it is not a 9, the first ball picked makes no difference to the probability of picking a 9 on the second pick.
- So it is with the MHP. We can note that, if the host opens a legal goat door randomly, it makes no difference to the probability of winning by switching which door he opens. In other words the conditional problem can obviously be treated unconditionally. Martin Hogbin (talk) 22:01, 8 December 2009 (UTC)
- I think the problem here is the right wording. Of course one may use symmetry arguments, as BTW I did before, but you do this in order to calculate the conditional probability (for the car to be behind the picked door!) in an easy way, as its value is the same as the unconditional. Nevertheless the probability of interest is the conditional one!! Nijdam (talk) 11:09, 9 December 2009 (UTC)
- OK, I accept what you say but my suggestion is that we ignore the issue of conditional/unconditional for the first section where we treat the problem in the simplest manner possible, showing why the answer is 2/3 and discussing the reasons why people get it wrong.
- After that we could say that 'strictly speaking' the problem is one of conditional probability but for the symmetrical case we can ignore this fact. Finally, in in a separate section, we should discuss the necessarily conditional case, the Morgan scenario, where the player knows the non-random host door opening strategy (but not the initial car placement). 12:06, 9 December 2009 (UTC)
- Well Martin, as you may have noticed from our lengthy former discussion, neither Rick nor me insist on mentioning the word conditional, but we somehow want to make clear that after a door has been opened new probabilities are at stake. And that a simple solution, along the lines "the car is in 1/3 of the cases behind the chosen door, also with 2/3 it is behind one of the other doors and hence (???) after opening of one of them with 2/3 behind the remaining closed door", is not complete. Nijdam (talk) 16:09, 9 December 2009 (UTC)
- For the symmetrical case the probability that the car is behind the chosen door remains 1/3 after a legal random goat door has been opened. Most people assume this, and there are several-easy-to-understand reasons why it is so. In my opinion for the symmetrical case we can just state this fact in the starting section, to be discussed in more detail later. Martin Hogbin (talk) 17:55, 9 December 2009 (UTC)
- As I said about wording, what does it mean if you say: "the probability that the car is behind the chosen door remains 1/3"? There is your and many other's problem. What do you mean with probability? A probability always remains the same! What actually is the case - as we have discussed over and over - the probability changes, without changing value. Nijdam (talk) 11:44, 10 December 2009 (UTC)
The Meta Paradox of The Monty Hall Problem Paradox
Selvin poses the MHp. He solves it unconditionally at 2/3 vs 1/3 if you switch. The problem is hailed as a great paradox.
vos Savant prints a letter inspired by Selvin in a general interest USA Sunday newspaper supplement. She solves it unconditionally at 2/3 vs 1/3 both when you made your choice, and when the switch is offered. Because Monty's actions don't impart usable knowledge to the contestant. It's a sleight of hand. Nothing happened.
All heck breaks out. Tens of thousands of letters, including over 1,000 from PhDs tell her she's wrong. And they are certain!
vos Savant soothes the savage beasts with logic and smarts. The unconditional solution carries the day. The problem is, again, hailed as a great paradox.
This group, "J. P. Morgan, N. R. Chaganty, and M. J. Doviak are Associate Professors and R. C. Dahiya is Professor, all in the Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia" (284 The American Statistician, November 1991, Vol. 45, No. 4 (C 1991 American Statistical Association) develops the argument that the problem is only properly solved using a conditional problem statement. Their criticisms, etc. rest on this: That when faced with 2 goats, the host must decide which goat to reveal. This rests on the assumption (presumption, invention) that the contestant might somehow gain usable information as to the location of the car in this particular instance of the game by Monty's actions. It's left unstated whether Monty's actions would be shared with the contestant. And if they are shared, what method is used. But it's clear: in this instance of game play they claim, the subject contestant could be armed with more useful information that the average contestant.
- The only problem is, their 'assumption' is not consistent with the first words of the MHP problem statement: "Suppose you're on a game show...", as Hosts don't tell contestants where the car is hidden. Actually, some Wikipedia editors have found a math error in the paper, and are in communication with the publication. Oh, and "Richard G. Seymann is Professor of Statistics and Business Administration, School of Business, Lynchburg College, Lynchburg, VA 24501" (1991 American Statistical Association The American Statistician, November 1991, Vol. 45, No. 4 287) wrote a paper that spoke only about Morgan's paper. It was included in the very same issue of the journal. It's weird. Is it a disclaimer, a clarifier? It's sure not an endorsement.
Others come out with papers supporting Morgans criticisms, including Gillman in 1992 and Grinstead and Snell 2006.
Others continue publishing unconditional papers. (It seems likely that if 3 Wikipedia editors plus Seymann find fault with the paper, so too would members of the Professional Mathematics Community. And as professionals, they don't make a big stink about it. They just ignore the paper and continue publishing articles that rely solely on the unconditional problem statement.)
So, has the Professional Mathematics Community decided that Morgan is right, and Selvin was a hack? I don't think so. Before, during and after Morgan's paper, respected, credentialed reliable Mathematics professionals continued to publish articles solving the MHP unconditionally. I don't know that any of these professionals in either camp have attacked or counter-attacked anyone else's paper. It looks to me, that in the Professional Mathematics Community nothing happened. No usable information was gained. Perhaps Morgan's paper, like Monty revealing a goat is just sleight of hand, imparting no usable knowledge? It's possible. Most published MHP articles say nothing of Morgan or conditionality.
Which brings us, finally, to the Meta Paradox. The Wikipedia editors are arguing, essentially, over whether or not solving the unconditional problem is 'enough'.
Suppose you are given a story problem about a game show. The Professional Mathematics Community agrees heartily that this is a delightful paradox which can be 'proved' or 'solved' using an unconditional problem statement. Maybe not even requiring formal probability notation. Symbolic notation is often used. Then "J. P. Morgan, N. R. Chaganty, and M. J. Doviak are Associate Professors and R. C. Dahiya is Professor, all in the Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia" come forth and say it must be solved conditionally, based on the arguments set forth in their paper. You are then offered to stay with the unconditional solution being complete, or you may switch to the conditional solution.
Many people are fooled by this paradox, and accept the switch. Because they don't realize that like Monty revealing the goat, no new usable information has been revealed by this paper. Nothing happened. Glkanter (talk) 18:27, 8 December 2009 (UTC)
- I like this way of looking at things! Except that there is no law against studying conditional Monty Hall problems, and quiz-players should also realize what they get when they go with the unconditional solution. So I think that something useful did come of the Morgan et al. contribution. Now we just need a reputable mathematician to publish a peer reviewed paper on the Monty Hall paradox paradox, and then wikipedia editors can write articles on it. Gill110951 (talk) 05:36, 22 December 2009 (UTC)
- Glad you like it, thanks! It's a free country. They can do what they want. It just adds no value to those simply wanting to understand the MHP paradox. As the article is written, quite the opposite! There's an old salesperson's saying, 'Don't close past the sale' (it may actually be 'Don't sell past the order'). I may have had a professor say something like, 'Don't over-solve the test problems. Solve it and move on'.
- Maybe you could look at my 'Huckleberry' section and tell me how his approach was insufficient, and how his results would have been improved by someone explaining to him the 'equal goat door constraint'? Glkanter (talk) 05:48, 22 December 2009 (UTC)
OK, I'm Paraphrasing slightly...
When asked how he was able to sculpt the venerated 'David', Michelangelo replied, 'It was easy really. I removed everything that didn't look like David'.
A comment from a newcomer to this discussion
Wow, this thread is long! And I haven't even looked at the archive(s?). I thought it might be worthwhile to make a comment as a person who has not been following this thread before now. I just, in the last couple of days, read the article and a big portion of the discussion.
My comment is simple: Please, I am not attacking anyone; I am just making a general, honest, respectful IMO comment. (And yes, I am schooled in mathematics.) I agree with those that say the article is too long, very unwieldy, and often downright confusing. I think the article as it now stands is almost worthless. I agree with those who say: Just state the "standard" problem as most people assume it is stated, and give a simple explanation as to why it is correct. Then meander off into the conditional and unconditional ponderings, the Bayesian statistics, etc.
I started out reading the article, with expectation of fun. I was already familiar what the "Monty Hall problem", and I understood it, at least in its more obviously stated form (based on the generally accepted assumptions). As I read on I thought, "Whaaa???" Much of the -- sorry, but most -- of the article is a murky mess, and even those who are somewhat probabilistically astute I think would have difficulty making sense of some of it. I'll cite just one example: The section titled "Popular Solution" is, IMHO, poorly written and confusing. Frankly, it's not clear what the author is meaning to get across in several places (even though I understand exactly what it is that he/she is intending to say). It needs to be rewritten, as does much of the rest of the article. Not tweaked, but rewritten. This sort of muddled presentation is just not necessary, and it is not worthy of the standards of Wikipedia. This stuff is not string theory or Gödel's incompleteness theorems in ZFC. This is introductory-level probability, albeit a very subtly tricky example of it.
I never seen an article on Wikipedia that has created such a WikeWar as this article has. It apparently has no resolution in sight. Anyway, I'm all out of suggestions -- if I have even made any.
Finally, just for fun, I wanted to mention a somewhat similar conditional-probability problem which I haven't seen anyone else mention. (It is not relevant to this article, nor should it appear in it; it's just related.) You play a "flip three coins game". The person I am gambling with shakes up three fair coins in a canister and spills them onto the table top. I am not allowed to see the coins initially before I make my choice; the canister shaker (my opponent) hides the coins from me. The rules are, the shaker peeks at the coins on the table and he has to tell me what the "majority" coin is. There will be either a majority of heads (3 heads or 2 heads) or a majority of tails (3 tails or 2 tails). Then, having been told what the majority is, I must guess what the third coin is -- heads or tails. If I get it right I get paid a dollar by the shaker; if I get it wrong, I pay him two dollars. Most people would think this a stupid gamble on my part; they will assume that the guess as to the heads-tails of the third coin has a 50-50 chance of being right. But it's easy to see (though it is initially counter-intuitive to many people) that if you always guess the opposite of the majority, you will win 3/4 of the time. Just write down all combinations: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT, and it's obvious. That would make a cool bar game. Even offering your opponent the 2-to-1 payoff, you would still win, on average, in the long run 25 cents per play:
1/4*(-2) + 3/4*(1) = 1/4 of a dollar per coin-shake, in the long-run average.
Good luck with your little war. Worldrimroamer (talk) 23:53, 8 December 2009 (UTC)
- There's no war. Just good faith disagreement. Likely to end soon, anyways. Is there any reason this editor's comments should not be reflected in our quorum building? Glkanter (talk) 01:18, 9 December 2009 (UTC)
- I'm afraid I don't understand what you mean. Who is the "this editor" to whom you refer? I tried to state at the outset that I was not targeting my comments at any particular person. I did not compile a ledger of who wrote what. I was just making general comments, in the hopes that they might be a small but helpful contribution. —Preceding unsigned comment added by Worldrimroamer (talk • contribs) 03:10, 9 December 2009 (UTC)
- Er... the obvious guess is that the third coin is the same as the majority, which is a bloody good guess: it is correct 3/4 of the time, both conditionally on the majority being heads, and conditionally on the majority being tails, and unconditionally on the nature of the majority. I shall try this in the pub to see if my population is smarter than Worldrimroamer's. Gill110951 (talk) 05:45, 22 December 2009 (UTC)
Sorry for being so cryptic. 'The editor' is you. As we try to bring this discussion to a close, we are attempting to build a Wikipedia Consensus to make changes very much as you described. So, I was asking the other editors if there is any reason your opinion should not be considered as part of the 'let's change the article' consensus, of which I am a part. It's all good. I think you'll find the discussion of the last few weeks most meaningful. Some extra Wikipedia Mathematics Project people have begun contributing, by request, and it's helped move things forward a great deal. Again, sorry for being unclear. Glkanter (talk) 03:27, 9 December 2009 (UTC)
- Glkanter, thanks for your reply. I understand now what you meant. And by referring to "editing wars" I did not mean to be denigrating. Perhaps I should have used a different term. I just thought it was very interesting that this "simple" little topic has stirred up so much discussion. It does my heart good to see that there are people that care about esoterica like this (at least, in the eyes of the general public it would seem like esoterica). Best regards to all ... Worldrimroamer (talk) 17:48, 9 December 2009 (UTC)
- Worldrimroamer, thanks for you contribution. New thought is always welcome here. One thing I might explain is that the article as it is now (more or less) in not the end result of the pages of discussion that you have seen. It is little changed in principle from the original FA version.
- There are basically two broad factions of editors here. Those who want to keep the article as it is (or perhaps as it was about a year ago) and those who want to change the article to reflect more or less what you say, namely that The Monty Hall Problem is a simple mathematical puzzle that most people get wrong. What the article needs (according to the pro-change editors) is an initial section that concentrates on a simple description of the puzzle with normal 'puzzle assumptions' (all choices are random unless specified etc). This should be followed by some simple, convincing solutions that show why the player has a 2/3 chance of winning by switching.
- The anti-change editors believe that the proposed changes are not justified by the available sources and that making them would jeopardise the article's FA status. Of particular importance is a paper by Morgan et al. which claims that the problem must be treated as one of conditional probability. This has the effect of changing the simple problem that most people get wrong to a complicated problem that most people are bored by. I strongly dislike the Morgan paper, you should get a copy and see what you think.
- It the moment, apart from a few minor changes, the article is still built around the Morgan paper. The editors who want change have refrained from drastic editing as fought to get a consensus for change here. So far although there is a majority for change there is far from general agreement.
- There are some things that are generally agreed on such as the game rules (the host always offers the swap and always opens a unchosen door to reveal a goat) the fact that overall the player has a 2/3 chance of winning by switching, and that it is assumed that the player has not studied replays of old shows to gain statistical information. I might add that, although there are strong opinions on both sides discussion has generally remained civil and there has been no edit warring. Martin Hogbin (talk) 11:57, 9 December 2009 (UTC)
- Thanks, Martin. Yes, I understand what you mean about the basic disagreement. It sounds like a rather intractable situation to me. I hope you guys can work something out. As I told Glkanter in the post immediately above, I should perhaps not have used the term "edit wars". I just meant that it was impressive how much intense interest has been evidenced in this discussion. I think that's a good thing. I just wish that the article were not so ... opaque? IMO, Wikipedia should be accessible both to the experts in the field, as well as to the (curious and smart) not-so-expert people. There's room for both. I'll butt out now and wish you luck. You may need it. :o) Best regards. Worldrimroamer (talk) 17:48, 9 December 2009 (UTC)
__________________________________________________ —Preceding unsigned comment added by Worldrimroamer (talk • contribs) 17:51, 9 December 2009 (UTC)
- The links are at the top of this talk page - as of the last FARC the article looked like this. Since then the "Solution" section (that arguably took the POV that the "unconditional" solution does not address the question as asked) has been split into a "popular solution" and "probabilistic solution" in a more NPOV manner. Saying this is "relatively unchanged" understates the situation fairly dramatically. What this extended discussion is about is furthering this change, to make the article effectively take the POV that a "conditional solution" is an unnecessary nuisance - i.e. that the POV presented by the aforementioned Morgan et al. paper is invalid. Saying the article in its current form, or even as of the last FARC is "built around the Morgan paper" is (IMO) factually false.
- No article is ever finished and improvements are always welcome. This extended discussion is about whether additional changes are necessary to undo the POV some editors think was present in the version at the last FARC. -- Rick Block (talk) 15:35, 9 December 2009 (UTC)
Building Consenus - Mediation
At day's end, it will be 7 days since Rick requested comments from all the editors we could think of who had shown an interest in the MHP, plus a general request to the WikiProject Mathematics page. Three days ago, Rick requested mediation assistance. To date there have been no volunteers.
How and when do we keep moving forward toward a consensus? Glkanter (talk) 13:56, 9 December 2009 (UTC)
- If you look at the list of pending cases at Wikipedia:Mediation Cabal, the oldest was opened on Nov 20. I don't know if they treat the backlog as a strict FIFO queue, but it seems like 2-3 weeks might be a fairly reasonable amount of time to wait for a response. -- Rick Block (talk) 14:42, 9 December 2009 (UTC)
Rick, based on the following, I don't see this procedure as being any help whatsoever to this particular group of editors. Is there some other benefit to this that I do not understand? What other path can the clear consensus take toward gaining 'permission' from the minority view to move forward with editing the article?
- "The Mediation Cabal is a bunch of volunteers providing unofficial, informal mediation for disputes on Wikipedia. We do not impose sanctions or make judgments. We are just ordinary Wikipedians who help facilitate communication and help parties reach an agreement."
You made some claim a couple of days ago about this legitimately created good faith consensus violating NPOV, which I'm sure each member of the consensus would dispute strongly. Is this the issue you want mediated? The claim seems far fetched, certainly, to me. Or is this still an issue from your point of view? Glkanter (talk) 13:01, 10 December 2009 (UTC)
- I have two concerns that I believe mediation may help:
- 1. I am less than convinced that the users Martin has identified as "for change" agree with what you and Martin are thinking they agree with. For example, in the section below Colincbn says that the Solution section of this version "seems to have nothing in it about the conditional solution at all". I believe you and Martin disagree with this. So, is Colincbn for the change you're suggesting or not? To some extent, I think many of us are talking past each other and not necessarily understanding what others are saying.
- That point is rather more easily resolved. Why not ask those in the list if they are in the right section. I have asked people to sign to confirm that they are in the right section or to move themselves if they are not. You are free to ask any editor to check that they are in the right section if you think that I have got it wrong. Martin Hogbin (talk) 22:20, 10 December 2009 (UTC)
- Rick, re: 'talking past each other', I sure would appreciate some closure on the very first section I started when I returned: 'Is The Contestant Aware?' You're last response was 'yes, but', and I've asked you to clarify, as neither of us wants me mis-interpreting your intent. Thank you. Glkanter (talk) 04:45, 11 December 2009 (UTC)
- Martin - this is a perfect example. You seem to not be understanding my point, but I'm puzzled how to make it more clear. If I try to clarify I suspect you'll think I'm arguing with you about the "consensus". A mediator presumably wouldn't have this issue and might be able to convey the point I'm making in a way that you wouldn't take as an argument. Actually, I'm saying the same thing I've said in some other threads lately which is that without talking about specific changes it's very easy to miscommunicate. -- Rick Block (talk) 03:56, 11 December 2009 (UTC)
- 2. I personally have been trying to play two roles here, i.e. as a proponent for one "side" in this discussion (per my comment above, I'm not sure there are only two sides) and (since I am an administrator) as an authority on Wikipedia policies and procedures. You, in particular, seem to believe I am not acting in good faith and that everything I say reflects an advocacy of a POV.
- Well, it's evident now why these discussions take 6 years and never get anywhere. Rick, under who's auspices were you alone chosen to act "...as an authority on Wikipedia policies and procedures" for purposes of these discussions? Were the other editors advised of your dual role? I sure wasn't, and I am greatly distressed by this revelation. Is this common to have an entrenched protagonist also serve as some sort of 'junior mediator'? I can see all kinds of conflict from this, and have personally witnessed and been the recipient of this conflict of interest in your discussions for 14 months. Have you considered not continuing this dual role? More than ever, I'm certain we need to move beyond the mediator cabal level to declare the consensus in favor of the proposals. Glkanter (talk) 16:25, 12 December 2009 (UTC)
- Both of these are areas where I think an uninvolved mediator could help. -- Rick Block (talk) 17:26, 10 December 2009 (UTC)
- Please do not attribute motivations to me or Nijdam or anyone other than yourself. I am plenty willing to budge and have done so in the past. Whether you are willing to budge is up to you. Informal mediation is the next step in the dispute resolution process, see Wikipedia:Dispute resolution. Formal mediation comes next, but my understanding is informal mediation is generally treated as a prerequisite. -- Rick Block (talk) 17:58, 10 December 2009 (UTC)
- When I first read MHP and the explanations, in the first follow-up column in "Ask Marilyn" years ago, I believed she was right about the answer, but thought her explanations were ridiculous. The best apparent solution offered was to use a simulation--for example, with playing cards. I immediately grabbed a deck of cards and tried the simulation, and within a few minutes saw that it was obvious that the simulation would lead to a 1/3 stay, 2/3 switch ratio for winning over the long run. Obvious, obvious! That was obviously because, of the 2/3 cases in which the prize was not card #1 (i.e., behind door #1), it would be card #2 half the time, and card #3 half the time. So obvious!
- This explanation was very unsatisfactory because in the actual tricky puzzle, the actual door opened is actually identified. Door #3 is opened! What then? The simulation (like other explanations) did not specifically address this scenario, but instead included alternate scenarios in which a completely different door (#2) was opened. Sure, we can agree that if either of the non-chosen doors might be opened, the odds of switching are double the odds compared to staying--but what is true once one of the non-chosen doors is actually opened? That is what makes the puzzle an interesting, tricky puzzle! That's why people much smarter than I got fooled!
- Eventually I worked out a reasonable solution + explanation (c) and let it go. This Wikipedia article brought the irritation back. I contributed with some silly stuff, got bored, and left.
- I'm back in for a minute to argue:
- (a) even though Morgan et. al. are very wrong and beside the point overall, they ought not be ignored, because they made a nice (misguided, but nice) argument that the player can't be sure the odds of switching are 2-to-1, since for all we know, a host choosing between two losing doors might prefer one to the other. They rightly acknowledge that you should still switch. They don't add a lot to the discussion beyond that. Their contribution is mostly "too clever by half" and not useful. It is reasonable for the player to play with the assumption that the host is no more likely to open one losing door than another.
- (b) the usual explanations given for why you double your odds by switching are worded as if the actual puzzle said something like this:
- Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1. You know at least one of the other two doors has a goat behind it. The host, who knows what's behind the doors, opens one of the other two doors (No. 2 or No. 3),showing you a goat. He then says to you, "Do you want to switch to the closed door you didn't orignally pick?" Is it to your advantage to switch your choice?
- I don't enjoy the fact that this is treated as an argument about"conditional" versus "unconditional" thingies. Those words seem too complicated to me. Too, I believe Morgan et. al. are fundamentally wrong. But given all that, Rick Block is closer to reality on this topic than is the average person who fundamentally understands the problem. The usual explanations given are not explanations that fit the actual puzzle as it was actually worded! That matters! Simple31415 (talk) 03:24, 13 December 2009 (UTC)
- I agree with some of your ideas, disagree with some. Mainly whether "The usual explanations given are not explanations that fit the actual puzzle as it was actually worded! That matters!" As an analyst (problem solver), I'm always looking for shortcuts. And that strictly means find 'differences that make no difference'. I think the open door fits that description. Please look at the section I recently created, Let's Say Some Huckleberry Played Repeatedly and tell me where the contestant did not properly maximize his situation using a simple solution. Glkanter (talk) 05:17, 13 December 2009 (UTC)
- Fascinating! Is there any particular reason why you returned now to say, 'Rick Block is closer to reality on this topic than is the average person who fundamentally understands the problem? Perhaps you could tell us which aspect of his argument you support, particularly in view of your comment that Morgan's contribution is mostly, "too clever by half", which I agree with totally although I might add, 'and, in fact wrong'.
- Also, there is no "the question", although it is true that the most often quoted problem statement is that by Whitaker. Perhaps you should read it, it says, '...the host, who knows what's behind the doors, opens another door, say No. 3...'. Note the words that I have put in italics. It is quite obvious that it is not intention of the questioner to ask about the probability given that the host opens a specific door but to ask about the probability when he opens one of the other doors to reveal a goat. The door numbers quoted throughout the question are just examples. Nobody can imagine that Whitaker wanted to know about only the specific case where the player chooses door 1 and the host opens door 3!
- In fact the MHP is, as we say in the article, a probability puzzle. Its main interest is that it is a simple problem that nearly everyone gets wrong. We should therefore treat the problem and solution simply without even mentioning the terms conditional/unconditional, at least for the first section of the article. Martin Hogbin (talk) 10:26, 13 December 2009 (UTC)
I don't think the informal mediation is going to be of any value. It requires a volunteer mediator, of which none have come forward yet. It could takes weeks before one comes forward, it could be never. Whatever the mediator comes up with is non-binding. It has no teeth. In the meantime, all sorts of edits are being made to the article without any discussions whatsoever.
I suggest we request Mediation immediately, recognize the consensus for the proposed changes, and stop being hostages to this 6 year long filibuster. Glkanter (talk) 14:40, 14 December 2009 (UTC)
- I just read about Informal Mediation. If I read it correctly, any Wikipedia editor can offer to mediate. No prior approval of the volunteer mediators occurs. I just don't see where continuing to wait, perhaps forever, for this is of any benefit. Glkanter (talk) 14:46, 15 December 2009 (UTC)
Most recent FARC version
Can somebody please explain specifically what they don't like about the version of the article as of the last FARC, i.e. this version? I think the article is actually much worse now than it was then and it might be easier to proceed from this version than the current one. Just a thought. -- Rick Block (talk) 20:40, 9 December 2009 (UTC)
- I do not like any of the 'Solution' section including the diagram and the mention of conditional probability. We need a simple solution, followed by some convincing explanations. Martin Hogbin (talk) 21:45, 9 December 2009 (UTC)
- Up to "The reasoning above" does not mention conditional probability, and says something very similar to what is in the initial paragraph of the current version (doesn't it?). Do you like the large diagram in the current version? It seems isomorphic to me (it varies the player pick rather than the car location - but both are based on a specific concrete example). Are you thinking there's something wrong with the image in the FARC version? -- Rick Block (talk) 23:06, 9 December 2009 (UTC)
Rick Block states above: "I think the article is actually much worse now than it was then..." This was how the Solution section began as of the last FARC:
- Solution
- The overall probability of winning by switching is determined by the location of the car.
until it was deleted with this diff:
http://en.wikipedia.org/w/index.php?title=Monty_Hall_problem&diff=next&oldid=247025116
So, I think the newer version is a whole lot better.
But this new section is just another filibustering technique by Rick. Glkanter (talk) 12:29, 10 December 2009 (UTC)
- The excerpted quote above is a perfect example of the muddlement of this article. I refer to the quote: "The overall probability of winning by switching is determined by the location of the car." Huhh??? What does this mean? Again, let me emphasize, I have no idea who it is that was being quoted, which quote I just repeated. I don't mean to attack anyone in particular. But I have to ask ... First, what is the difference between the "probability" and the "overall probability"? Second, what does the probability have to do with the location of the car? I don't get it. The location of the car has nothing to do with the probability. If the writer has a point to be made, then he/she should try to state the point non-cryptically. Sorry for this rant, but this kind of thing is very frustrating ... End of rant. Shalom. Worldrimroamer (talk) 04:26, 10 December 2009 (UTC)
OK. So far, we have "delete the first sentence". Fine. Let's assume this change, which makes the Solution section look more like this version. And Martin's issue with the image (which he hasn't clarified) and his issue with anything that says "conditional probability". The referenced version has one sentence that says "conditional probability" in the 2nd paragraph below the figure basically identifying what the "subtly different" question is actually called. Possibly we could delete this sentence, although I'm not sure I see the point. Anybody else have specific gripes with this version? -- Rick Block (talk) 05:25, 10 December 2009 (UTC)
- Rick, I started discussing this in October, 2008. I have no interest whatsoever in comparing and contrasting to a May, 2008 version. I think it is unreasonable to request that we do so. It has the effect of negating discussions that led to all the edits to the article since the FARC version of May, 2008. Is that your intent? Glkanter (talk) 12:29, 10 December 2009 (UTC)
- Well, in-spite of my preference for the popular solution to be the main focus of the article I still feel that the conditional solution should be mentioned in the Variants section. The FARC version linked above seems to have nothing in it about the conditional solution at all. Also I like the current lead much better. Personally I think that simply moving the conditional section to the Variants heading would be the best compromise as that way no information will be lost from the article but the flow will still proceed from the most common understanding of the MHP towards more in-depth analysis. I would be more than willing to make the change myself as an editor who has only been part of this discussion for a short time. How does that sound to you guys? Colincbn (talk) 07:28, 10 December 2009 (UTC)
- Colincbn - You apparently mean something different by the "conditional solution" than at least Glkanter - which is why I would really prefer we talk about specific changes. I gather you are referring NOT to the entire "Probabilistic solution" section, but only the paragraph in that section starting with "Morgan et al. (1991) and Gillman (1992) both show a more general solution ..." and not the figure that follows. Is this right?
- Sorry about that, I'll try to be more specific but I might get my terminology wrong as I'm no mathematician (I'm a history, religion and biology guy). Basically I figure the conditional problem statements referenced to Morgan would be best explained in a single subsection under the variants section. ie: take the first paragraph of the Probabilistic solution section, that starts with "Morgan et al. (1991) state that many popular solutions are incomplete...", as well as the third paragraph that starts with "Morgan et al. (1991) and Gillman (1992) both show a more general solution ..." (as you said), and put them together in the new Conditional problem subsection. This might require some rewording of course. I might also suggest moving the probability tree image (File:Monty tree door1.svg) to the new subsection along with giving it any other useful visual aids or expanded explanations that are available. If the section gets too big we can add a "main article" link at the top of that subsection and continue expanding in a separate article that deals with the conditional problem extensively. I figure that way we get to keep all the information currently here (which I am in favor of) as well as restructuring the layout to flow from the most common interpretation to more advanced and in-depth analysis including variations and more focus on the Parade version of the question (which I am also in favor of). I think this is a fairly good compromise as no one can claim undue weight or anything like that, but we also get to cover the most prominent variants to the popular solution. Colincbn (talk) 15:52, 10 December 2009 (UTC)
- Colincbn - You apparently mean something different by the "conditional solution" than at least Glkanter - which is why I would really prefer we talk about specific changes. I gather you are referring NOT to the entire "Probabilistic solution" section, but only the paragraph in that section starting with "Morgan et al. (1991) and Gillman (1992) both show a more general solution ..." and not the figure that follows. Is this right?
- Colincbc, I think your suggestions have a lot of merit, at the appropriate time, as I posted just a few minutes ago. I would like to say however, I don't agree that the additional sections beyond the current 'Popular Solution' section are a "flow from the most common interpretation to more advanced and in-depth analysis". I consider those sections as unsupported criticisms and confusing different problems that might be of interest to some people. And they're not part of the Monty Hall problem paradox itself, but the published literature that only uses the MHP as a starting point. Glkanter (talk) 16:06, 10 December 2009 (UTC)
- While I can certainly see your point, my main concern is finding a compromise. I figure there are some things that will always be disagreed upon, and both sides will need to accept some things they don't like, but its the only way to really progress. The way I see it if neither side can have an article they think is 100% perfect both sides getting one they feel is 80% perfect is the best way forward. (Now we just need to figure out where that 80% is...) Colincbn (talk) 16:33, 10 December 2009 (UTC)
- Yes, a compromise is appropriate. Based on these discussions, I modified my proposal a couple of days ago. http://en.wikipedia.org/wiki/Talk:Monty_Hall_problem#Glkanter.27s_suggestion Glkanter (talk) 16:41, 10 December 2009 (UTC)
- Glkanter - I think comparing to a version that passed a Featured Article review is entirely appropriate. I'm talking about considering replacing just the "Popular solution" and "Probabilistic solution" (not the entire article) from the current version with the "Solution" section from this older version. I think it's clear not many people like the structure of the two current sections under "Solution" (even me). Choosing a different starting point might make progress easier. -- Rick Block (talk) 15:06, 10 December 2009 (UTC)
Colincbn - can you comment directly on the idea of using this version of the Solution section? I think it may well provide a reasonable compromise. -- Rick Block (talk) 18:04, 10 December 2009 (UTC)
- Correction: "good starting point for a reasonable compromise". -- Rick Block (talk) 18:20, 10 December 2009 (UTC)
- As a starting point definitely. I do like the Cecil Adams explanation and the two smaller images on either side of it in the current Popular solution section as a good "layman's" explanation though. And I imagine that we will need to move the paragraph that starts "A subtly different question is which strategy is...". Possibly replacing that paragraph with something that reiterates that the solution presented is only for the unconditional version and that the Parade version has spawned variants with a "see below" link. Colincbn (talk) 23:37, 10 December 2009 (UTC)
Who decides?
Consider 3 so-called 'variants':
- Instead of goats, behind 2 doors are cows.
- Instead of 1 car & 2 goats, it's 2 cars and 1 goat.
- The host may have a 'bias', or 'behaviour' or 'method' to determine which goat to reveal when the contestant selects the car. This may or may not be somehow made aware to the contestant.
Who decides when a 'variant' is no longer the Monty Hall problem? Glkanter (talk) 12:39, 10 December 2009 (UTC)
- Reliable sources decide. -- Rick Block (talk) 18:05, 10 December 2009 (UTC)
- In mathematics two problems are considered the same if they are isomorphic (see isomorphism). For example, the MHP and the Three Prisoners problem are isomorphic so even though they have different names and somewhat different descriptions they are actually the same problem. If you're asking for opinions you're on the wrong page (see /Arguments).
- The isomorphism with the Three Prisoners problem is an interesting point that has already been raised. The Morgan style argument is never heard concerning the TPP. It is never suggested that the warden may have had a preference for one of the prisoners. This is if course because Gardner specified that he chose randomly if he had a choice. He obviously did this specifically to avoid the Morgan style argument and to make the problem simple. That is exactly what we should do with the MHP, especially as the Morgan scenario is now shown to be just a contrafactual conjecture. Martin Hogbin (talk) 10:14, 11 December 2009 (UTC)
- No, what we should do is adhere to FUNDAMENTAL Wikipedia policy and neutrally represent what reliable sources say. To be clear, the specific point I"m talking about is whether a conditional probability analysis (using the assumption of equal probability of host choice between two goats) needs to be presented - more or less like it was in this version of the Solution section - not the generalization where the host preference is left as a variable. -- Rick Block (talk) 17:47, 11 December 2009 (UTC)
- That is absolutely fine and it is exactly what I want to do. We have a number of sources who treat the problem in a simple manner (Selvin, vos Savant, Devlin etc) and we have a number of sources that treat it in a more complex manner (Morgan etc). It is therefore quite reasonable, and advantageous for most readers, to start with a section that treats the problem and its solution and 'Sources of confusion' and 'Aids to understanding' in a simple manner. We can then treat the problem in a more complex manner for those interested in such things. Note that a simple treatment of a subject followed by a more detailed one cannot be described as a content fork, it is standard practice. The problem is that, when this has been suggested, you tell us that Morgan say that the sources that treat the problem simply are wrong or inapplicable, that they present 'false solutions'. This is an obviously pro-Morgan POV that should not be permitted.
- No, what we should do is adhere to FUNDAMENTAL Wikipedia policy and neutrally represent what reliable sources say. To be clear, the specific point I"m talking about is whether a conditional probability analysis (using the assumption of equal probability of host choice between two goats) needs to be presented - more or less like it was in this version of the Solution section - not the generalization where the host preference is left as a variable. -- Rick Block (talk) 17:47, 11 December 2009 (UTC)
- The fact is that we do have reliable sources that do treat the problem simply (do not distinguish between goat doors) and there is no reason not to represent these sources neutrally (not as false or incomplete or flawed) in the article. Martin Hogbin (talk) 18:33, 11 December 2009 (UTC)
- I do not like the diagram you refer to because it shows the choice of two goats as separate pictures, horribly complicating the problem for a beginner. We desperately need to keep it simple; the problem is plenty complicated enough for most people. If we do not do this we fail in our fundamental purpose of informing our readers.Martin Hogbin (talk) 18:33, 11 December 2009 (UTC)
- And I'm being somewhat non-responsive since I think one of the problems we've had on this page is too much focus on opinions and not enough focus on what reliable sources say. The three fundamental Wikipedia policies concerning content are WP:V, WP:OR, and WP:NPOV. If anyone hasn't read any of these they really should (how about right now?). What WP:V means is that everything an article says has to be sourced to a WP:reliable source. What WP:OR means is basically the converse of WP:V, i.e. you're not allowed to add content based on your own opinion. In the extreme, this is the case even if you know with absolute certainty that what you're saying is true but you can't find a reliable source to back you up. WP:NPOV means basically two things. First, that Wikipedia MUST fairly (neutrally) represent what reliable sources have to say. Second, if reliable sources disagree that Wikipedia articles are not allowed to take sides. In the aggregate, these policies mean that even though every editor has opinions and even though a collection of editors might have a collective opinion about content issues Wikipedia doesn't give a damn about these opinions. -- Rick Block (talk) 03:29, 11 December 2009 (UTC)
Repeated text
Why does the entire Krauss and Wang text appear twice in the first bit of the article? Isn't the article long enough without this repetition? RomaC (talk) 14:48, 10 December 2009 (UTC)
- Please see WP:Lead section. The lead is meant to be a concise, standalone overview of the entire article. The K&W problem definition was added to make the problem description unambiguous (even in the lead). I would be fine with deleting it and presenting only the Parade description. I predict others will object to this. -- Rick Block (talk) 15:44, 10 December 2009 (UTC)
- Actually I kind of like the FARC lead better, it clearly states that the problem as stated in Parade is ambiguous. The K&W problem is in the beginning of the Problem section as an example of an unambiguous way of stating the problem anyway so I don't think it is entirely necessary in the lead. But I would rather work on consensus of the other suggested changes above before diving into those waters... Colincbn (talk) 16:10, 10 December 2009 (UTC)
- Unlike the Parade statement of the problem, Selvin's statement is not very well known. It's also considerably longer. Because Jstor makes the first page of any reference available as a preview, it can be viewed online here. -- Rick Block (talk) 17:39, 10 December 2009 (UTC)
- I like the solution though. Selvin's answer is published in a reliable source and I would like to see it in the article (but in the form of a pretty diagram). It treats the opening of either of the two unchosen boxes by the host as equivalent. Martin Hogbin (talk) 22:28, 10 December 2009 (UTC)
- Unlike the Parade statement of the problem, Selvin's statement is not very well known. It's also considerably longer. Because Jstor makes the first page of any reference available as a preview, it can be viewed online here. -- Rick Block (talk) 17:39, 10 December 2009 (UTC)
- Isn't it funny how we've argued so much about vos Savant's failings and intents with the problem statement, and we never, since I've been around, discussed Selvin, et al's. I'm going to go read that page right now! I guess collaboration really can work. Who knew? Rick, thank you very much for the link! Glkanter (talk) 22:37, 10 December 2009 (UTC)
- It says it all - it gives a simple solution (showing that the contestant has a 1/3 chance of winning if they stick and a 2/3 chance if they swap) and notes that the contestant's chance of having the car in their original box is unchanged at 1/3 after Monty has opened a door. Not only that but it was published in the same peer-reviewed journal as Morgan but it is not followed by a highly critical comment. Looks like a winner to me. Martin Hogbin (talk) 23:43, 10 December 2009 (UTC)
- Isn't it funny how we've argued so much about vos Savant's failings and intents with the problem statement, and we never, since I've been around, discussed Selvin, et al's. I'm going to go read that page right now! I guess collaboration really can work. Who knew? Rick, thank you very much for the link! Glkanter (talk) 22:37, 10 December 2009 (UTC)
- From a sourcing perspective, it is important to note that this is a letter to the editor, not an article. Articles from peer reviewed journals are usually the most reliable sources. Letters to the editor are more like primary sources. -- Rick Block (talk) 17:18, 11 December 2009 (UTC)
- You might also be interested in Selvin's second letter in vol 29 #3. I can't find a pdf of the journal page online, but there's what appears to be a faithful copy here. Although I haven't compared it to the copy I have of the printed journal page, it's clearly missing a "/" on the 4th line of the conditional probability expansion. BTW - references to both of these have been in the article for several years. -- Rick Block (talk) 05:35, 11 December 2009 (UTC)
- Thanks for that. It does not, in my opinion, discount the solution given the his original letter. The interesting point, when you look at that history of the problem, is that Monty actually never offered the swap and, if Selvin's account is accurate, it was the contestant who suggested it. Martin Hogbin (talk) 09:56, 11 December 2009 (UTC)
- You might also be interested in Selvin's second letter in vol 29 #3. I can't find a pdf of the journal page online, but there's what appears to be a faithful copy here. Although I haven't compared it to the copy I have of the printed journal page, it's clearly missing a "/" on the 4th line of the conditional probability expansion. BTW - references to both of these have been in the article for several years. -- Rick Block (talk) 05:35, 11 December 2009 (UTC)
- The other point which becomes apparent is that Monty, who was the only person who knew for sure his door opening policy, has clearly stated that he knew the contestants chance of holding the car remained at 1/3 after he had opened his door. That seems to me to rule out Morgan's conjecture that he might have had a preference for one door or the other. Martin Hogbin (talk) 09:56, 11 December 2009 (UTC)
Thanks, Rick, for finding a solution that resolves my concern (an overly long intro) Butwhatdoiknow (talk) 00:02, 21 December 2009 (UTC)
There's A Difference Between A Logical Argument and OR
The first section I created after my return is called 'Is the Contestant Aware'. http://en.wikipedia.org/wiki/Talk:Monty_Hall_problem#Is_The_Contestant_Aware.3F
All I ask is this:
- "Has it been agreed by the editors of this article that regardless of how Monty handles the 'two goats remaining' situation, the contestant has no knowledge of the method?"
It's a pretty simple, straightforward, and incredibly relevant question.
But neither Rick or Nijdam have given me a straight 'yes' or 'no'.
So, we're asking for mediation, but we haven't even stated our underlying reasons for disagreement. Because without their responses, any logical argument I might make to a mediator can be refuted, for most any reason.
And, as the section name indicates, using your noggin is not the same as OR. But that seems to be the reason for not replying.
So, enough of the 'well, I didn't say that per se', and let's get to the bottom of this. No mediator can help us if we're not being forthright. Glkanter (talk) 05:11, 11 December 2009 (UTC)
- Despite your claim to the contrary, this has already been answered. See, for example #Is The Contestant Aware?, above. And again (for what is it the 4th time?) this is a question for the /Arguments page, not here. Whether the editors of this article agree or don't agree about this should have no bearing whatsoever on what the content of the article should be. In fact, from a Wikipedia policy perspective an unreferenced logical argument is precisely WP:OR. The content of the article must be based only on what reliable sources say. Would it help if a mediator said this rather than me? -- Rick Block (talk) 06:25, 11 December 2009 (UTC)
- We disagree. And your choosing to not respond to my very relevant talk page questions is seen, by me, as dodging the truth. You interpret what all these Wikipedia policies & guidelines mean one way, sometimes I'll interpret them another. So, no, it will take the 'formalist' of procedures to convince me that 'endorsing wrongness' is a Wikipedia policy. The source material has conflicts. Editors resolve conflicts of content all the time in the real world. I'm just saying sometimes these conflicts are resolved using logic, often known as common sense. OK. Don't answer. Let's just move the process forward, and the mediation cabal has nothing to offer me. Can we move on to the next level? Glkanter (talk) 08:54, 11 December 2009 (UTC)
- Rick, despite your claims to the contrary, you have never answered the question. You have argued for why it shouldn't be answered based on Wikipedia policy, and why you think (and this is your POV) it doesn't matter, but you have not answered the question. Part of that is that it isn't quite the right question, but you have been evasive about why that is so. That is also POV.
- No reference uses non-standard, specific knowledge of the game protocol (whether a door is always revealed, and if it always reveals a goat), or a specific asymetric probability for the two uncertainties (car placement and host choice between two goats) to answer the question posed by the MHP. This cannot be denied. (Rick - feel free to try: just point out such a reference. If you don't, it validates my statement.) The so-called "conditional solution" includes terms to represent those concepts, but whenever they use that result to answer the question, they make at least three assumptions: A door is always opened, a goat is always revealed, and the car placement is treated as a uniform choice. For good reason - the question cannot be answered "yes" or "no" without those three assumptions. A brain teaser that asks a "yes/no" question has to have a "yes/no" answer. The fourth assumption - that the choice of the door to open is treated as a uniform choice - is made in some references, but not others. But those others don't use any knowledge of that probability to answer the question, either. They say it is irrelevant to the question. Either way, no knowledge that is not explicitly in the possession of the contestant is used. This is the point you need to agree to, or not: that every reference eliminates any reliance on knowledge that is neither specifically stated nor clearly assumable, before answering the question "should she switch?"
- Once you admit to that, and I don't see how you can deny it, we can see that the so-called "conditional solution" is only a tool that is used to generalize the MHP; it is not the MHP problem itself. Because the aspects unique to it are always removed before answering the MHP. Yes, this seems to contradict some sources (but not all, or even the majority - just those that try for a rigorous approach, and make a debatable interpretation about the meaning of door numbers). My point - and I hope Glkanter's, although he isn't phrasing it the same way - is that it only seems that it contradicts these sources. And that assuming there is a contradiction is POV because the sources do not (again, just find me one that uses it to answer the MHP, and I'll back off of this) use specific knowledge not available to the contestant. They either use the contestant's (assumed) knowledge, or say it is irrelevant.
- So it is not OR, or inconsistent with any source, to say the problem has to be addressed in the SoK of the player only. All of the sources do the same thing, after they have created that more general tool. They just don't come out and phrase explicitly that "We are now putting the solution into the SoK of the contestant because that is the only useful SoK." They just do it. All of them. The false impression that specific doors are meant never factors into any source's answer of the MHP itself. It never factors into any explanation about why the common intuition, that the remaining doors should be equivalent, is incorrect. So the conditional solution does not aid our readers in understanding why switching is the correct strategy, and it is not necessary to use it to explain the result. It does add extra depth to the various possibilities, but only after the reader has understood that revealing a door places conditions on the remaining door but not on the chosen door. It is not the MHP. It will not be diminished, or ignored, by moving it to after the discussion of alternate strategies. But it does get in the way of understanding, on an intuitive level, why switching works. JeffJor (talk) 18:02, 11 December 2009 (UTC)
- Nice job, JeffJor. But not quite perfect from my POV.
- Morgan uses some hazy allusion to contestant awareness to discredit the reliably sourced published 'simple' solutions. This is the nucleus of all of Rick's arguments, that Morgan says the 'simple solutions' are inadequate. No published source directly refutes Morgan, so, ipso facto, Morgan is the Uber Reliable Source. Which in Rick's world is a neutral POV. So, Morgan is relying on the contestant having some aura of knowledge. As someone else wrote on this problem, elsewhere in Wikipedia, 'I choose not to discuss a problem where the host and contestant are mind reading or are in collusion'.
- I get tired of typing it, and you guys have to be tired of reading it, but any 'host behaviour' is the opposite of the problem statement, 'Suppose you're on a game show...'. Now, you're on a street with a hustler and a card table, some shells and a pea. It is meaningless vis-a-vis the MHP paradox.
- There's little point in repeating and continuing the arguments. Both Rick and Nijdam have not given a straight answer to 'Is the Contestant Aware'. Which by the way, Mr. JeffJor, is phrased perfectly fine for my needs, thank you. I think I've asked courteously and tried to explain why it's important. When we get in front of someone to argue for the consensus, I am prepared to explain that this refusal to answer forthrightly (remember the 'meow, meow' answer to this very question?) is consistent with the filibustering, personal attacks on my grasp of the subject matter (Even got one of those from Rick just today. Everyone knows this has been decided in our favor for weeks, and he still says I don't know what I'm talking about. What does that tell you about his grasp of the subject matter, or perhaps his intellectual honesty?) and other passive-aggressive discussion techniques I've experienced personally for 14 months, and countless editors have put up with for 6 years or so. Glkanter (talk) 23:55, 11 December 2009 (UTC)
- Excuse me, but what question are you thinking I haven't answered? I answered Glkanter's questions about a week ago (with this edit), in the section I linked to above.
- If what I wrote before is not clear, what I think is that the contestant knows what is given in the problem statement, no more and no less. The specific problem statement varies, so what the contestant knows also varies. If, for example, we're given the K&W statement of the problem but without the host's protocol for choosing between two goats (I believe this is precisely the problem Morgan et al. call the "vos Savant scenario"), the contestant does not know the host's choice is uniform. This means a specific player who has initially chosen door 1 and has seen the host open door 3 does not have enough information to know her precise probability of winning by switching. It is something between 1/2 and 1, and it is unconditionally 2/3. I agree there is a sense in which this means the probability is 2/3, but I think this is actually the answer to a different question than what nearly everyone interprets the MHP to actually be asking and that this is the precise reason the MHP is a paradox. I also agree that most people who ask the question intend the answer to the "conditional" question (the one that most people think the question is asking) also be 2/3. IMO, this means the host's preference for choosing between two goats should be specified as part of the problem statement.
- However, in addition to the above I also believe that for editing purposes Wikipedia doesn't give a damn what I think and insisting that this question is important or that anyone answer such a question demonstrates a profound misunderstanding of Wikipedia's fundamental content policies.
- JeffJor - One reference is Morgan et al. Another one that uses exactly the same approach and makes exactly the same points is Gillman. Another one that distinguishes the "unconditional" and "conditional" questions is Grinstead and Snell. I definitely do not claim that the host preference must be treated as a variable (are you thinking I'm insisting on this?), but that the "Solution" section is really not complete without an analysis using conditional probability and that the distinction between conditional and unconditional probabilities is a central issue at the heart of the MHP. Martin has argued for a long time that a conditional probability analysis would be inaccessible to most readers - essentially that we need to "dumb down" the article. Per Wikipedia:Make technical articles accessible, we should start with an unconditional explanation, proceed with a conditional explanation, with a picture. Hmmm. This sounds exactly like the Solution section in this version. You have seemingly ignored the questions I asked you above at the end of #What "the conditional problem" and "the unconditional problem" mean. I am interested in your response. -- Rick Block (talk) 04:53, 12 December 2009 (UTC)
- Rick: Saying "Yes we are limited to the player's knowledge BUT the player can wonder about knowledge she doesn't have as though it were useful" is a non-answer. It is straddling the fence on exactly the point you refuse to acknowledge, by answering it both ways. If, for example, Morgan's formula had turned out to be 3/(4+3q) instead of 1/(1+q)? Such "wondering" would not be useful. Switching would be helpful with only some possible qs, but the question as it was asked would still have a definite answer. By assuming q=1/2 and P(C1)=P(C2)=P(C3)=1/3 because the player cannot assume anything else for them.
- Morgan, et al, say "In general, we cannot answer the question ... unless we know the host's strategy." So they don't answer the MHP question based on their (incomplete - ignoring P(C1)) formula. The only answer to the MHP question that they give is based on what they call "the vos Savant scenario" and the fact that q doesn't matter. Gillman wrote the assumtion of symmetry for car placement into their problem statement - at least they didn't ignore it like Morgan - but also only provide an answer to the MHP question based on standard game protocol and ignoring q. Grinstead and Snell don't address the MHP's question when they present the so-called conditional solution. So you have not provided the referencecs I asked for. Once again, there is no source that uses the unique properties of the so-called conditional solution to answer "Should the contestant switch?" Those properties only become important if non-standard game rules apply, or if assymetric probabilities are known by the contestant to exist.
- The article version you linked is still treating the possibility, that the doors could be treated differently, as something the player could use. It says "A subtly different question is which strategy is best for an individual player after being shown a particular open door." That answer is also "switch, based on an assumed q=1/2 and P(C1)=P(C2)=P(C3)=1/3" in every source that addresses it. Because they don't say that there are other values of those parameters that can be used. Any mention of it belongs after mentioning variants, and providing the "bayesian" solution. Because they don't address, in any way, the issues that make the MHP controversial. They only address ways that variations of the MHP, from what was intended by vos Savant and Selvin, can become more intersting as a mathematic (as opposed to logic) problem. JeffJor (talk) 20:46, 12 December 2009 (UTC)
- Jeff - Are you saying you consider any analysis using conditional probability to be a variant? Again, for about the 4th time in the last few days, I'm NOT saying we need to include the host preference (q) generalization in the initial analysis. I'm happy moving that to a variant section (I don't think there's anyone arguing against this, so maybe I'll do that right now). What I'm not happy with is introducing a POV which favors an unconditional solution by omitting any mention of conditional probability. A conditional solution assuming uniform car placement and random host selection between two goats belongs in the initial Solution section. Not doing this is what I'm saying would be counter to NPOV. IMO, the figure in the existing 'Probabilistic solution" section showing the symmetry belongs in the initial solution section. I think there should also be an explanation that there is a difference between the "conditional" question and the "unconditional" question very early, although perhaps not necessarily in the initial Solution section. -- Rick Block (talk) 00:31, 13 December 2009 (UTC)
- Rick - I never said that using conditional probability was a variant. I clearly said that using different probabilities based on door numbers is the variant. The problem can be solved with conditional probability, but by the tree G&S use as their figure 4.4. Morgan's criticism of the solutions from the Parade affair are mostly wrong, since they apply to the variant where different probabilities can be considered. Solutions that do not use conditional probability ARE NOT WRONG, as Morgan says; they are only wrong FOR MORGAN'S VARIANT WHICH IS NOT THE MHP. And those solutions Moragn dismissed, which are correct, are easier for lay people to understand. There is no need for G&S's conditional solution early, but it can be included. I don't think it helps anybody who needs to rely on the initial section. But there is no place for Morgan's formulation there.
- About my recent edits: It is you who is not sticking to what the sources actually said. MvS did not say that "letting the host choose a door with the car" was the only strategy that was not a part of her problem. She said it was the most significant, of of all the conditions she assumed were defined by her answers. That also incldues anything that would prevent her 2/3 answer from being correct. Every single one of her answers makes the assumptions that render Morgan's formula useless. You have to realize that she needs to compact her column into very few words, and so is not addressing the problem rigorously. And shouldn't be expected to. The only mention of anything having to do with assymetric probabilities CAN ONLY APPEAR in the "variants" section.
- Morgan, at al, do not, EVER, claim that their formula answers the question in the MHP. In fact, they say it doesn't. With emphasis added: "In general we cannot answer the question 'What is the probability of winning if I switch...' unless we either know the host's strategy of are Bayesians with a specified prior." This is why the question you keep refusing to give a direct answer to is important. It says we don't "know the host's strategy" and are not "are Bayesians with a specified prior." The direct answer you shodl give means that, by their own admission, MORGAN DOES NOT APPLY. Anything they conclude applies only to their variant, and only then if one of these conditions they describe holds. It does not help to "wonder" what the possibilities might be, except in the case where every answer to that wondering says "switch." WHICH WAS MORGAN'S THESIS. JeffJor (talk) 18:08, 14 December 2009 (UTC)
- Glkanter: I've given up on getting Rick to stop using Morgan. It is a reference, and it claims to address the MHP (it doesn't, it addresses what they changed the MHP into), so he will forever stand behind his "Wikipedia Policy" arguments to say it must be included. (I can find an internet reference that claims the answer is 4/9 and has not been discredited, mainly becasue it can't be understood, but we won't quote it because we know what is wrong with it. Why we can't do the same with Morgan, since they misquoted the problem into something which is documented to be not the intent, I can't fathom.) The fact is, it is an interesting treatment, but of a variation. And the only reason I said your question wasn't well-asked, was because it allowed Rick to give his non-answer. We need to include the fact that it is only useful to "wonder" about other knowledge if the contestant can actually get it. And just one last comment for Rick: although you will say this is OR that contradicts Gillman, it isn't. It is a fact that Gillman glosses over because of the altered nature of his MHP. The q=1/2 approach is not just the equivalent to "announcing the switch strategy before a door is opened," it is equivalent to "announcing the switch strategy without knowledge of how the host chooses a door." Under Gillman's modificaiton, "before" is the time that is not known, at least until he assumes q=1/2 so that he can answer the question. JeffJor (talk) 21:06, 12 December 2009 (UTC)
JeffJor, oh, yeah, you've made the right decision. We have clearly demonstrated a consensus. Rick is going to use every method he can devise to extend his filibuster. There have been a lot of folks prior to us who ultimately made the same decision, to quit arguing with Rick. But we have proven the argument unlike anyone before us. "Suppose you're on a game show..." End of discussion on who's POV for the doors, and no more host behaviour. They used to argue 'little green men from space's' POV as being the MHP. I kid you not. And, no more host behaviour means no more Morgan. Of course, kmhkmh is still arguing the definition of a game show, but won't answer the 'Is The Contestant Aware?' question.
As far as the details, Morgan and his ilk get mentioned, they're published. But no more bad mouthing the Devlin solutions. Did you read my Huckleberry section? Please re-read my modified proposal. So, we just have to navigate Wikipedia's consensus processes. Rick is the king of that crap, so we'll learn as we go up the chain, whatever it is. Do you have any experience with that? So, a few of us will continue working together the straight path to improving the article. Glkanter (talk) 22:53, 12 December 2009 (UTC)
Let's Say Some Huckleberry Played Repeatedly
Let's say some huckleberry played repeatedly. They play for $1 per play, rather than the car.
Huckleberry has figured things out using the Combining Doors solution. He doesn't understand, in fact, he is not even aware of the term 'conditional' as it applies in this instance, as he is less educated than others.
So, with just the brains and common sense he was born with, he wins 2/3 of the time. His method worked fine. Perfectly, really. Nobody could give Huckleberry ANY ADDITIONAL INFORMATION OF ANY CALIBER WHICH CAN PRODUCE A BETTER RESULT. And if this were repeated endlessly, it would always work out to 2/3.
There are other solution techniques which may produce the SAME result, but Huckleberry doesn't understand that the 'equal goat door constrain must = 1/2'.
It was an elegantly simple Paradox for Selvin, and vos Savant, and Huckleberry. 1/3 = 1/3. And always did. Glkanter (talk) 18:03, 11 December 2009 (UTC)
Parallel Universe Experience: Same random distribution of cars. Huckleberry makes all the same door choices. Monty picks the exact opposite equal goat door as he did above. Huckleberry always switches. There is NO CHANGE TO THE 2/3 OUTCOME, EVEN IF YOU PLAY FOREVER. And the individual play outcomes are identical. Huckleberry wins or loses the same exact instances regardless of which goat door Monty opens. Glkanter (talk) 14:50, 12 December 2009 (UTC)
I Guarantee It. Old Paradoxes Are Like New Technology.
“New technology goes through three stages:
First, it is ridiculed by those ignorant of its potential.
Next, it is subverted by those threatened by its potential.
Finally, it is considered self-evident.” –unknown —Preceding unsigned comment added by Glkanter (talk • contribs) 04:39, 12 December 2009 (UTC)
- Looks like I started this section just in the nick of time: Moved conditional analysis involving host preference q to variant section.
- There's an old saying: "If you see a parade, get in front of it." I think Rick must have heard this old saying, too. Maybe Rick will argue for the proposed changes when we get to mediation as well? Glkanter (talk) 05:23, 13 December 2009 (UTC)
Moved conditional analysis involving host preference q to variant section
I think this is at least one of the changes that has been argued for, and I haven't seen anyone argue against it (and those of you who think this is what I have been arguing against are simply incorrect), so I've moved the paragraph about the Morgan/Gillman generalization introducing the host preference as a variable q to the Variants section. If anyone is arguing about this, feel free to revert. -- Rick Block (talk) 00:51, 13 December 2009 (UTC)
- I think that is a good move but it still does not address the fundamental issue that many people here are concerned about, which is that the Monty Hall problem is a simple problem that most people get wrong. This article needs to reflect that fact, based on the many reliable sources that treat the problem simply (I am not going to mention the c-word). Martin Hogbin (talk) 17:50, 13 December 2009 (UTC)
- I understand your point, but based on other comments there seems to be some confusion about what those of us who favor presenting a "conditional approach" as an equally valid POV are saying. Rather than argue generalities, I think it is helpful to see specific changes. My stance is that a single solution section, more or less like (and about as long as) this one is sufficient. I think it would be very helpful if you (or anyone) could draft a specific proposal (not just an outline, but actual content). With the amount of text on this page, we could have 5 or 10 specific proposals by now. -- Rick Block (talk) 18:45, 13 December 2009 (UTC)
- Rick, you wrote this above: "..."conditional approach" as an equally valid POV...".
- Hey Rick, we talked about this before:
- "...The total probability must be 100%, and before the host opens a door it's surely 1/3 player's door vs. 2/3 for the other two doors (so 2/3 of the players who decide to switch before the host opens a door will win) but the only thing that keeps it that way after the host opens a door is that pesky equal goat door constraint. The best way I know to show that it CAN change is to contrast the problem as stated with a different problem (i.e. the aforementioned "host opens lowest numbered door possible" variant)...Rick Block (talk) 00:24, 27 October 2008 (UTC)"
- "False. The only thing 'keeping it that way' is that pesky law of probabilty that the outcomes must = 100%. Glkanter (talk) 03:46, 27 October 2008 (UTC)"
- "...The total probability must be 100%, and before the host opens a door it's surely 1/3 player's door vs. 2/3 for the other two doors (so 2/3 of the players who decide to switch before the host opens a door will win) but the only thing that keeps it that way after the host opens a door is that pesky equal goat door constraint. The best way I know to show that it CAN change is to contrast the problem as stated with a different problem (i.e. the aforementioned "host opens lowest numbered door possible" variant)...Rick Block (talk) 00:24, 27 October 2008 (UTC)"
- The more things change, the more they stay the same, eh? Still using some so-called 'variant' to explain away the simple solutions. That doesn't even make sense from a mathematical standpoint. Oh, but now (per your paragraph above) you say they're 'equally valid'.
- By now, your only remaining argument is that 'people could get confused', because the MHP only works with the particular set of premises given. The part about why it works may be true. The part about confused readers is your personal interpretation, and not supported by at least one editor who includes the problem in his course work. And it's certainly not Wikipedia's job to teach probability in the MHP Paradox article. Anything like that belongs in what I call the 'Diversions' section, if at all.
- And there is no point whatsoever in creating mock articles until the consensus for the proposals has been 'certified' to your satisfaction. They are not ambiguous in any way regarding the minimal value of the 'conditional' approach. Anything else is just a waste of time, and feeds your filibuster. Glkanter (talk) 12:46, 14 December 2009 (UTC)
- Hey Rick, we talked about this before:
Proposed unified solution section
Here's a proposal for a unified solution section that I suggest replace the current two solution subsections. I offer this partly as an example of what I mean by a specific suggestion, and partly to show what I think would be a sufficient, NPOV, solution section.
According to the problem statement above, a car and two goats are arranged behind three doors and then the player initially picks a door. Assuming the player's initial pick is Door 1 (vos Savant 1990):
- The player originally picked the door hiding the car. The game host must open one of the two remaining doors randomly.
- The car is behind Door 2 and the host must open Door 3.
- The car is behind Door 3 and the host must open Door 2.
Players who choose to switch win if the car is behind either of the two unchosen doors rather than the one that was originally picked. In two cases with 1/3 probability switching wins, so the probability of winning by switching is 2/3 as shown in the diagram below. In other words, there is a 2/3 chance of being wrong initially, and thus a 2/3 chance of being right when changing to the other door. This result has been verified experimentally using computer and other simulation techniques (see Simulation below).
Another way to understand the solution is to consider the two original unchosen doors together. Instead of one door being opened and shown to be a losing door, an equivalent action is to combine the two unchosen doors into one since the player cannot choose the opened door (Adams 1990; Devlin 2003; Williams 2004; Stibel et al., 2008).
As Cecil Adams puts it (Adams 1990), "Monty is saying in effect: you can keep your one door or you can have the other two doors." The player therefore has the choice of either sticking with the original choice of door, or choosing the sum of the contents of the two other doors, as the 2/3 chance of hiding the car hasn't been changed by the opening of one of these doors.
As Keith Devlin says (Devlin 2003), "By opening his door, Monty is saying to the contestant 'There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3. I'll help you by using my knowledge of where the prize is to open one of those two doors to show you that it does not hide the prize. You can now take advantage of this additional information. Your choice of door A has a chance of 1 in 3 of being the winner. I have not changed that. But by eliminating door C, I have shown you that the probability that door B hides the prize is 2 in 3.'"
Another way to analyze the problem is to determine the probability in a specific case such as that of a player who has picked Door 1 and has then seen the host open Door 3, as opposed to the approach above which addresses the average probability across all possible combinations of initial player choice and door the host opens (Morgan et al. 1991). This difference can also be expressed as whether the player must decide to switch before the host opens a door or is allowed to decide after seeing which door the host opens (Gillman 1992). The probability in a specific case can be determined by referring to the expanded figure below (note the case where the car is behind Door 1 is now the middle column) or to an equivalent decision tree as shown to the right (Chun 1991; Grinstead and Snell 2006:137-138). Considering only the possibilities where the host opens Door 3, switching loses with probability 1/6 when the player initially picked the car and otherwise wins with probability 1/3. Switching wins twice as often as staying, so the conditional probability of winning by switching for players who pick Door 1 and see the host open Door 3 is 2/3—the same as the overall probability of winning by switching. Although these two probabilities are both 2/3 for the unambiguous problem statement as presented above, depending on the exact formulation of the problem the conditional probability may differ from the overall probability and either or both may not be able to be determined (Gill 2009b), see Variants below.
A formal proof that the conditional probability of winning by switching is 2/3 using Bayes' theorem is presented below, see Bayesian analysis.
I have tried to remove any POV-ish statements in the above. If there's anything left that does not sound NPOV please either just fix it or discuss here. The idea is to present both a plainly correct unconditional solution (it's basically vos Savant's from her second column) as well as a plainly correct conditional solution, without expressing a preference for either treatment. -- Rick Block (talk) 19:12, 14 December 2009 (UTC)
- And I don't think you did a good job of removing your POV, Rick. It's all through the second paragraph, as you try to make it sound like what we want while leaving Morgan in it. My criticisms will address that, and hopefully suggest NPOV corrections, soon.
- The 3-point options at the start, and the following diagram, present the unconditional solution (e.g., Door #2 gets opened in some cases), yet seemingly claim to follow from the K&W statement. To use this approach (which we should), this part needs to attribute it to the MvS statement, and not use door numbers. (This is consistent with MvS's answer, which refers to "the first door" and "the second door".) The following preserves her use of examples while not requiring door numbers:
- According to the Parade problem statement, a car and two goats are arranged behind three doors. The player initially picks one, and the host always opens a different door with a goat, choosing at random if necessary (Seymann). (Aside: Seymann does not say the MvS problem statement is ambiguous, as the article currently claims. Seymann says the assumptions should be inferred from intent, and intent is quite clear from her solutions and following columns. Seymann is chastising Morgan, et al, for not using this clear intent. Seymann does say the host chooses randomly, as an "agent of chance".)
- The 3-point options at the start, and the following diagram, present the unconditional solution (e.g., Door #2 gets opened in some cases), yet seemingly claim to follow from the K&W statement. To use this approach (which we should), this part needs to attribute it to the MvS statement, and not use door numbers. (This is consistent with MvS's answer, which refers to "the first door" and "the second door".) The following preserves her use of examples while not requiring door numbers:
- The second paragraph does not match the decision tree it says it matches. It is trying to address the conditional solution without assymetric probabilities, yet duplicates the diagram's unconditional treatment. So, it should cite G&S only, not Morgan, because this "symmetric conditional" treatment is far closer to their treatment than Morgan's. I'm not going to try to draw a tree, but it is in G&S as Figure 4.4 (the existing tree misrepresents it). We can include the "disallowed" options that the existing tree does, opening Door #2, by making it clear how they are disallowed by G&S. Make them a different color, add a column that says "didn't happen." Alternately, it could just trim the diagram down, like G&S trim thier tree down.
- Another way to analyze the problem is with conditional probability, as the specific case where the player has picked Door 1 and the host has opened Door 3. This contrasts with the approach above which addresses the average probability across all possible combinations of initial player choice and door the host opens. The probability in this specific case can be determined by referring to the decision tree as (will be) shown to the right (Grinstead and Snell 2006:137-138). Considering only the possibilities where the player chooses Door #1 and the host opens Door 3, switching loses with probability 1/18 and wins with probability 1/9. But these possibilities comprise only 1/6 of the total possibilities, so their probabilities must be divided by 1/6 in accordance with Bayes Rule. Thus we get the same overall probability of winning by switching as before. And while these two probabilities are both 2/3 for the problem statement as presented above, if there is reason to beleive that the host prefers to open one door over another, the conditional probability may differ from the average probability that disregards which doors are chosen (Grinstead and Snell 2006:137-138, see Variants below.
- The second paragraph does not match the decision tree it says it matches. It is trying to address the conditional solution without assymetric probabilities, yet duplicates the diagram's unconditional treatment. So, it should cite G&S only, not Morgan, because this "symmetric conditional" treatment is far closer to their treatment than Morgan's. I'm not going to try to draw a tree, but it is in G&S as Figure 4.4 (the existing tree misrepresents it). We can include the "disallowed" options that the existing tree does, opening Door #2, by making it clear how they are disallowed by G&S. Make them a different color, add a column that says "didn't happen." Alternately, it could just trim the diagram down, like G&S trim thier tree down.
- I left it as "there is reason to believe" rather than the correct "the player has reason to believe" to keep what Rick thinks is POV out. The content of this section now matches G&S in every way, except it is a paraphrasal. G&S is aimed at students of probability, where we want to aim at non mathematicians. JeffJor (talk) 16:17, 15 December 2009 (UTC)
- The problem is that the diagram is still complicated by the fact that it shows the host opening one of two doors. You are stuck in the world of door numbers. I would prefer this diagram (with pretty pictures) and some explanation:
1/3 1/3 1/3 You choose a goat You choose a goat You choose a car The host opens a door to reveal a goat The host opens a door to reveal a goat The host opens a door to reveal a goat You Stick You Swap You Stick You Swap You Stick You Swap You get a Goat You get a Car You get a Goat You get a Car You get a Car You get a Goat
- So, you're OK with the words but not the diagram? Comments from other might be nice, but one thing I like about the diagram is that it SHOWS the symmetry (the diagram is symmetric). -- Rick Block (talk) 19:52, 14 December 2009 (UTC)
- I think this whole thing is a nightmare giant step backwards into an abyss. How did the Combined Doors solution get eliminated? Why? There is no need to include 1/6 in any explanation of the paradox. 'Simulations' as 'proofs'? Please. And I don't understand why editing is being discussed prior to the consensus being recognized. Just more filibustering. Enough, already! Glkanter (talk) 20:45, 14 December 2009 (UTC)
- Glkanter - Where did combined doors go? Well, I think vos Savant's solution is clear enough for nearly anyone. Most of the folks who talk about "combining doors" offer it as an aid to understanding, so it could certainly go there (and, let me just say, for one who is so against variants of the problem, your attachment to this explanation is incredibly incongruous). 1/6 is included because it is what the unconditional probability of the player's initially selected door becomes in the cases where the host opens each door, just like the sources say (the probability tree is directly from Chun, 1991). The forward reference to the Simulation section is there because some people trust simulations more than logic. If you don't want it, let's just delete it and see if anyone objects. I'm trying to work toward a specific compromise we can all live with. -- Rick Block (talk) 02:58, 15 December 2009 (UTC)
- Binary choices don't offer 'compromise'. The simple solutions are more than sufficient. Let's get the Formal Mediator to acknowledge the consensus to make the proposed changes. I think a lot of what is taking place lately does not represent a good faith effort at recognizing the consensus that clearly exists. Did Huckleberry get it right or not? Is the contestant aware? These are direct yes/no questions, not 'yes, but'. You are THE LAST person who should be leading the editing effort. Maybe you haven't accepted it yet, but your 6 year crusade was misguided. Simply wrong. Just like I've been trying to tell you for 14 months. I'd appreciate it if you'd stop all this mis-direction, and let the consensus edit the article as proposed, finally. Glkanter (talk) 03:25, 15 December 2009 (UTC)
Any other comments? I'm particularly interested in comments from the folks Glkanter is considering part of the "consensus", i.e. JeffJor, Colincbn, Boris, and Melchoir. -- Rick Block (talk) 14:29, 15 December 2009 (UTC)
- Which of your various roles are you in when you ask this question, Rick? Editor, Owner, FA Shepard, or Junior Moderator? Glkanter (talk) 14:37, 15 December 2009 (UTC)
- Glkanter - I see you've edited in the combining doors solution. Does this mean you're more or less OK with a single solution section? I'd prefer for the combining doors bit to be in an aid to understanding section (it's even phrased that way "Another way to understand the solution ..."). Do you consider this a deal breaker? I'm curious what others think about this. -- Rick Block (talk) 19:33, 15 December 2009 (UTC)
- Jeff - Aside from some minor quibbles (which I don't have time to enumerate at the moment - I will later, although perhaps not until tomorrow), I'm more or less OK with your version of the conditional analysis. Not minor - I assume you don't mean for the aside to be in the article. Please say what you think about including the "combining doors" bit in the solution section. -- Rick Block (talk)
- Martin - Is Jeff's version of the image more to your liking? -- Rick Block (talk) 19:33, 15 December 2009 (UTC)
- Anyone else have any comments about all of this? -- Rick Block (talk) 19:33, 15 December 2009 (UTC)
- Rick, you continue to be confused. There are no 'deals'. And you are NOT the deal maker. You chose to not be part of the consensus. It makes no sense, then, for you to be proposing what the consensus wants. I wish I had a way to just make you stop. I don't, of course. I have, over 14 months used the Combined Doors solution as the published simple solution to support my agrument with you and the others. You all continue to post, even today, that I don't understand the problem, or worse, actually. But, Devlin's solution was right and proper all along. And it's in the article that way now. Only YOU would think of removing it. Certainly not me. And I don't foresee a 'Unified solution' section, or a 'disclaimer' of any sort. It's not correct. That's why I proposed a brief section summarizing the 6 years and soon-to-be 10 archives of arguments, and why the consensus supports the simple solution. Glkanter (talk) 19:56, 15 December 2009 (UTC)
- Here's my interpretation of Wikipedia policy, speaking as an admin. Consensus applies to changes to articles (even specific changes), not to groups of editors. No editor is more or less a part of the consensus for any individual change than any other editor. Trying to cut off discussion about a specific suggested change is classic disruption. -- Rick Block (talk) 21:09, 15 December 2009 (UTC)
But Rick, you are not part of the consensus. You are against the proposals. Or has that changed, and mediation is no longer required? Glkanter (talk) 21:30, 15 December 2009 (UTC)
- And if you honestly believe I am attempting to violate disruption, I suggest you follow the procedures necessary to stop me. Otherwise, I'll consider the threat just one more of your endless filibustering techniques, not offered by you in good faith. Glkanter (talk) 21:40, 15 December 2009 (UTC)
Martin's suggestion
- I would like to see 'combining doors' back at the start of the article along with my suggestion above. I think that most people will agree that these are the most convincing explanations of the basic problem (where the issue of conditional/unconditional is deferred to a later section). These diagrams should be accompanied by new wording specifically relevant to the explanations shown in the diagrams. This should be followed by 'Aids to understanding' and 'Sources of confusion' relating to the simple solutions - this is what the Monty Hall Problem is all about.
- After the above we should explain why some formulations of the problem are, strictly speaking, conditional (with reference to the Morgan paper) and why this fact is not so important for the symmetrical case.
- Then we should have a section on variants, the most important of these being 'the host chooses any unchosen door randomly', but including the Morgan scenario, 'we are aware of the host door opening policy'. We can then have more on sources of confusion etc as it relates to the more complex cases. That is what I would like to see. Who agrees? Martin Hogbin (talk) 15:18, 15 December 2009 (UTC)
- Just to add, I would not object to a brief and mildly worded disclaimer at the start of the simple solution to the effect that some academic sources insist the problem must be treated conditionally but, for simplicity and clarity, these issues are discussed in a later section. Martin Hogbin (talk) 15:22, 15 December 2009 (UTC)
- The so called "combining doors solution" is in a certain sense the worst of all. Let me try to make this clear to you. Choose door 1 and start from this situation: then with prob. 1/3 the car is behind door 2 and also with 1/3 behind door 3. Formally: P(C=2)=P(C=3)=1/3. That's why we can say: P(C=2 or C=3) = 2/3 (doors combined). But there is no immediate logical reason why this should lead to the prob. 2/3 that the car is behind door 2 when door 3 is opened. We know P(C=3|H=3)=0, but this does not imply that P(C=2|H=3)= 2/3. Only if one reasons that from symmetry it follows that {H=3} is independent of {C=2 or C=3}, one may conclude that also P(C=2 or C=3|H=3) = P(C=2 or C=3) = 2/3, and hence P(C=2|H=3) = 2/3. Even then as you may note is the prob, of interest the conditional prob. I'm pretty sure however most of the readers do not understand this independency, but simply have no idea of the different prob.s and follow a wrong way of reasoning. BTW also some referred sources do! Nijdam (talk) 22:25, 15 December 2009 (UTC)
- I accept what you are saying. The issue is addresses in words by Adams, who says (without proof) '...as the 2/3 chance of hiding the car hasn't been changed by the opening of one of these doors'.
- On the other hand, three sources use this approach and I think it is intuitively acceptable to many people. What about having footnotes for the two simple solutions saying something along the lines of, 'There are some important complicating issues here which are discussed later but, as it happens, if the host chooses randomly they do not affect the result'. This has the advantage of not affecting the simplicity of the explanation but drawing attention the the complication issues for those interested. Again, this approach is not uncommon in maths text books. Martin Hogbin (talk) 10:00, 16 December 2009 (UTC)
- Being a lay person, I'm curious, what was the outcome of all the work Boris did? He said his mission here was finished. Devlin, Adams, etc. are published, and a consensus is that their approach has NOT been mathematically refuted by Morgan and the others. There's no complication with the MHP, the complication is with the 4 confused sources which do NOT address the MHP. I've suggested a brief paragraph after the simple (only) solutions that summarizes the lengthy discussions that have taken place as the appropriate way to recognize Morgan and the others. Glkanter (talk) 11:29, 16 December 2009 (UTC)
- Again, I'm not sure where to insert comments. (1) On my "aside": Yes, Rick, it was meant to be left here, not included in the article. That's why it is an aside. I didn't want you to change the text again to match your POV of what Seymann says, or to continue thinking than nobody had ever refuted Morgan's assertion that the door numbers were intended. That was refuted when Morgan published. (2) I'm not a great fan of the "combining doors" explanation, because it is a (very slightly) different problem. But not anywhere close to as different as Nijdam claims. The only difference is that you shold get the better of the two prizes, not both. (For Nijdam: the Host's required behavior, revealing an unchosen door with a lesser prize, is logically equivalent to the switching player receiving the better prize of the two unchosen doors. That is indeed an "immediate logical reason" for the 2/3 probability applied to the combination. But Carol Merrill should lead a goat out of the combined doors, in such a way that you don't know where it came from, for it to be the same problem. It is when we translate the combined-door probability back to the original, unopened Door #2 that it can be associated with one door only). I just don't know of a source that explains it my way. Delvin comes close, since he used empty doors instead of goats. But it is a good way over the intuition bubble, as Delvin explains; so the only reason to not use it is if we have too many explanations. I'm not sure which we are proposing keeping. We need at least one basic solution (as I did above), and one explanation for why it works (like combining doors). I don't consider the conditional probability one I listed "basic," but I'm not going to fight including it. That does, however, limit what other "basic" solutions we could use, because we lose readers if we over-explain. (3) We need a disclaimer with the full Morgan conditional statement. Not before. The disclaimer is that some advanced treatments consider it a Door 1/Door 3 issue; but that that was never the intent of the originators, has been denied by the originators, and is not even universally accepted. Then add that even those who use it remove the dependency on those door numbers before addressing "Should the contestant switch?" Essentially, the full conditional solution is a tool only. JeffJor (talk) 17:17, 16 December 2009 (UTC)
- Nijdam, how does this question differ from the issue you and Boris analyzed? Boris declared his mission finished. I'm beginning to think that like Rick, you are simply using various filibustering techniques to avoid recognizing the legitimate consensus for the proposed changes. Why don't you ever answer my questions? I imagine Socrates would have. Glkanter (talk) 12:08, 17 December 2009 (UTC)
OH MY GOD!!!
Holy cow. I went camping this weekend and I woke up in the tent in 33 degree F. cold trembling in a cold sweat, having a nightmare about goats and shiny cars and numbered doors and a genetic-cross monster whose body was that of a water buffalo and the head was that of Thomas Bayes, and everyone was throwing food about and nobody knew whether everything or anything is random or predetermined and people were capriciously changing their minds in the middle of the game conditionally and unconditionally, and I decided that I would just run off and try to re-read "Kant's Critique of Pure Reason", and then maybe shoot myself. That might be easier. :o) Worldrimroamer (talk) 02:04, 15 December 2009 (UTC)
- OK, folks ... please don't fuss at me. I was only kidding. Worldrimroamer (talk) 02:08, 15 December 2009 (UTC)
Why Wait To Edit The Article & FAQs As Per The Proposals Supported By The Consensus?
There's plenty of editing of the article currently taking place, including by Rick Block, who is not part of the consensus.
Why couldn't the consensus just go ahead and begin fixing the article, so that it is in line with the proposals? Glkanter (talk) 11:51, 15 December 2009 (UTC)
- I think the article is OK through the Popular solution. Then the Probabilistic solution begins all the trouble with the gibberish and double talk about Morgan's stuff.
- Let's put a paragraph right after the Popular solution that discusses why the Morgan solution, while published, is considered to be discussing a different problem.
- Then we replace nearly all the text of the Probabilistic solution, and replace it with how to do the conditional solutions.
- Beyond that, it's just chatter, that hopefully most readers won't need. Unless there's something in all that which denigrates the Popular solutions, I would just let it be for now. Glkanter (talk) 12:17, 15 December 2009 (UTC)
- The article is not protected. Anyone can make whatever change they'd like, and to a large extent making a change and seeing that it is not reverted is how consensus is demonstrated (see WP:BRD). I think you should also really read WP:BATTLE. -- Rick Block (talk) 14:46, 15 December 2009 (UTC)
How do we resolve the inconsistency between the Contestant's POV in the MHP, and the "unknowns'" POV in all the host variant stuff? Especially that large table. The whole thing makes no sense to me. Glkanter (talk) 15:08, 15 December 2009 (UTC)
- Are you suggesting this section is not a neutral recounting of what reliable sources say? Or is your problem that you don't understand it? -- Rick Block (talk) 15:40, 15 December 2009 (UTC)
- Like many things in the article it is inconstant from a logical standpoint. The MHP, as you know, is from the contestant's POV. All these so-called 'variants' are solved from some 'unknown' person's POV. So that's inconsistent. Even worse, since it's not the contestant's POV, it's not the MHP. I just don't see much value there. Maybe a section with links, and that's it.
What has to be considered the MHP??
I very much like this question to be answered first. (BTW several sources mention the MHP to be equivalent to the three prisoners problem.) Nijdam (talk) 22:35, 15 December 2009 (UTC)
- I created a "construction" site, where we may step by step build the article, until we come at a point we don't find agreement on. —Preceding unsigned comment added by Nijdam (talk • contribs) 16:41, 16 December 2009 (UTC)
- Nijdam, have you joined the consensus? Is mediation still necessary? What conclusions did you and Boris reach? Please address my points from this diff Huckleberry & Awareness and these questions before we invest all the time and effort. Glkanter (talk) 16:51, 16 December 2009 (UTC)
- I think everyone would like an answer to that question but it is unlikely that there will be agreement on the subject. I would say that there are several formulations of the problem. I would personally like the MHP be treated initially as a mathematical puzzle, formulated in such a way as to make the solution simple and not depend on conditional probability. In my opinion the MHP is fundamentally a simple problem that most people get wrong, this is undoubtedly its most notable aspect. This was obviously Gardner's intention with the TPP although you might still argue that this problem is still, strictly speaking, conditional. The difficulty with implementing my preferred approach here is that there are no sources that specifically treat the problem in that way, unless we can find one.
- Without doubt, the most notable problem statement is Whitaker's but, as you are well aware, this leaves so much unsaid that it can be interpreted in many different ways. In my opinion vos Savant answered the question correctly (but failed to make clear exactly what the question was) whilst Morgan interpreted the question too strictly (and still ambiguously) and then answered their interpretation (for the most part) correctly.
- The only unambiguous problem formulation that I know of in the literature is the Krauss and Wang version that we quote. This is exactly equivalent to the TPP but this suffers the problem that, although the effect of conditionality is negated by the host's random choice, it can still be argued (as you do) that the problem is still strictly one of conditional probability and thus, unfortunately, it is not amenable to a simple solution.
- So, I regret that I have failed to answer your very important question. The lack of an answer is the cause of much of the argument here. I think we just have to argue it out as best we can to create the best article possible. Martin Hogbin (talk) 11:00, 16 December 2009 (UTC)
- Here is the Monty Hall problem:
- "Suppose you're on a game show..."
- The symmetry is a premise. And needs no disclaimer or footnoting. Anybody that disagrees should tell me what's wrong with Huckleberry's approach and must answer, incorrectly, that 'the contestant is aware' to Is The Contestant Aware? Glkanter (talk) 11:22, 16 December 2009 (UTC)
- Here is the Monty Hall problem:
- From Wikipedia's perspective, the MHP is whatever reliable sources say it is. WE don't need to (in fact, we don't get to) decide. What we do need to do is say what reliable sources say about whatever they consider it to be.
- As I thought I made clear above, reliable sources do not answer the question of what exactly is the MHP. We have at least Selvin's original statement, Whitaker's question, vos Savant's partial formulation, Morgans misiquotation and subsequent incomplete formulation, and several formulations in K&W. The only one that is unambiguous is the one we quote from K&W, but K&W make no claim that this is The MHP. Thus reliable sources do not answer the question and we must decide what the subject of our own article is to be here.
- Martin - you keep saying it's a simple problem that most people get wrong and that approaching it as a conditional probability problem is (more or less) a nuisance. If it's so simple, why (in your opinion) do most people get it wrong? My answer to this is that they try to solve it conditionally (what was it in the K&W experiment - 35 out of 36 subjects consider only the case where the player has picked Door 1 and the host has opened Door 3), which means to me that we can't fully explain it without addressing this issue.
- Yes we can, there is no evidence anywhere that anyone considers it important which door the host opens. K&W's main point is that giving door numbers just confuses the issue, and I agree. Martin Hogbin (talk) 18:14, 16 December 2009 (UTC)
- Regardless of the above, since reliable sources address it both ways the article must also. Not to do so would imply a POV that the "unconditional" approach is better or more correct. Avoiding this POV is the reason I'm suggesting we go back to a single Solution section. JeffJor seems to be OK with this (since he's engaged in editing a proposal for such a suggestion). I can't tell if Glkanter is OK with this or not. You (Martin) are at least currently saying you're not OK with this (is that right?). So, directly, would both of you (Nijdam, too) be OK with a single solution section more or less like JeffJor's suggestion? -- Rick Block (talk) 15:04, 16 December 2009 (UTC)
- Rick, I didn't realize I hadn't made my position clear. The 'Solution section' will contain simple solutions only. No disclaimers, no footnotes, no 'buts'. This is what the consensus has agreed to. Your NPOV threats are just that. They are intimidation, and an attempt at prior restraint. Just more filibustering, as usual. And Rick, nobody is buying your NPOV act. The current article has such a heavy Rick Block/Morgan bias it's laughable. The FAQs alone make me want to vomit. Just getting to NPOV from your extremes will be an accomplishment. I can't conceive of the article ever being so POV-ed in the other direction as great as you have accomplished. I agree, as you posted on the Mediation Cabal page, this discussion is totally out of control. How many times does the consensus have to tell you your interpretations are inconsistent with the consensus? Glkanter (talk) 15:20, 16 December 2009 (UTC)
- Whoa, whoa, whoa. Rick, you have a sad way of imnposing your own POV in every possible way that lets you keep it in the article. It is downright insulting. I am not "OK" with considering the Morgan "conditional" approach as addressing the MHP. It does not. Seymann says it does not. Nobody, not even Morgan, says it does address the actual question. But I am a realist, and I recognize that you will not let it be separated AS IT SHOULD BE, because you feel justified in your POV because your misread those sources and think they say something they do not.
- So I am "OK" with trying to improve the article in a way that can be accomplished. That means inculding your un-intended, not-MHP, POV in the article. Specific door numbers were never intended to be important, and any possible importance implied by the "reliablle sources" you quote, who demonstratably misrpresented the MHP, was for rigor only. It gets removed by those sources before actually addressing the MHP with their formula. IT HAS NOTHING TO DO WITH THE MHP. IT IS NOT REPRESENTATIVE OF THE MHP. IT DOES NOT HELP IN UNDERSTANDING WHY THE MHP IS CONTROVERSIAL. All it does is extend the thought problem, in a way that is interesting only to mathematicians. The body of the article needs to address the problem as seen by the general populace, not pedantic mathematicians who proved one assumption (of two similar ones that are normally made) was unnecessary. While interesting, being unnecessary doesn't explain the unintuitive nature of the problem. JeffJor (talk) 22:46, 16 December 2009 (UTC)
- It seems like we must not be talking about the same thing here. I'm talking about a conditional probability analysis, like the one you wrote above (the paragraph starting "Another way to analyze the problem ..."). Which I think is exactly the way the general populace, not pedantic mathematicians, see the problem. What are you talking about? -- Rick Block (talk) 23:07, 16 December 2009 (UTC)
- Jeff - if you can reply I would appreciate it. My assumption was that since you wrote the above paragraph that you would be OK with putting it (or something like it) in the article. If that was not your intent I'm sorry to have misinterpreted. -- Rick Block (talk) 01:53, 17 December 2009 (UTC)
- Rick - Just to be clear, there are four categories that different sources have used to approach the MHP. I will call them "Unconditional Approach," "Symmetric Conditional Approach," "Asymmetric Conditional Approach," and "Reduced Asymmetric Conditional Approach." UA, SCA, ACA, and RACA for short. And there can be different kinds of RACAs, depending on what gets "reduced," by which I mean eliminating the importance of a condition.
- The UA supports either MvS's explanation, or Devlin's combined doors, or anything similar. It really doesn't utilize the doors in any specific way. It can be done rigorously (G&S do it in their first solution not attributed to MvS), but usually isn't. Non-rigorous solvers use symmetry to reduce G&S's twelve cases to four. The SCA looks at specific doors, but assumes that any uncertainty must be uniformly assigned as in G&S's second solution. Some UAs - those that mention door numbers - look like SCAs but are really using numbers only as examples. The "tree" version as it currently exists did this. You can tell because it has four cases that sum to P=1, not twelve cases as in G&S. ACA is stated in K&W (not Morgan, Gillman, or G&S since all ignore car placement bias) as Equation 1. But it is not useable to answer the question "Should she switch?" because of the placement bias. Morgan, Gillman, and G&S use an RACA where they make one reduction - they eliminate placement bias by assumption. K&W discuss several ways of reducing ACA by assumption - their no/one/two door solutions. Morgan's thesis is not that ACA is proper (although they mistakenly assume it), but that you don't need to assume anything to reduce the importance of host choice. They reduce it by making it unimportant to the quesiton, and that is still a reduction. Eventually, every source that considers an ACA reduces it to a question (not a probability value) that is answered independent of door numbers.
- So, what I think is that UA needs to be the primary focus of the article. SCA can be used, or not; but if you insist on it, it needs to be done properly as G&S did. That's why I changed the probabilities in the discussion I wrote of it. I don't think you will let anybody take it out of the body completely, so I left it and did my best to improve it. But it really does not help non-students of probability. They don't understand that conditional probability depends as much on what is removed as what is left. The terms in the SCA formula can show this, but the non-student will not understand how to read such a formula. So it really doesn't help them, it just makes the article look like a textbook that they don't want to read. And any ACA is addressing a variant of the MHP where asymetric probabilites need to be considered. Few people think the problem says that, and fewer still think they can be used. Certainly not any of the references, who always reduce ACA completely. So I don't see any benefit from that to the general public. There is benefit to students of probability, but that benefit is not directly related to the MHP itself. It only shows how you don't have to make assumptiosn to reduce all of the conditiosn that might affect a strategy. So it can be included AS LONG AS IT IS CLEARLY SEPARATED FROM THE ACTUAL MHP DISCUSSION. And it needs to be made clear that it is a non-standard interpretation of the problem, and that the parts that make it diffewrent are never used directly to answer the MHP.
- I hope this helps you and Nijdam. It is the ACA and RACA that belong as a variant. The SCA is "better" solution in the sense of rigor only; but it does not satisfy any need the article has, except rigor. Since the reason the MHP is enigmatic at all has nothing to do with rigor, but with intuition, we really should pay more attention to the issues that surround intuition. JeffJor (talk) 15:35, 17 December 2009 (UTC)
- I don't need help. What you call UA, suggesting an approach to the MHP, is just a solution of a specific simple version, not the K&W-formulation, and in my opinion also lacking the characteristics of the MHP. This is also admitted by G&S: This very simple analysis, though correct, does not quite solve the problem that Craig posed. The SCA and ACA are both solutions to the MHP (K&W version). Nijdam (talk) 12:55, 20 December 2009 (UTC)
- (1) Nijdam, elsewhere you had asked for my interpretation. When I "hoped it helped" you, I meant "helped you understand my interpretation." (2) But you apparently do need my help, because you think something other than UA/SCA is "the MHP." As has been clearly demonstrated through references,UA/SCA is indeed what was intended as "the MHP" by the originators, whether or not some others misinterpreted it. And all of the controversy in the general public surrounding those puplications stem directly from, and only from, it. Anything else is a distraction. The fact that some sources disagree on this point only proves that we need to handle that disagreement by separating the issues. Craig's problem can solved, as written. This is clearly admitted by Seymann. It is G&S's alternate interpretation that cannot be solved. They only supply a solution to an isolated example that they do not claim is as a valid solution the problem itself. And in fact, G&S is the model for how to do this. They separate their approaches the exact same way. JeffJor (talk) 13:04, 21 December 2009 (UTC)
- Nobody can solve the problem that Craig posed because it is not clear exactly what it is. Morgan interpreted it in one particular way that makes the problem clearly conditional. In fact he probably just wanted to know, 'What is the probability you win if your strategy is to switch?', as Morgan put it. Martin Hogbin (talk) 15:32, 20 December 2009 (UTC)
- Incorrect, Martin. Read Seymann's comment to Morgan. It was not supposed to be a rigorous prob=l;em statement, it was a "fun" puzzle in a newspaper. And there are clear assumptions that can, and should, be made; which were reinforced by MvS herself. The controversy had nothing to do with any of those possible ambiguities. JeffJor (talk) 20:55, 20 December 2009 (UTC)
- Jeff it would probably help our cause if you were to stop jumping down my throat at the first opportunity. I was the first, I believe, to bring to the attention of editors here Seymann's commentary, which points out that the problem can be interpreted in different ways. I am agreeing with you that Whitaker probably just wanted the simple unconditional problem answered, with normal 'puzzle' assumptions (as correctly made by vos Savant). That is why I quoted from Morgan's example of an unconditional statement of the problem. Even that 'most reliable of sources' gives the problem to which the simple solutions, including vos Savants are the answer; that is what I have quoted from. Martin Hogbin (talk) 21:23, 20 December 2009 (UTC)
- It would help if we were all on the same page. Any ambiguity in the Parade statement is unimportant to what makes the MHP controversial. Considering such is what leads some people to think the Morgan analysis should be part of the main issue. The point is, that the Whitiker statement is sufficient for the vehicle in which it was published. Seymann does not say it "can be interpreted in different ways." He says, and I quote, "Simply put, and quite clear considering her suggestions for simulation procedures in her two later columns, the host is to be viewed as nothing more than an agent of chance who always opens a losing door, reveals a goat, and offers the contestant the opportunity to switch to the remaining, unselected door." There is nothing vague about this. Craig's problem can be solved, without having to allow for any alternate host strategies.
- Rick (and others) keeps trying to inject his POV by suggesting Seymann thought it was ambiguous, and Seymann did not. And MvS said explicitly that the vast majority of the controversy in the letters had nothing to do with such possibilities. Your comment came in the middle of an "edit battle" where Rick reworded the article to inject that POV, and I removed it. As long as they keep trying, NPOV requires it be squelched. It is only this way that we can eliminate POV from the article. JeffJor (talk) 23:11, 20 December 2009 (UTC)
- I will leave you to your lone battle as you seem determined to pick fights with those who essentially agree with you. I was arguing that Craig's question should be treated in a manner sympathetic to its origin long ago. Martin Hogbin (talk) 00:12, 21 December 2009 (UTC)
- Incorrect, Martin. Read Seymann's comment to Morgan. It was not supposed to be a rigorous prob=l;em statement, it was a "fun" puzzle in a newspaper. And there are clear assumptions that can, and should, be made; which were reinforced by MvS herself. The controversy had nothing to do with any of those possible ambiguities. JeffJor (talk) 20:55, 20 December 2009 (UTC)
- Nobody can solve the problem that Craig posed because it is not clear exactly what it is. Morgan interpreted it in one particular way that makes the problem clearly conditional. In fact he probably just wanted to know, 'What is the probability you win if your strategy is to switch?', as Morgan put it. Martin Hogbin (talk) 15:32, 20 December 2009 (UTC)
- I'll note that I didn't "reword" anything. Butwhatdoiknow's complaint was the length of the lead. There was a pending suggestion on the talk page to drop the K&W problem statement in favor of the Parade problem statement. He dropped the Parade one. I simply flipped this to the (existing) paragraph about the (much better known) Parade one, with one minor change to make it less POV (in the same direction as JeffJor's subsequent edit). Here's a diff to an earlier version [2]. -- Rick Block (talk) 05:32, 21 December 2009 (UTC)
- Rick said "I didn't 'reword' anything ... with one minor change to make it less POV same direction as JeffJor's subsequent edit." Ignoring the self-contradiction, your rewording was in the opposite direction. You added "Some of the controversy was because the Parade version of the problem leaves certain aspects of the host's behavior unstated." You said this as though it was a significant portion of the controversy, which is what MvS specifically denied and what you have no support for. The only thing that contributes to the controversty soem have called "The Parade affair," which is the controversy meant here, is the unintuitive result. You left out the parts that said the problem statement was perfectly clear for the forum in which it was published, which is the "direction of my edits." JeffJor (talk) 12:46, 21 December 2009 (UTC)
- Sorry, wrong diff. The version I started with was this one. This is the correct diff [3]. I changed "A well-known, though ambiguous (Seymann 1991), statement of the problem was published in Parade magazine:" to "A well-known statement of the problem was published in Parade magazine:". The "Some of the controversy" sentence is in both what I started with and your subsequent edit. In general, references do not belong in the lead (see WP:LEADCITE). -- Rick Block (talk) 15:09, 21 December 2009 (UTC)
- I don't care what the diff is, Rick. If you edit a paragraph that has two known flaws in it, and take out only one flaw, you are tacitly approving the other. There is no documented evidence that the MvP statement generated that controversy. She denied it was a signifcant factor. Others inserted the possibility (not the controversy), by failing to interpret the Whitiker statement as it was clearly intended. That means no alternate host strategies, and no probabilities that differ by door number alone. So you can solve the problem by UA/SCA, which are equivalent as G&S said. The only place where any controversy has arisen surrounding anything besides UA/SCA is HERE. That isn't NPOV. It is OR. It does not come from any of your references, because they never said what q was. They only said there ewas a potential for it to make a difference, but it didn't affect the answer. JeffJor (talk) 17:29, 21 December 2009 (UTC)
- I agree intuition is the problem and that rigor is not the answer. IMO (this is WP:OR) the problem statement deliberately forces the reader to think about the specific conditional case where door 1 has been picked and the host has opened door 3, so the player is now looking at two specific closed doors and an open door. And, yes, this case is used as an example.
- The salient features of this case (which apply to any other) are 1) the player doesn't know where the car is with certainty, and 2) there are only two possible choices. The "equal probability" assumption (cf. Falk or Fox and Levav) strongly leads people to the conclusion that the odds must be 50/50 in this case, and therefore any other equivalent case. Note that this reasoning starts with a specific conditional case, and then extends to the unconditional answer not the other way around.
- The unconditional solutions ignore the specific conditional case the problem statement has forced the reader to think about, and jump straight to the (correct) unconditional answer. However, they NEVER reconnect back to the original conditional case—that is, these solutions do not address the mental model most people construct which led them so convincingly to their initial 50/50 conclusion. This is basically a bait and switch approach, leaving people with two choices - trust their "equal probability" intuition, or believe a solution that seems to be true but doesn't specifically address why or how their intuition failed. I think this is precisely what leads to many of the arguments over this problem. Most people are very reluctant to abandon what they see as an intuitively obvious answer. The unconditional solution approach tries to lead people to a different mental model. The other alternative is to address the conditional case head-on, and explain why even in this case the odds are 1/3:2/3. I would like the article to do both of these, in one solution section. As Boris says "The coexistence of the conditional and the unconditional can be more peaceful". -- Rick Block (talk) 21:27, 19 December 2009 (UTC)
I'm Ready for Formal Mediation
I suggest we quit waiting around for the informal mediation. There may never be a volunteer.
Formal Mediation is the next step, then arbitration.
Is there a second to my motion? Glkanter (talk) 15:26, 16 December 2009 (UTC)
Is Rick's diff out of control consistent with our efforts to find an unbiased informal mediator? Or has he made that impossible?
- "==Please help==
- "The situation at talk:Monty Hall problem is really getting out of control..." Rick Block (talk) 15:34, 15 December 2009 (UTC)"
Glkanter (talk) 20:22, 16 December 2009 (UTC)
- I've asked the chair of the mediation committee what a reasonable length of time to wait for an informal mediator might be before proceeding with a formal mediation request. See User talk:Ryan Postlethwaite#Mediation question. -- Rick Block (talk) 02:06, 17 December 2009 (UTC)
Wiki-Ego
I can see why someone who has a great deal of time, effort and pride invested in Wikipedia would be protective of his work. Especially if the only Featured Article which he personally 'sheparded' through the review process was at risk of being dramatically revised. Revised so much, that the FA designation would likely be at risk.
But there is a difference between understanding and condoning. In addition to all the filibustering that takes place on the talk pages, these phrases were used when requesting (supposedly) un-biased assistance as per Wikipedia policy:
- "...This is a featured article that has been through 2 FARCs..." - Request to Mediation Cabal
- "...The situation at talk:Monty Hall problem is really getting out of control..." - 2nd request to Mediation Cabal
- "...At least one of the other editors involved is agitating to proceed with formal mediation..." - 3rd request, sent to Ryan Postlethwaite, Chairperson of the Mediation Committee
So, I can see why an owner of an article would reject all good faith efforts at improved clarity. I just don't agree with the continual passive-aggressive intellectual dishonesty that I have witnessed throughout the 14 months I've been active on this article. I've got a lot of time, effort and pride invested in this article, too. And I'm part of a legitimate consensus for making the proposed changes. That's why I point out, without hesitation, when I think another editor is not behaving in good faith. These aren't personal attacks. They are a recognition of why the article has been so wrong, for so long, despite the efforts of so many 'agitating' editors who disagree with the "shepard's" POV.
Some people will argue that this discussion is out of line. But everything I wrote is supported by diffs. Why fear the truth? I would reply that the criticisms would come from those who favor the status quo for the article. Glkanter (talk) 13:05, 17 December 2009 (UTC)
- This rant is nothing but another in your continuing series of disruptive edits. Please stop. -- Rick Block (talk) 13:43, 17 December 2009 (UTC)
- Call it 'disruptive' if you want. I call it honest. There's been no edit warring by anybody. No profanity on the article's various talk pages. Nothing disruptive at all, other than your endless filibusters, despite a consensus to proceed. Just things you find uncomfortable. All these discussions show good faith by many people that they're looking for a proper Wikipedia solution. The only 'disruption' was Dicklyon's unprovoked savage violation of my MHP talk page new section. I noticed you didn't admonish him at the time. Not a peep out of you, the self-appointed Monty Hall problem Admin and Mediator. But you told a buddy, whom you may have been trying to recruit to mediate this dispute, that 'I chased him away'. Dicklyon takes no responsibility whatsoever for his own reprehensible actions, and you defend him. That's intellectually dishonest. And you know it. Glkanter (talk) 15:12, 17 December 2009 (UTC)
Does anyone object to Formal Mediation?
"Mediation is a voluntary process in which a neutral person works with the parties to a dispute. The mediator helps guide the parties into reaching an agreement that can be acceptable to everyone. When requesting formal mediation, be prepared to show that you tried to resolve the dispute using the steps listed above, and that all parties to the dispute are in agreement to mediate. Mediation cannot take place if all parties are not willing to take part. Mediation is only for disputes about Article Content, not for complaints about user conduct."
Please indicate below:
I am willing to take part in Formal Mediation
- Glkanter (talk) 15:03, 18 December 2009 (UTC)
- Rick Block (talk) 16:03, 18 December 2009 (UTC)
- Martin Hogbin (talk) 16:34, 18 December 2009 (UTC)
- JeffJor (talk) 16:35, 18 December 2009 (UTC)
- Gill110951 (talk) 13:22, 20 December 2009 (UTC)
- Colincbn (talk) 02:26, 22 December 2009 (UTC)
- Nijdam (talk) 17:13, 11 January 2010 (UTC)
I am not willing to take part in Formal Mediation
Thank you. Glkanter (talk) 15:03, 18 December 2009 (UTC)
Nijdam posted an edit to this page nearly 24 hours ago. How much longer do we wait for him to indicate his decision? Glkanter (talk) 11:56, 21 December 2009 (UTC)
Nijdam posted an edit to this page over 48 hours ago. Still no comment or signature on this Formal Mediation. Can we move on without his signature? Is anybody else ready to move this forward? Glkanter (talk) 13:11, 22 December 2009 (UTC)
- I thought we were proceeding. This seems much more important than the RfC below. Martin Hogbin (talk) 13:15, 22 December 2009 (UTC)
- Well, we were. But there's no Formal Mediation unless everyone agrees to it. I see us as stuck, for no apparent reason.
- Yes, that RfC is a distraction. Unfortunately, for me, anyways, it has to be dealt with seriously. I knew this was coming the minute he vandalized my talk page edit, and begged Dicklyon to let me edit my own section as I had written it. All to no avail. So it goes. Glkanter (talk) 13:23, 22 December 2009 (UTC)
The parties that need to agree are the parties listed in the request for mediation. As far as I know this does not exist yet. After creating such a request you notify the named parties about it, see Wikipedia:Requests for mediation/Guide to filing a case. Pre-agreeing to mediation here is nice, but ultimately irrelevant. The parties named in the informal request were Martin, Jeff, Glkanter, Nijdam, Kmhkmh, Father Goose, and myself. I've been assuming Glkanter or Martin were working on a formal request. If this is not the case I'd be willing to write one up, although if someone else would prefer to do this that's fine with me. I might suggest Martin. -- Rick Block (talk) 17:09, 22 December 2009 (UTC)
- It would have been nice of you to present this view either when you signed, or instead of signing, back on the 18th. Why go to all the effort of creating a request until Nijdam, and Kmhkmh indicate they will go along with the decision? I have serious doubts that Nijdam will agree. How hard can it be for them to say? Just more stalling of the inevitable. Excellent job this time, Rick! Glkanter (talk) 17:23, 22 December 2009 (UTC)
- Based on the level of hostility you've exhibited toward me, I thought you would prefer someone else write up the mediation request. That's what this edit (from last Thursday) meant, where I provided a link to the appropriate procedure. I assumed you'd read this and that you or Martin were working on it. Per below, Martin is OK with me writing it up. Are you? -- Rick Block (talk) 20:09, 22 December 2009 (UTC)
- Rick, I am not familiar with the procedure, and would be quite happy for you to do it. If you want me to do it let me know and give me some clues what to do. Martin Hogbin (talk) 17:56, 22 December 2009 (UTC)
Informal mediation still a possibility
I note that User:K10wnsta has offered to serve as an informal mediator for the case at Wikipedia:Mediation_Cabal/Cases/2009-12-06/Monty_Hall_problem#Discussion.
If nobody had an objection to proceeding with the case at that venue, it would probably allow things to get under way sooner.--Father Goose (talk) 04:19, 24 December 2009 (UTC)
Re: Wikipedia:Mediation_Cabal/Cases/2009-12-06/Monty_Hall_problem
In my brief overview (I haven't delved into archives), the discussion appears to have remained civil and, more important, cooperative in seeking a means of negotiation. If everyone is willing to excuse the sluggish response to your request for informal mediation (blame it on the holidays ;) ), I'd be happy to work with you in resolving the dispute. However...
You waited over two weeks for assistance at MedCab and, procedurally, are justified in pursuing formal mediation. If someone has already applied significant effort in preparing for that, I understand if you wish to continue in that direction.
--K10wnsta (talk) 20:29, 25 December 2009 (UTC)
- There are 11 archives to this discussion, covering 6 years. While technically 'civil', it is very contentious. I appreciate your offer to help, but I'm not sure there would be a result worth the time investment you would need to make. Honestly, if I may, your original comment about this puzzle being 'mathy' did not create confidence in this reader. And I'm the least Mathematics educated person on this talk page.
- But, this is just one person's opinion, offered in good faith. I'd hope I can support whatever the consensus decides on your generous offer. Glkanter (talk) 17:33, 26 December 2009 (UTC)
- Hehe, well, the 'mathy' description stemmed from reading about a dispute involving 'mathematical sources' and 'conditional probability' in what appeared to be an article about the host of a campy game show. I couldn't fathom how the subjects were related (and even questioned my recall of the host's name). It was certainly not intended to express any personal disdain for mathematics - in fact, I enjoy and excel in most math-related fields (notably algebra, geometry, and statistics).
--K10wnsta (talk) 21:07, 28 December 2009 (UTC)
- Hehe, well, the 'mathy' description stemmed from reading about a dispute involving 'mathematical sources' and 'conditional probability' in what appeared to be an article about the host of a campy game show. I couldn't fathom how the subjects were related (and even questioned my recall of the host's name). It was certainly not intended to express any personal disdain for mathematics - in fact, I enjoy and excel in most math-related fields (notably algebra, geometry, and statistics).
- As one of the long term involved editors I would welcome some mediation. I am not sure what your understanding of maths and probability is like but the Monty Hall problem has been described as the world's most tenacious brain teaser. It will therefore be necessary for you to first get your head round the basic problem, if you are not already familiar with it. Note that nobody here disagrees with the basic numerical answer under the 'standard rules'.
- If you proceed with mediation, it would be interesting for you to start by reading the article (or possibly a previous version ) through to see how good an understanding of the problem it gives you before consulting other sources or talking to anyone about it, as the debate is essentially about how well this article addresses the basic problem. You currently have the advantage of seeing this article as a newcomer but once you have been drawn into the debate you will quickly lose that viewpoint. This suggestion is not intended to be an attempt to 'get in early' with my POV. Perhaps someone on the 'other side' could confirm that they would be happy for you to take this approach. Martin Hogbin (talk) 11:02, 28 December 2009 (UTC)
- I have read the article and now understand how Monty Hall could be associated with conditional probability (see my reply above). I haven't yet delved into the actual dispute as I prefer remaining sequestered from it until we get past the informal formalities (eg. all interested parties agreeing to participate in the mediation process).
--K10wnsta (talk) 21:07, 28 December 2009 (UTC)
- I have read the article and now understand how Monty Hall could be associated with conditional probability (see my reply above). I haven't yet delved into the actual dispute as I prefer remaining sequestered from it until we get past the informal formalities (eg. all interested parties agreeing to participate in the mediation process).
Kanov Is Wrong
The article states (without citation) that Kanov stated that in the "Ignorant Monty" case, swapping still yields a 2/3 chance of winning - but a quick simulation of all cases reveals this to be wrong: suppose I pick door 1, and Monty opens door 2 without knowing what is there but reveals a goat (all other permutations are equivalent to this): the car will now be behind either door 1 or door 3 with a 1/2 probability. --New Thought (talk) 09:44, 19 December 2009 (UTC)
- You are quite right. Because of all the argument here nobody has noticed a simple error. There seems to be a section based on a the supposed opinion of a mysterious Kanov. I will remove this unless someone can explain why I should not. Martin Hogbin (talk) 10:54, 19 December 2009 (UTC)
- I've wondered about this, going back to the summer. If I recall correctly, Marilyn vos Savant says it's 1/2 because of the plays that get eliminated by Monty revealing a car. I might suggest that once the contestant is faced with the two doors and a revealed goat, it's the same 1/3, 2/3 as the original MHP. Then I have to figure out how this is consistent with Deal or no Deal, which says there is no advantage to switching.
- Vos Savant is correct. If Monty chooses any unchosen door randomly you have to decide what to do if he reveals a car, asking the player whether she wants to change after a car has been revealed is pointless. Easiest would be to replay those games from the start. Games where Monty reveals a car are therefore discounted. These games can only be ones where the player has originally chosen a goat because, if the player has originally chosen the car, the host cannot reveal it. Thus in the 'Ignorant Monty' case we remove some games where the player originally chose a goat but none where she originally chose the car, thus her chance of having originally chosen the car goes up. Martin Hogbin (talk) 18:56, 19 December 2009 (UTC)
- I've wondered about this, going back to the summer. If I recall correctly, Marilyn vos Savant says it's 1/2 because of the plays that get eliminated by Monty revealing a car. I might suggest that once the contestant is faced with the two doors and a revealed goat, it's the same 1/3, 2/3 as the original MHP. Then I have to figure out how this is consistent with Deal or no Deal, which says there is no advantage to switching.
- Martin - I don't think you're addressing the issue. I believe the confusing scenario is a specific show, say last Tuesday's, where Glkanter was the contestant. On this show, he's initially picked a door, say Door 1, and Monty has forgotten where the car is. He says "Oh dear, I've forgotten where the car is. I hope this works out OK - Carol, please open a random unchosen door". And, fortuitously, the door that is opened, say Door 3, reveals a goat. There is no decision about what to do if the car is revealed, because the car simply wasn't revealed. Glkanter's initial choice has a 1/3 chance of having been correct. The chance the car is behind the open door is clearly 0. The other one must have a 2/3 chance.
- That is the issue I addressed. I explained why the probability that the player has chosen the car increases when Monty reveals a goat by chance. Even for the one-off case the fact that Monty has chosen randomly but in fact revealed a goat means that the player is more likely to have chosen the car. Martin Hogbin (talk) 20:15, 19 December 2009 (UTC)
- Martin - I don't think you're addressing the issue. I believe the confusing scenario is a specific show, say last Tuesday's, where Glkanter was the contestant. On this show, he's initially picked a door, say Door 1, and Monty has forgotten where the car is. He says "Oh dear, I've forgotten where the car is. I hope this works out OK - Carol, please open a random unchosen door". And, fortuitously, the door that is opened, say Door 3, reveals a goat. There is no decision about what to do if the car is revealed, because the car simply wasn't revealed. Glkanter's initial choice has a 1/3 chance of having been correct. The chance the car is behind the open door is clearly 0. The other one must have a 2/3 chance.
- Glkanter - is this more or less what you're thinking? -- Rick Block (talk) 19:41, 19 December 2009 (UTC)
- The original sources are correct. "Kanov" is presumably the name of the anonymous editor who put this in the article (yesterday). I've reverted this change.
- Glkanter - the probability this is talking about is precisely the one applying to the contestant faced with two doors and a revealed goat (in a case where the host has randomly, but successfully, opened a door revealing a goat). Perhaps Martin or JeffJor could explain to you why the "combining doors" solution (or any of the other unconditional solutions) do not apply, and why the probability is indeed 1/2 in this case. This is not a sarcastic suggestion - I could try to explain it but I doubt that you'd be willing to listen to me. -- Rick Block (talk) 17:24, 19 December 2009 (UTC)
Even though this problem is always described as "counter-intuitive", I find it interesting that EVERYONE on Earth understands the problem intuitively if you look at it another way: When you watch Deal or No Deal, the only reason it's suspenseful is because the person opening a case does NOT know if there's a big number inside that case. If you were on a Monty Hall Problem game show, and picked door #1, and the host said "I'm going to open a door now... hmmm... number 2" (ignorant monty - or at least from the player's POV, you must assume ignorant monty), you would be worried and suspense-filled that he might open the door with the car. When he doesn't, you feel relief. However, if Monty said, "Now, let me open a door with a goat in it... number 2" you would feel no suspense. He has told you the door has a goat, you know it's a goat, and it has no suspense. This is because there is no risk in him opening a door. He will always open a goat door. If your odds of having a goat behind your original selection improved, you'd be excited after he revealed a goat, but because he knows it's a goat, you feel no more excited about your first choice than before he opened the good. This is an example of how people DO intuitively understand this, but then don't recognize the ramifications of this feeling when offered the choice to switch observe below:
- Interesting. Martin Hogbin (talk) 20:19, 19 December 2009 (UTC)
Here is an analysis of all cases when the car is behind Door number 3 (logic dictates that there are tables for the car behind behind doors 1 and 2 that have identical probabilities (for the appropriate doors). The number at left is the door you choose; the number at the top is the door Ignorant Monty opens. The result is whether you should switch ("y" or "n"). "c" represents Monty revealing the car.
1 | 2 | 3 | |
---|---|---|---|
1 | y | c | |
2 | y | c | |
3 | n | n |
1,1 2,2 and 3,3 are greyed out, because he can't open the door you chose. As you can see, there are two cases where switching nets you a car, and two cases when it does not. There are also two cases where he reveals the car ("c") and you are (presumably) not offered a choice, as the car location is now known. Ignorant Monty has a 1/3 chance of revealing a car and ending the game. ONCE that does not happen, there are four possible cases left, 1/2 of which require switching to win, 1/2 of which require keeping to win. This is the conditional probability of "What is the probablity that switching will win GIVEN that Montry did not reveal the car?" The absolute probability is absolutely true - even with ignorant Monty, switching will win you the car 1/3 of the time - 1/3 of the time staying will win, and 1/3 of the time Monty will reveal the car, and you will not get the option.
Regular Monty has 0 chance of revealing a car. While regular monty has a decision to make SOMETIMES (if you select the car, he must pick which goat to reveal), as long as his pick is random, the result of his pick are both the same: you should still not switch, (so the conditional probability of winning by switch IF monty randomly selects one door or the other is 0 in both cases - you can't win by switching). Thus, if you picked right the first time, don't switch. If you picked wrong the first time, DO switch. Therefore, 1/3 of the time, don't switch, 2/3 of the time, switch.
This is true in the ignorany monty case also: If you picked wrong (2/3), do switch. If you picked right (1/3) don't switch. However, half of the time when you pick wrong (half of 2/3 = 1/3), Monty reveals the car, and you don't get to make a choice. Therefore, IF you get the option to switch (only 2/3 of the time will you get this far), then the odds are even between keeping (1/3) and switching (1/3) (the other third is monty reveals the car). TheHYPO (talk) 19:47, 19 December 2009 (UTC)
- As a PS: I thought I'd explain the difference in why one is conditional and one is not: remember that if you have four cases: in order to say that any of them has a 1 in 4 chance of occuring, there MUST be an equal chance of each occuring. In the original monty hall problem (let's say car is behind door 3):
- If you pick door 1 (1/3 chance), he MUST open door 2 100% of the time (thus, also 1/3 chance).
- If you pick door 2 (1/3 chance), he MUST open door 1 100% of the time (thus, also 1/3 chance).
- If you pick door 3 (1/3 chance), he could open doors 1 or 2 (if he picks randomly, 50% chance of either).
- As you can see, your choice of doors all have an equal 1/3 chance of occuring, there are four 2nd step cases ([you:1 monty:2], [you:2, monty:1], [you:3, monty:1], [you:3, monty:2] with DIFFERENT probabilities of occuring (1/3 each for the first two - both of which say "switch", 1/6 each for the second two - both of which say "don't switch"). Thus some people claim that logically, two of those four 2nd step cases say "switch" and two say "stay" - that's 50/50. But two cases occur half has often has the other two. In the Ignorant Monty problem, all 6 cases in my table above are equal probability (1/6). This is because when you pick a "wrong" door, he has two options, not one. so your 1/3 choice results in two 1/6 choices for Ignorant Monty (one of which reveals the car and ends the game). If he DOESN'T reveal a car, you're left with four cases with initial probability of 1/6, and thus, each case NOW has a 1/4 chance (two win by switch, two lose by switching, thus 1/2 chance of winning by switching.) TheHYPO (talk) 20:08, 19 December 2009 (UTC)
Technology required new section.
I believe I correctly summarized vos Savant.
Let's re-apply some things we've learned: 'Suppose you're on a game show...' Still true? Contestant's SoK? 'Random' would equal Deal or No Deal. 'He's drunk' or 'forgetful' might not be communicated to the contestant. Then it's still the MHP from the contestant's SoK.
What exactly is the revised problem statement? —Preceding unsigned comment added by Glkanter (talk • contribs) 20:04, 19 December 2009 (UTC)
- The probability from the contestant's point of view depends on the contestant's knowledge of the game rules. If the contestant is told the host knows what is behind the doors and will always choose a goat then the probability of winning by switching is 2/3 from the contestant's POV (SoK). If the contestant knows the host is choosing another door randomly (and then is relieved to see a goat revealed - see comment above) the probability of winning by switching is now 1/2. Is that your understanding? Martin Hogbin (talk) 20:28, 19 December 2009 (UTC)
By 'random' I mean 'car or goat revealed by Monty'.
I don't thìnk your summary or Rick's summary reflect my thoughts on this puzzle. Have I been obtuse? Why summarize me at all? —Preceding unsigned comment added by Glkanter (talk • contribs) 21:03, 19 December 2009 (UTC)
- It was meant to be an explanation of why the probability of winning by switching is 1/2 if the host chooses an unchosen door randomly (that is to say he might choose a car or a goat). You seemed uncertain as to whether you agree with this statement. Do you agree? Martin Hogbin (talk) 21:16, 19 December 2009 (UTC)
- If the contestant is informed (that is, it's a premise of the puzzle) that the host is opening doors randomly, and may reveal a car, then it's Deal or No Deal. Rick had a very elaborate scenario for the 'drunk' or 'forgetful' Monty. What is communicated to the contestant prior to his decision? Is this still a game show, then? How is it stated as premises?
- I'm just pointing out that 'random', or 'forgetful' still require 'formalized' problem statements, which may be different. Absent that, either, or any answer may be correct. I'm not real good at multi-tasking. I just had some thoughts that could have developed into something. But until we have the underlying MHP squared away, I find this personally distracting. Glkanter (talk) 21:27, 19 December 2009 (UTC)
Many more words
Until now only Boris has shown the derivation of a solution in formulas, using symmetry. This leads to the conclusion - as I BTW showed a million comments ago - that the conditional probability we are interested in is equal to the unconditional and hence may be easily calculated. It doesn't show the conditional probability is not needed. All others come with words, words, .... Nijdam (talk) 17:25, 20 December 2009 (UTC)
- Yes, Nijdam, too many words. How about your signature agreeing to Formal Mediation? Glkanter (talk) 17:33, 20 December 2009 (UTC)
- There are so many words here because people don't explicitly state the assumptions they use to get their solution. Conditional and unconditional probabilies are equal and easy to find by symmetry in a special, nice, symmetric case. I took a look at the Selvin paper. I like the intro very much indeed, I don't like the solution. He does not say in advance what assumptions he is making, you can only guess them by studying his proof. He enumerates the cases and solves the problem by counting. This means that he is assuming that all cases are equally likely. This means that he is assuming the car-key is hidden uniformly at random, that the quiz-player chooses a box uniformly at random independently of the location of the key, and that the quiz-master opens a box uniformly at random out of those available to him, given the previous two choices. Why I don't like Selvin's solution? Because it depends on his strong assumptions. We only need to assume that the first box you pick has 1/3 probability of having the key, in order to guarantee that always switching gives you 2/3 probability of ending with the key. Proof: everytime you would have got the key without switching you don't get it with switching, and vice-versa. I guess that most players think that they have a 1/3 chance of picking the right box first time. Whether or not this is true could be empirically verified. This is both real and theoretical game theory. Gill110951 (talk) 06:06, 22 December 2009 (UTC)
How About A Temporary Editing Freeze On The Article
I don't understand why all this article editing is taking place without being discussed.
While we 'old guys' are working towards a formal WP solution, newer people are editing at will.
This seems unproductive, not good for the article or readers, and distracting.
Any support for a temporary freeze? Is this even plausible? Thanks. Glkanter (talk) 12:25, 21 December 2009 (UTC)
- I think it would be wise for editors to wait, as after mediation there may well be major changes (I hope) and they would the be wasting all their effort. Martin Hogbin (talk) 15:31, 21 December 2009 (UTC)
- I guess I am "a newer person, editing at will". My excuse: when I see factual incorrectness or incompleteness in the existing article I make small edits - I don't touch the main structure. I obviously won't/can't object if those contributions get thrown out later. What I do like is the draft construction page, http://en.wikipedia.org/wiki/Talk:Monty_Hall_problem%5CConstruction That seems to me to be a very useful step: make a fresh start aiming to accomodate the various opinions which are around. It is precisely because there are so many different ways to formulate a Monty Hall problem that it is so attractive. Gill110951 (talk) 19:35, 21 December 2009 (UTC)
Rick Just Filed This RfC On Me
He's expecting Dicklyon to 2nd it. I see a lot of unintended irony here. I had just created a new section on the talk page with 3 edits. Then, here's what I call Dicklyons's unprovoked vandalism on my talk page edits:
It's all right here: Is This Chronology Correct?
So, if anybody wants to put in a good word for me, I'd be much obliged. Please note, I'm pretty sure I will get this promptly dismissed, but any support is appreciated. Glkanter (talk) 04:25, 22 December 2009 (UTC)
- Anyone unfamiliar with this process might want to review Wikipedia:Requests for comment/User conduct/Guidance2. It is certified now which means it won't be closed until the criteria at Wikipedia:Requests for comment/User conduct/Closingis met. Anyone is welcome to comment. -- Rick Block (talk) 05:41, 22 December 2009 (UTC)
On this RfC/U, Rick Block and Dicklyon are trying to make a case that I am disruptive, don't edit the article often enough(?),incivil, interrupt consensus building, chase other editors away, contribute nothing of value, too aggressive with my POV, have bad breath, etc. I'm holding my own on the RfC. It's gotten pretty ugly. So, if anybody would like to drop a supportive word about good ol' Glkanter, now would be a good time. By reading the RfC, you will also learn a lot about the inner thought processes of some well known editors. Thanks. Glkanter (talk) 22:22, 26 December 2009 (UTC)
Please see this new section on the Arguments Page
Thank you. Glkanter (talk) 15:10, 23 December 2009 (UTC)
I'll Bet That 'Paradox' and 'Game Theory' Are Mutually Exclusive And Opposites
I think there are 2 POVs regarding how to 'cherish' the MHP paradox.
Some of us, including myself, love the simplicity. Nothing happens. Heated Arguments over 1/2 vs 2/3 ensue. More than once, even.
Other people like the complexity, and 'what ifs' that the MHP could be with just a little tweaking. The permutations can approach Game Theory scenarios.
Since it was a great paradox before Morgan and conditional, I consider the 'simplicity' people the ones who accurately support how Selvin's MHP paradox should be presented in the Wikipedia Article. Glkanter (talk) 15:35, 23 December 2009 (UTC)
There's Only 2 Things Being Debated Anymore
1. The simple solutions are not solving the correct problem.
2. Morgan's paper, published in 1991, can claim to recognize and describe the Monty Hall Problem Paradox, first published by Selvin in 1975, equally as well (and equally importantly) as Selvin's original paper, which relied only on simple solutions.
I'd like to see the people arguing in support of those 2 arguments come out and directly say it. Once you clearly state your positions, the other editors, using reliably published sources can then address your objections to the proposed changes. Glkanter (talk) 18:32, 23 December 2009 (UTC)
This Is Why They Can't Be Represented in the Article 'Equally'.
Simple solution is not a solution at all
"This is the same topic discussed in more detail three sections down (about the subtly different question), and indeed Morgan et al. argue the "simple" solution is not a solution at all." -- Rick Block (talk) 16:28, 26 October 2008 (UTC) —Preceding unsigned comment added by Glkanter (talk • contribs)
- Many sources do give simple solutions but you try to use one source to veto all others by saying, 'Morgan et al. argue the "simple" solution is not a solution at all'. This is your POV but it is not what sources (note the plural) all say. Some sources give the simple solution as the correct one. These sources should be properly represented in the article. Martin Hogbin (talk) 13:42, 24 December 2009 (UTC)
- I have no idea why Glkanter reposted this old quote. I think we all agree the POV of the article should not be that the unconditional solutions are incorrect. On the other hand, the article does need include the POV expressed by Morgan et al., and Gillman, and Grinstead and Snell (this is their POV - and whether any editor here agrees with it or not is completely irrelevant) that the unconditional solutions are addressing a slightly different problem than what they think the problem is. I think the only question here should be how best to do this in an NPOV manner. What I hear you (and Glkanter and Jeff) arguing is that they're wrong (sorry, per WP:OR and WP:V Wikipedia doesn't care what you think about their POV), or that their POV should be excluded (sorry, per WP:NPOV Wikipedia must include all significant views). I'd be delighted to work toward a more NPOV treatment. -- Rick Block (talk) 17:27, 24 December 2009 (UTC)
- Is it typical for a FA article to need a '...more NPOV treatment.'?
- This is yet another example of exactly the kind of disruptive behavior Wikipedia:Requests for comment/Glkanter is about. I'm offering to help you achieve your goal. What would you say you're doing? I'd call it trolling. I can't speak for anyone else, but I'm extremely tired of it. Please stop. -- Rick Block (talk) 19:53, 24 December 2009 (UTC)
- Actually, Rick, this is another example of you claiming whatever fits your current needs. Here you sound the alarm about the potential for the article to have a POV Last Paragraph. As if the Wikipedia world as we know it would collapse if that happened. But, when you acknowledge that the article currently has a Morgan POV (above), you're not quite as concerned about fixing it in a timely manner. Glkanter (talk) 22:47, 24 December 2009 (UTC)
@Rick, you seem to be putting up an Aunt Sally (Strawman argument). You seem to be implying that I want to remove the POV of Morgan and others who agree with them from the article. That is not the case. I have always suggested that the article should start with the simple non-conditional solutions and then, after discussing these thoroughly, move on to the conditional case discussed by Morgan and others. It is clear, from your reposted quote above (I had not noticed that it had been reposted) that you believe that the Morgan paper should somehow veto or overrule all other sources no matter what they say. Martin Hogbin (talk) 22:54, 26 December 2009 (UTC)
- Am I somehow not being clear here? What I believe is that the article should represent as a POV what it is that Morgan et al. (and Gillman, and Grinstead and Snell) say. What they all say is that the unconditional solutions don't exactly address the problem. Morgan et al. go so far as to say the "simple" solution is a "false" solution. In the quote above, I'm saying that Morgan et al. say this, not that I think this POV should veto or overrule all other sources. Whether you agree with what they say or not, do you at least agree that this is what these sources say? I assume you understand that saying that these sources say the simple solution is no solution is not the same as the article taking this as its POV. -- Rick Block (talk) 00:21, 27 December 2009 (UTC)
- I am fine with Morgan's POV in the appropriate place but not in the simple solution section. Martin Hogbin (talk) 17:51, 3 January 2010 (UTC)
- There's a huge dichotomy between what you write above, Rick, and how you've edited and protected Morgan's POV throughout the article, even down to the FAQs. Plus, it contradicts your long standing and still existing arguments that Morgan's, as the sole peer-reviewed source, is the prevailing POV, and deserves prominence in the article. Plenty of sources apparently don't find Morgan's arguments all that convincing, as they keep publishing simple solutions. I'm not publishing them, professionals are publishing them. It's not my POV, it's the sources' POV. That's why there is an 'editing' function required more so for some articles than others. Editing means more than proper footnoting. And sometimes it means telling a story as it happened, chronologically. That's not POV, that's editing. Glkanter (talk) 18:18, 3 January 2010 (UTC)
I Guess I'd Better Start Editing The Article
In the RfC that Rick Block and Dicklyon filed on me RfC Glkanter one of the 'complaints' was that I argue on the MHP talk pages too much, at the expense of actually editing the MHP article. The associated 'remedy' was that I modify the MHP article more frequently and discuss my reasons for doing so less often.
Now, that's no reason to slap me with an RfC, but the point is well taken. I've asked for a 'freeze' on the article of some sort at least twice in the last couple of weeks. Meanwhile, some editors just make edits without discussing them first.
So, consistent with my stated understanding of the various literature on the MHP, and in accordance with Rick's criticism/suggestion as conveyed via Wikipedia's formal RfC procedure, I will begin to thoughtfully edit the article as I understand the consensus has approved. Glkanter (talk) 15:51, 24 December 2009 (UTC)
How about I start with the FAQs on the talk page? That looks like pure Morgan POV, a clear violation of NPOV. Anybody want to clean it up, or should I just delete it? Glkanter (talk) 16:47, 24 December 2009 (UTC)
- I'm sorry, but how do you find the FAQ a "clear violation of NPOV"? Would it help if it said "according to these sources" a couple of times? There is no particular requirement that talk page FAQs adhere to NPOV, but I'd be happy to work with you to make this more NPOV if it bothers you (which it clearly seems to). -- Rick Block (talk) 17:39, 24 December 2009 (UTC)
- No thanks. I'm going to use the RfC as an opportunity to learn. Since you feel I should be sanctioned because I've only made '6 article edits out of about 1000 talk page edits', I'll go it alone, without all that 'discussing' you find so offensive from me. Glkanter (talk) 22:39, 24 December 2009 (UTC)
Here's another one. Id like to change the 'Simple solution' heading to something like 'Original Paradox explanation' or 'Selvin's Proof' or 'vos Savant's Popular Solution'? I'd like to get the point across concisely that it was this level of understand from which all the excitement about the paradox came. Not to be confused with the 'conditional solution' or, non-solution without the equal goat door constraint being equal to exactly 1/2, that came out some 15 years later. Glkanter (talk) 16:03, 25 December 2009 (UTC)
Then a transition section that says 'For many people, this is all the understanding they need, and was Selvins and vos Savant's point. Others may want to continue further into this article...' And as long as there's no bad-mouthing the 'original' solutions, you 'conditional' guys can pretty much do what you want with the article from there. Glkanter (talk) 16:10, 25 December 2009 (UTC)
FAQ page boilerplate
This is most of the 'greeting' to the talk page of the FAQs. Probably only seen by other editors.
- "This page is an FAQ about the corresponding page Monty Hall problem."
- "It provides responses to certain topics being brought up again and again on the talk page, sapping many editors' time and energy by forcing them to respond repeatedly to the same issues. The FAQ addresses these common concerns, criticisms, and arguments, and answers various misconceptions behind them."
I think this can be improved. Anybody mind if I take a shot at it? Glkanter (talk) 23:32, 26 December 2009 (UTC)
- This is standard boilerplate from Template:FAQ page. Are you suggesting changing the standard boilerplate (used on over 100 pages) or replacing the standard message with something custom for this page? -- Rick Block (talk) 00:29, 27 December 2009 (UTC)
- It's a variant of template:FAQ, one of many templates intended for use on talk pages. See Wikipedia:Template messages/Talk namespace. Template:FAQ2 is another version. - Rick Block (talk) 01:58, 27 December 2009 (UTC)
- No, there is no rule against changing the text in templates. I'm sure there are plenty of people who have these templates on their watchlists. If you make a change anyone objects to they'll revert it. Whatever change you make will show up on every talk page the template is used on, so don't change the text to be less generic. -- Rick Block (talk) 05:01, 27 December 2009 (UTC)
- I'm saying if you edit template:FAQ page the text will appear on any page that transcludes this template (anything marked as "transclusion" here). This is a feature of the MediaWiki software used to run this site. This is both theory (in the sense that it is a known feature of the software) and something I have personally experienced, many hundreds of times. -- Rick Block (talk) 18:50, 27 December 2009 (UTC)
I appreciate your help with this, Rick. I'm suggesting we would edit . What then? Glkanter (talk) 19:27, 27 December 2009 (UTC)
- If you edit Talk:Monty Hall problem/FAQ and change
<noinclude>{{FAQ page}}</noinclude>
- to something else, e.g.
<noinclude>blah blah blah</noinclude>
- "blah blah blah" will only show up on the MHP FAQ page. However, because the text you're talking about is inside the "noinclude" tags it does not appear when you're viewing this page (Talk:Monty Hall problem), even if you click the "show" link at the top of this page (scroll up to the top of this page and try it!). The bottom line is you only see this text if you're editing the FAQ page (and previewing your edit), or directly viewing the FAQ page as opposed to the talk page (there's no link to it, so I'm not sure how this would happen). I might suggest that whatever you think of this text, it's not worth worrying about. -- Rick Block (talk) 19:56, 27 December 2009 (UTC)
Variants
Variants - Slightly Modified Problems section.
Since the MHP is from the contestant's POV, there should be some narrative about what the POV's in this whole section represent. Are they the contestant's? Is it a premise in each different problem that it's no longer the contestant's POV? What about addressing the Monty Hall problem from 'not-the-contestant's POV' for comparison purposes? This would be beneficial to the readers, I believe. Glkanter (talk) 16:55, 27 December 2009 (UTC)
- Can we call this "state of knowledge", not "POV" (to distinguish from the local Wikipedia meaning of POV)? In all cases what is meant is the probability given everything included in the problem statement. This is perhaps most literally the SoK of the puzzle solver, but presumably matches the contestant's SoK as well. The "MHP" is also from the puzzle solver's SoK, so there's really no difference. If this is not clear it wouldn't hurt to try to clarify it, but I don't think anyone should be confused about this since it is how mathematical word problems are universally treated. If it's important to the problem to take some particular perspective, the problem says to. For example, in vos Savant's "little green woman" scenario [4] if the player has picked door 1 and the host has opened door 3 we (the puzzle solver) know the probabilities are split 1/3 (door 1) and 2/3 (door 2) but the question is what are the little green woman's chances of randomly picking the door with the car, not what is the probability the car is behind door 1 or door 2. -- Rick Block (talk) 19:28, 27 December 2009 (UTC)
- In Selvin's and vos Savant's MHP, what the reader knows and what the contestant knows are both consistent with "Suppose you're on a game show..." Every host/producer decision is described as 'random'. That's no longer true with the 'variants' where the reader becomes aware of some host bias. The contestant, of course, cannot. Hence, I disagree with your above explanation.Glkanter (talk) 19:37, 27 December 2009 (UTC)
Editing the MHP FAQs
Rick, the current text includes this:
- "The point of introducing this variant is to show the difference between the unconditional and conditional questions. In this variant, these questions have different answers exposing the difference between unconditional and conditional solutions."
I still disagree that using a different problem is a means of challenging a particular problem. Originalists would argue that all you've demonstrated is the difference between puzzles with different premises. I would further argue that with the contestant being aware of Monty's left door bias, this is no longer the MHP about a game show that Selvin and vos Sovant made so famous. Glkanter (talk) 06:38, 27 December 2009 (UTC)
Are there 3 published solutions?
Selvin's - simple: 2/3 & 1/3, always switching doubles your likelihood of getting the car
Morgan's - conditional, no symmetry: between 1/2 and 1 (?), never to your disadvantage to switch
Morgan's - conditional, with symmetry: 2/3 & 1/3, always switching doubles your likelihood of getting the car
Have I summarized the above properly? Glkanter (talk) 11:03, 27 December 2009 (UTC)
If so, maybe the article could transition from:
Simple, to conditional - with symmetry (they are equivalent), to conditional - no symmetry (leftmost door variant). Glkanter (talk) 11:30, 27 December 2009 (UTC)
- This is the current structure of the article, so I don't get what you're suggesting (change the article to be like the article?). The conditional with symmetry solution dates to Selvin as well. -- Rick Block (talk) 20:24, 27 December 2009 (UTC)
- Rick, rather than make controversial changes or deletions to the article's text, I am trying to make it clearer to the reader how the whole 'Morgan' controversy started. I thought Morgan's whole point was that Selvin and vos Savant overlooked something? So, I'm just suggesting to actually add a 3rd solution section, for increased overall clarity. Glkanter (talk) 20:41, 27 December 2009 (UTC)
Some questions for you all
In an attempt to see exactly who thinks what I have set up some questions on User:Martin_Hogbin/Monty_Hall_problem/dissenters. Everyone is welcome to add their answers. Please comment briefly only in the comment section and have discussions about the questions on the associated talk page.
Whether we have external mediation or not I am sure it will help if everyone answers the questions on this page. I am trying to determine of we have two distinct camps, a single axis of opinion, or just randomly scattered views on the subject. Are there any other questions that editors feel will help sort out the differences of opinion here? I have just added a few extra ones. Martin Hogbin (talk) 11:36, 28 December 2009 (UTC)
Is the term 'Variant' as used in the MHP a common usage?
I disagree with your recent reverts to the article, Rick.
I just checked the Morgan paper, and they do not use the word 'variant' or any derivative of it when describing the problems.
I think this is an uncommon usage, and does not clearly indicate to the reader exactly what is being described. I don't think adding 'Slightly Modified Problem' to a heading, and replacing 1 instance of 'variant' in the article also with 'slightly modified problem' is 'pointy'. Different than your POV, perhaps, but that does not necessarily make it, or any other edits I may make in good faith, 'pointy'. Glkanter (talk) 19:45, 27 December 2009 (UTC)
- Yes, "variant" is common usage. Pointy was referring to the dates. Sevlin's 2nd letter has a conditional solution, so saying the "probabilistic" solution dates from 1991, or is not the "original" solution, or addresses only a variant is a clear attempt to diminish this solution which is not only a violation of NPOV but is factually false. I would appreciate it if someone (anyone) would revert this change. If Glkanter reverts again without further discussion here I'll report him to Wikipedia:Administrators' noticeboard/Edit warring for edit warring. -- Rick Block (talk) 20:15, 27 December 2009 (UTC)
What's Wrong With Adding Dates For Clarity?
Explain the problem to me please. Glkanter (talk) 01:45, 28 December 2009 (UTC)
- Per above, the date you're adding for the conditional solution is simply wrong. Both the "popular" (unconditional) and conditional solutions date to 1975, both to Selvin. If you're going to add dates, you need to add 1975 for both which makes it completely redundant. You seem to be trying to insert your completely made up chronology (#Is This Chronology Correct?) into the article. -- Rick Block (talk) 04:31, 28 December 2009 (UTC)
- So this whole time 'conditional v unconditional' has really been 'Selvin v Selvin'? No way. That's never been your stated intent. Or the way your POV article is written. It's always been 'Morgan v Selvin/vos Savant'. Adding two simple dates to two headings makes it clear where it all started. And it wasn't with Morgan. So the dates help the reader, and do not hurt the article. Your response is not credible based on your previous arguments for many years. Glkanter (talk) 12:56, 28 December 2009 (UTC)
- Rick, you have already acknowledged a Pro-Morgan POV in the article. And offered to help me edit. Why must you continue to confound my very modest efforts at improving the article by removing this POV? Clarifying what a so-called 'variant' is and adding dates are non-antagonistic efforts to improve the article. They just happen to be different than your preference. Your actions, especially calling out to 'anyone' for 'revert' help seem to show ownership issues. Just let me edit out the POV in good faith, OK? Glkanter (talk) 13:20, 28 December 2009 (UTC)
- This whole time, conditional v unconditional has always been different. In Selvin's second letter (which I've previously pointed you to, to refresh your memory there's a copy here), he says he received "a number of letters" including several who "claim my answer is incorrect". Like vos Savant, he says "The basis to my solution is that Monty Hall knows which box contains the keys" but unlike vos Savant he goes on to say "and when he can open either of two boxes without exposing the keys, he chooses between them at random" (emphasis added). Also unlike vos Savant he goes on to present a solution using conditional probability which he calls an "alternative solution" to the solution in his first letter "enumerating the mutually exclusive and equally likely outcomes".
- The issue Morgan et al. address is that vos Savant's solution, and her subsequent defense of it, and the popular discussion at the time (1990 and 1991), completely overlooked the critical assumption that makes the unconditional and conditional solutions the same, i.e. that the host must choose between two boxes (two goats doors) randomly. Selvin knew this and acknowledged it in his second letter. Martin Gardner knew this and addressed it in his version of the Three Prisoners problem. vos Savant blew it, and both Morgan et al. and Gillman called her on it. That's what the "Morgan controversy" is about. Morgan et al. and Gillman both examine the consequences of omitting this assumption, in the process showing why it's critical and how the unconditional and conditional solutions are different. The unconditional solution is NOT saying that every player who switches has a 2/3 chance of winning, but that the average across all players is 2/3. The conditional solution shows that the chances are the same (2/3) for each player only if the "equal goat" assumption is made. Even without it, players who switch will win on average 2/3 of the time (and if they switch they're never worse off), but to say a player who picks door 1 and sees the host open door 3 has a 2/3 chance of winning by switching is a conditional statement and requires this assumption. The assumption can be explicitly part of the problem description (as per the Krauss and Wang version) or implicitly assumed because of symmetry or the principle of indifference, but the statement is still a conditional statement. I don't think ANYONE here (other than you) has ever argued against any of this.
- What I said was "I'd be delighted to work toward a more NPOV treatment". This is not saying that I think the article has a pro-Morgan POV, but that I'm acknowledging that you think it does and I'd be happy to work with you to make it address whatever concerns you have. Adding dates (even if they weren't wrong) is not improving the article or addressing any POV concern. What you seem to be doing is trying to introduce an anti-Morgan POV. That's not how it works. Please read WP:NPOV again, specifically WP:STRUCTURE. You have said repeatedly [5] you want an unconditional solution first and foremost, followed by a disclaimer like "The Monty Hall problem is unconditional. That is the whole paradox; the rest is the explanation; go and learn." This would be sort of the exact opposite of editing out POV. -- Rick Block (talk) 19:08, 28 December 2009 (UTC)
- Let the readers decide amongst the published papers. Just don't cloud the story with unnatural euphemism's like 'variant'. The 3 separate solution sections approach accomplishes my goal, along with dates clearly highlighting the history and clarifying what the heck a 'variant' is. That's not a POV, that's shedding light on the controversy. It just weakens your POV, so you demonize it.
- But anyway, you argue whatever side of the coin is convenient for you each day. What's the point in going around further?
- I've asked user:K10wnsta to drop by and say what he might be able to do as an informal mediator. I'll wait to see what he says. Do not in any way take the fact that I haven't reverted your change (again) to mean I accept it or agree with it. I think it might be helpful if some other folks would comment on this specific change as well. -- Rick Block (talk) 23:39, 28 December 2009 (UTC)
Any support for Arbitration?
I think Rick Block and Nijdam are fillibustering and ownershipping against beneficial changes to this article.
I see no point in waiting for either form of mediation unless Nijdam indicates he will accept the findings.
Rick filed an RfC against me last week, the first item of which is 'only edited the article 1 time.' Now, as you've seen yesterday, every edit I make, he or Nijdam at his request, reverts.
If at least 2 people are with me, I'll proceed. Glkanter (talk) 17:31, 28 December 2009 (UTC)
- Be aware that mediation does not produce "findings". Its purpose is diplomatic -- to help the parties to find points of agreement. In the best of cases, the parties can agree upon a full course of action, thereby resolving the dispute.
- I am almost certain that the Arbitration Committee would not accept the case, since this is primarily a content dispute, and ArbCom rejects content disputes flat out. Just FYI.--Father Goose (talk) 06:20, 29 December 2009 (UTC)
- Father Goose, this is from the Formal Mediation page:
- "Mediation cannot take place if all parties are not willing to take part. Mediation is only for disputes about Article Content, not for complaints about user conduct."
- Since I asked on December 18th if there were objections, and would everyone indicate their agreement to take part, Nijdam has not responded. I felt that rendered the exercise useless.
- Father Goose, this is from the Formal Mediation page:
- The arbitration request would allege that Rick Block and Nijdam are using various filibustering and ownershipping techniques (for example, not replying to the Formal Mediation question) against beneficial changes to the MHP article, as desired by the consensus of editors many weeks ago. I fear some of them have lost interest because of the filibustering. Glkanter (talk) 11:01, 29 December 2009 (UTC)
- I think you will find that what Father Goose says is still true. They are unlikely to even consider the case. Martin Hogbin (talk) 19:12, 5 January 2010 (UTC)
- The arbitration request would allege that Rick Block and Nijdam are using various filibustering and ownershipping techniques (for example, not replying to the Formal Mediation question) against beneficial changes to the MHP article, as desired by the consensus of editors many weeks ago. I fear some of them have lost interest because of the filibustering. Glkanter (talk) 11:01, 29 December 2009 (UTC)
Mediation
Case link
I've re-opened the case at MedCab and volunteered to assume the role of mediator in a discussion aimed at resolving an on-going dispute here. Additionally, I've issued invitations to participate in the discussion to all involved parties listed in the mediation request. While anyone is welcome to offer input, I ask that those who participate do their best to be concise and refrain from assumption/presumption regarding other's perspectives.
As mediator, my primary goal is to step in as an uninvolved party and help find some common ground from which to proceed. It is not my task to pass judgment on anyone's opinion in the discussion and there is no 'right' or 'wrong' beyond that which is dictated by Wikipedia policy.
I have read the article and understand its subject matter and all it details. As I begin delving through the talk page archives, I'll open the discussion with a call for opening statements. If you feel any archived passages are significant in summarizing the situation, it would help to include links, but please conclude your first post with a Summary of Position (your opinion as it relates to the matter). And remember...concise ;-)
--
(talk) 05:28, 29 December 2009 (UTC)
- K10wnsta, what exactly are you trying to facilitate? The original ruckus which led to the request was that perhaps 3 editors were not willing to allow a large consensus to change the article. Their reasons have been debated extensively, and imho, found wanting.
- Rick wrote this on his informal mediation request:
- What would you like to change about this?
- "I would like the endless discussions to be settled. I would like the article to remain a featured article."
- What most of us want is some measure of the three change proposals to be reflected in the article in a prompt fashion, via consensus. Is this what you are working towards? In what way? Do you have agreement from all parties that they will honor whatever comes out of this? What form will the outcome of this mediation take? Glkanter (talk) 14:24, 3 January 2010 (UTC) Glkanter (talk) 17:05, 3 January 2010 (UTC)
Nijdam's position
I want the article clearly mention the remark made by some sources that the so called "simple solution" is not complete. It doesn't need initially mentioning the technical term "conditional probability". To make my point clear: the following resoning:
- The player, having chosen a door, has a 1/3 chance of having the car behind the chosen door and a 2/3 chance that it's behind one of the other doors. Hence when the host opens a door to reveal a goat, the probability of a car behind the remaining door must be 2/3.
is not complete and better should read:
- The player, having chosen a door, has a 1/3 chance of having the car behind the chosen door and a 2/3 chance that it's behind one of the other doors. When the host opens a door to reveal a goat, this action does not give the player any new information about what is behind the door she has chosen, so the probability of there being a car remains 1/3. Hence the probability of a car behind the remaining door must be 2/3.
Something alike holds for the so called "combined doors solution" and most of the other simple ways of understanding. That's all.Nijdam (talk) 08:36, 29 December 2009 (UTC)
I'd like to add that the (a) MHP always involves enumerated doors and a decision to switch offered to the player after a door is opened, seen by the player who has to decide. This is in my opinion and of many (most) sources the only relevant problem.Nijdam (talk) 11:48, 31 December 2009 (UTC)
Martin Hogbin's position
The MHP is essentially a simple mathematical puzzle that most people get wrong. At least the first part of the article should concentrate on giving a simple, clear, and convincing solution that does not involve conditional probability. All diagrams and explanations in this section should not show or discuss the possible difference that the door opened by the host might make, although I would be happy to include, 'this action does not give the player any new information about what is behind the door she has chosen' as in Nijdam's second statement above. The first section should give aids to understanding and discuss why many people get the solution wrong, without the use of conditional probability. The first section should be supported by sources which do not mention conditional probability
The simple solution section should be followed by an explanation of why some formulations of the problem require the use of conditional probability, with reference to the paper by Morgan et al. and other sources. It should also include the various variations of the basic problem and other, more complex, issues. Martin Hogbin (talk) 10:19, 29 December 2009 (UTC)
Glkanter's position
I want the article to clearly mention that the remarks made by some sources, that the so called "simple solution" is not complete, is not shared by all sources. It need not mention "conditional probability" beyond saying that due to the symmetry forced by being a game show, the simple solution is equivalent to the symmetric 'conditional solution'.
I think I agree with Nijdam on the text, although they are both OR. It's consistent with my 1st talk page edit, using an IP address in October, 2008:
- Monty's Action Does Not Cause The Original Odds To Change.
- When Monty opens a door, he doesn't tell us anything we didn't already need to know. He always shows a goat. It makes no difference to this puzzle which remaining door he shows. So it starts out as 1/3 for your door + 2/3 for the remaining doors = 100%. Then he shows a door, but we knew in advance that he was going to show a goat. The odds simply haven't changed following his action. They remain 1/3 for your door + 2/3 for the remaining doors (of which there is now just 1).
I'd like to see 3 solution sections: Selvin's simple solution of 1975, transitions to Selvin's symmetrically equivalent conditional solution of 1975 (where the discussion of the simple solution's criticisms occurs), transitioning to Morgan's conditional non-solution of 1991.
I'd like to see the word 'variant' either stricken, or augmented by 'slightly different problem'.
I'd like to see a lot of 'blather' removed from the article. Too much time and effort is spent in the various remaining sections explaining the conditional solution, for no real reader benefit. Glkanter (talk) 10:39, 29 December 2009 (UTC)
And the 'Variants - Slightly Modified Problems' section needs work. The MHP is from the contestant's state of knowledge (SoK). The versions in this section are not. This needs to be normalized for the reader in a few possible ways: An explicit statement that the contestant is aware of these new conditions (in which case these are no longer game show problems), or the explicit statement these problems are not from the contestant's SoK, and a comparison of the MHP from a non-contestant's SoK. Glkanter (talk) 13:14, 29 December 2009 (UTC)
Rick Block's position
First, I think the basic issue is an NPOV issue. The primary question is whether the article currently expresses a "pro-Morgan" POV, i.e. takes the POV of the Morgan et al. source that "unconditional" solutions are unresponsive to the question and are therefore "false" solutions - and, if so, what should the remedy be.
There are a variety of sub-issues we need to discuss but I think the main event is how the solution section is presented. I strongly object to splitting the solution section into separate sections (this was done some time ago, well after the last FARC), which inherently favors whatever solution is presented in the first such section. I mildly object to including the "combining doors" explanation in the solution section rather than in a subsequent "aid to understanding" section.
What I would like is for the article to represent in an NPOV fashion both a well-sourced "unconditional" simple solution (e.g. vos Savant's or Selvin's) and a well-sourced conditional solution of the symmetric case (e.g. Chun's, or Morgan et al.'s, or Gillman's, or Grinstead and Snell's) in a single "Solution" section, more or less like the suggestion above (see #Proposed unified solution section - somewhat modified just now). This follows the guidelines at Wikipedia:Make technical articles accessible, specifically most accessible parts up front, add a concrete example, add a picture, and do not "dumb-down".
Once we address this basic issue I think the other issues will be easier. -- Rick Block (talk) 19:43, 29 December 2009 (UTC)
Overview
- So, let's start at the very beginning...based on my analysis of the archives (*whew*), the positions stated here, and the current article lead, am I correct in understanding everyone agrees on defining the core Monty Hall problem as the one presented in Parade magazine in 1990 that reads:
--K10wnsta (talk) 22:01, 1 January 2010 (UTC):Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, 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?" Is it to your advantage to switch your choice?
- I'm not sure what you mean by "core". This is certainly a well known statement of the problem, but it is not very precise. What the the standard analysis is based on (per Barbeau) and how people generally interpret it (per Krauss and Wang) is consistent with the more explicit Krauss and Wang version from the "Problem" section of the article. In particular, the initial placement of the car is assumed to be random (and/or the player's initial choice is assumed to be random) and the host is constrained to always open a door deliberately revealing a goat, choose between two goats randomly, and always make the offer to switch. The effect of changing or omitting various of these assumptions is discussed in the "Variants" section. -- Rick Block (talk) 22:54, 1 January 2010 (UTC)
- That is the most often quoted and the most notable statement but, as Rick says, it leaves a lot out. Martin Hogbin (talk) 22:57, 1 January 2010 (UTC)
Hello K10wnsta. There are several editors here keen to get on with improving this article. Are you still intending to mediate? Martin Hogbin (talk) 10:08, 4 January 2010 (UTC)
- Yes, and I apologize for the delay. I have written and rewritten and deleted probably ten thousand words worth of questions, answers, and ideas to offer here and always end up backtracking and starting over because, honestly, you guys have covered every imaginable nuance and detail of the issue in past discussion here. It is not a decision I came to lightly, but I must wash my hands of this. This is an issue that I cannot help you to resolve.
- Prior to stepping away and recommending formal mediation (actually, I would recommend some binding arbitration, but we'll follow procedure), I would like to present to you my own perspective (or a third opinion) on the state of the article (and this is boiled down from three lengthy paragraphs):
- I was intrigued by the problem. Reading the article, I had the same reaction most people have when the problem is initially presented ('it must be a 50-50 chance'). The actual solution seemed outlandish. Once I read the explanation with the door diagrams, a light bulb went on and it made sense - it was better to switch. Much like the birthday problem, it's an intriguing solution that seems to defy logic. Once explained, it makes sense, and the novelty no longer exists. Everything else in the article is over-analysis, largely because it goes about solving variations of a problem that, once varied, lack the same novelty response that makes the Monty Hall problem notable to begin with.
- Again, I'm sorry for the delay in posting this, but it was something I really didn't want to give up on. I wish I could offer more than just an opinion on the state of things, but you guys are so far out in left field in your analysis of this, I just can't wrap my head around it. Sorry I let you down.
--K10wnsta (talk) 01:04, 5 January 2010 (UTC)
Methods of numbering doors
(I welcome you all back to my screen. This article has improved a lot over the last year.)
Morgan et al. (1991) seem to assume that the doors are statically numbered, having the same numbers through repeated experiment. Vos Savant however writes in her column: "You pick a door, say #1, and the host opens another door, say #3". This may mean that after a door is picked, we (always) call it #1, while the opened door is (always) called #3. Such dynamic numbering can make it easier to discuss and calculate the given options. The consequences of the assumption of Morgan et al. are further explained in this article under "Probabilistic solution - 1991".
The Morgan paper classifies solution F5 as "incorrect because it does not use the information in the number of the door shown". This is only true assuming statical numbering. In this context it is questionable why Morgan et al. quote vos Savant wrongly, writing "You pick door No. 1, and the host opens No. 3". Heptalogos (talk) 14:11, 30 December 2009 (UTC)
- The show the problem clearly refers to (Let's Make a Deal, hosted by Monty Hall) had doors with numbers on them. Here's a screenshot [6]. You could assume the doors are numberless and the numbers magically appear on them as they are used, but the question for editing purposes is what reliable sources say about the problem. Are you suggesting this "numberless door" interpretation is the predominant one used by reliable sources - or is this more like your own original research? If the latter, I'd suggest moving or continuing this thread at the #Arguments subpage. -- Rick Block (talk) 18:00, 30 December 2009 (UTC)
The Morgan paper is not about Monty Hall, but about a question in a column of vos Savant, starting with "Suppose you're on a game show". All exact information is, of course, in the paper, so no other sources are relevant. Heptalogos (talk) 19:44, 30 December 2009 (UTC)
- So, again, are you talking about what reliable sources say (if so, references would be helpful), or something you've thought of on your own (if so, please use the #Arguments page)? -- Rick Block (talk) 19:56, 30 December 2009 (UTC)
The source I mention, "Morgan et al. (1991)", is probably the most argued source in this article. It is in the article reference list mentioned as: "Morgan, J. P., Chaganty, N. R., Dahiya, R. C., & Doviak, M. J. (1991). "Let's make a deal: The player's dilemma," American Statistician 45: 284-287." Heptalogos (talk) 20:11, 30 December 2009 (UTC)
If you're suggesting to move or continue discussion about the specific arguments used in the Morgan source, to or on the arguments subpage, then that's fine with me. But I'm doing more than that, namely introducing a new element to the global dilemma, which is the method of numbering doors. Heptalogos (talk) 20:43, 30 December 2009 (UTC)
- I think the point that you are making is already covered by considering the player's initial choice to be random. In other words we can take it that either the Morgan paper refers 'only to the specific door numbers shown or that all the door numbers including the door opened by the host are examples only. Martin Hogbin (talk) 21:03, 30 December 2009 (UTC)
The Morgan paper is quite clear about the disctinction between conditional and unconditional. I quote: the unconditional problem, which may be stated as follows: "You will be offered the choice of three doors, and after you choose the host will open a different door, revealing a goat. What is the probability that you win if your strategy is to switch?" The distinction is made by opening a specific door, instead of "a different door". This is mentioned elsewhere in the paper several times. I agree that No. 3 is an example and might be No. 2 as well, but the paper assumes that there is an essential difference between "the open door" and "door No. x, which is open". To my opinion these are only labels which don't make any difference, unless of course one assumes that a specific door is labelled the same through repeated experiment. Heptalogos (talk) 21:40, 30 December 2009 (UTC)
- Yes, they're clearly assuming distinguishable doors with persistent numbering. Is the topic you're introducing, specifically "the method of numbering doors", discussed in reliable sources or is this a topic you're introducing based on your own personal knowledge? Once again, only in the former case is this an appropriate topic for this page. If this is the case, please provide references to sources that discuss this topic. There are plenty of disagreements here without delving into original research. -- Rick Block (talk) 22:47, 30 December 2009 (UTC)
I want this article to explain how the conditional probability could actually differ from the overall probability (I refer to chapter "Probabilistic solution - 1991"), when the distinction between both is made by information (a door number) which seems to have no statistic dependency or influence on the requested probability. To my opinion, the average intelligent reader of this paradox, who has no mathematical skills, still doesn't understand the necessicity of using the relatively complex method of conditional solution. I agree that the discussion about the necessicity itself should preferably be held elsewhere, but an elementary explanation key in the article should be, I guess, in the idea of a static door position through repeated experiment, whatever it means. The meaning of that I would like to be explained. Heptalogos (talk) 23:45, 30 December 2009 (UTC)
- I would actually prefer a combined solution section, like the one above (#Proposed unified solution section). It presents a conditional approach as an alternative, with a brief explanation of the difference between the conditional probability and the average probability, with a forward reference to the "Variants" section (at the moment called "Variants - Slightly Modified Problems"). Is this version more clear? -- Rick Block (talk) 01:02, 31 December 2009 (UTC)
I propose adding a link in external references
I propose adding an external link to http://www.opentradingsystem.com/quantNotes/Monty_Hall_problem_.html
The link in question contains derivation of solution in a general context developed on other examples. —Preceding unsigned comment added by Kaslanidi (talk • contribs) 20:27, 30 December 2009 (UTC)
I object, per WP:ELNO points 1, 4, and 11. - MrOllie (talk) 20:32, 30 December 2009 (UTC)
Yes, I Have A 'POV'.
My 'POV' is that this paradox twists peoples' brains a lot, just the way it is. Whatever 'is' means.
So a sequenced roll out of how the problem became published, then controversial twice would help the interested reader. What could make more sense then to describe the events roughly as they occurred, and beliefs/understandings changed, or maybe they didn't.
Let the reader decide for himself, or herself.
Yes, 'sequenced roll out' really means 'chronological'. Forgive me.
Pretty radical, eh? Just tell the story as the sources do, and let the reader draw his own conclusions. Whoda thunk it? Glkanter (talk) 20:35, 30 December 2009 (UTC)
- I'm not sure I understand what you're suggesting. There is already a "history" section. Are you suggesting pitching the entire article and starting over with a strictly chronological accounting that would be sort of an expanded version of the "History" section? Or are you simply saying the same thing you said above as your position in the "Mediation" section, i.e. that you want to see 3 solution sections? -- Rick Block (talk) 01:37, 31 December 2009 (UTC)
BEST WISHES TO YOU ALL
And a fruitful start (continuation) in 2010! Nijdam (talk) 11:45, 31 December 2009 (UTC)
- I would like to echo the last comment: Best wishes to all! And a fruitful start (continuation) in 2010. Gill110951 (talk) 19:36, 3 January 2010 (UTC)
What If Morgan Had Used A Different Variant?
Say, the 'forgetful' Monty?
Marilyn vos Savant says this is really a 'random' Monty, who might reveal a car.
I expand this to say, if it's random, then anyone, including the contestant could open the doors.
And if it's the contestant, then we're really talking about 'Deal Or No Deal'.
Does it makes sense to criticize the original solutions to the MHP based on an analysis of Deal or No Deal? Not in my book. Glkanter (talk) 17:21, 1 January 2010 (UTC)
- If Monty *might* reveal a car nothing much is changed: the player always switches (only she should be sensible about which door to switch to). If Monty might *not open a door at all* then of course things do change, though under some conditions they don't. Game theory solves this case. If Monty might or might not open a door, and if he might reveal a goat or a car when he does (he does know where the car is), then the minimax solution is the very boring game: the car is hidden uniformly at random, and Monty never opens a door; the player chooses her door uniformly at random, and thereafter never switches. With those strategies the player is guaranteed at least a probability of 1/3 of winning the car; the quiz-team is guaranteed a probability of at most 1/3 of losing the car. So this solution is the saddle-point or Nash equilibrium. (von Neuman's theorem says that there certainly is a saddle-point). I think the solution is unique too: if the player would use any other strategy then the quiz-team could decrease their car-losing probability from 1/3, and if the quiz-team would use any other strategy then the player could increase her car-winning probability from 1/3. Gill110951 (talk) 19:46, 3 January 2010 (UTC)
- I'm more interested in the idea that one can critique puzzle 'B', and find fault with puzzle 'A'. Which is what Morgan do, and every time I ask Rick to critique the simple proofs, much as I asked him about Huckleberry, he replies with a critique of a different puzzle. I don't get it. I think it's baseless. Glkanter (talk) 20:26, 3 January 2010 (UTC)
- Is Morgan et al. a reliable source by Wikipedia standards? Does this article say If, regardless of the host's action, the player's strategy is to never switch, she will obviously will the car 1/3 of the time. Hence, the probability that she wins if she does switch is 2/3. ... F1's beauty as a false solution is that it is a true statement! It just does not solve the problem at hand.?
- Is Grinstead and Snell a reliable source by Wikipedia standards? Does this book say This very simple analysis [as a preselected strategy, staying wins with probability 1/3 while switching wins with probability 2/3], though correct, does not quite solve the problem that Craig posed.
- Is Gillman a reliable source by Wikipedia standards? Does this article say This is an elegant proof [vos Savant's solution], but it does not address the problem posed, in which the host has shown you a goat at #3.
- Perhaps you don't understand or don't agree with what they're saying, but the beauty of Wikipedia is that it doesn't matter whether you understand it or agree with it so long as what the article says is NPOV and verifiable against a reliable source. We have 3 reliable sources here which all say nearly exactly the same thing. -- Rick Block (talk) 22:34, 3 January 2010 (UTC)
- So, did you check the sources of the popular solution? They are quite contradictory. Heptalogos (talk) 23:17, 3 January 2010 (UTC)
- Yes, I have checked. They present an unconditional solution without mentioning conditional probability as if the solution is responsive to the question that is asked. So, we have reliable sources that say one thing and other reliable sources that say the first bunch is not quite addressing the problem. I'm open to any reasonable suggestion for how to present this in an NPOV fashion, but not ignoring the second bunch of sources because some editors disagree with what they say. -- Rick Block (talk) 00:19, 4 January 2010 (UTC)
- I'm very sorry, you are right. I lost the context because I sometimes find it hard to understand Glkanter. The article is quite good on the solutions, very NPOV (without irony). Now let's hope some scientist attacks Morgan, Grinstead and Gillman in a very irrelevant, but reliable manner. Heptalogos (talk) 20:52, 4 January 2010 (UTC)
Another Straw Man from Rick. Both sides go in the article. Why not chronologically? I've been saying this for a week. Your's and Morgan's POV not dominating the current article? Don't make me laugh. Now, why not answer for yourself, as a sentient being, what does "Suppose you're on a game show...' mean? Without hiding behind Wikipedia's policies. It's OK, we're on a talk page. Glkanter (talk) 01:16, 4 January 2010 (UTC)
- Kind of ironic that you're complaining about straw men in a section titled "What If Morgan Had Used A Different Variant?", don't you think? I've asked you before, but where exactly do you see Morgan's POV dominating the article? As far as I can tell the only mention of this POV is in the "Probabilistic solution" section, the 4th paragraph in "Sources of confusion" and a paragraph in "Variants". -- Rick Block (talk) 02:53, 4 January 2010 (UTC)
- All is right with the world. As we enter our 15th month of this argument, I disagree with you on the meanings of both 'Straw Man' and 'ironic'. Your POV? It's in every word except the intro and the Simple solution section. You just don't see it for some reason. Glkanter (talk) 06:19, 4 January 2010 (UTC)
- What question is that? What does "assume you're on a game show" mean? I've answered this before. It provides a context for what is meant by "host" and "contestant" and "door". -- Rick Block (talk) 15:26, 4 January 2010 (UTC)
What is the Morgan scenario?
I have now shown that in order to get an answer (probability of winning by switching) of anything other than 2/3, Morgan have had to assume that we know that the producer places the car randomly, but we do not know that the host opens a legal door randomly. Is there anyone here who can justify that odd POV.? Martin Hogbin (talk) 10:12, 4 January 2010 (UTC)
- Yes. There will be at least one editor who will attempt to justify that odd POV. Glkanter (talk) 13:10, 4 January 2010 (UTC)
- It's not "Morgan's scenario", but Morgan's interpretation of vos Savant's scenario - perhaps "Morgan's vos Savant scenario" would be a better way to refer to it. It exactly matches the rules she sets up for the experiment she describes in her 3rd column, see [7]. I assume this is the "false simulation" they refer to in the introduction to their paper. The cups are labeled #1, #2, and #3. The host randomizes where the penny is placed. The contestant randomizes her pick. Then the host "purposely lifts up a losing cup from the two unchosen" (no mention of randomization in the case where the host can lift up either unchosen cup, even though both the initial placement and the player's pick are explicitly randomized). What is counted is overall success when not switching and overall success when switching, rather than success when not switching and success when switching for players who have picked cup #1 and have seen the host reveal what's under cup #3. This experiment is explicitly addressing the unconditional probability of winning by switching rather than the conditional probability - in a setup where they might actually be different (since the host is not required to randomize his choice of cup to lift up in the case where there is a choice). -- Rick Block (talk) 15:14, 4 January 2010 (UTC)
- It really is unfair to try to blame vos Savant for failings in the Morgan paper. Sure Morgan mention vS at the start of their paper, they also mention the "prisoner's dilemma" (where the warden does secretly toss a coin), and Mosteller's' solution. Later on they state that vS took it that the host never reveals a car, and it is clearly this rule that they describe as the vos Savant scenario.
- Regardless of what vos Savant or anyone else assumed, Morgan are under an obligation to firstly make consistent assumptions (for example that all unstated distributions are to be taken as random) and then to make clear the assumptions that they have made. They conspicuously fail in both of these respects.
- Morgan claim to have 'an elegant solution that assumes no additional information', clearly referring to the Whitaker's original question, rather than vos Savant's interpretation. Their solution does not live up to this claim.
- Finally, regardless of reasons or motives, Morgan do in fact consider the scenario that we know the producer places the car randomly but we do not know that the host chooses a legal door randomly. Their answer of 1/(1+q) is based on this scenario. This is not a reasonable or consistent assumption thus their answer of anything other than 2/3 is not valid. Martin Hogbin (talk) 16:11, 4 January 2010 (UTC)
- Rick, you are still misrepresenting MvS's statements, and that is an example of how you inject your POV into everything you write. Please stop taking her statements out of context. "Anything else is a different question" refers to all of the assumptions she made in her approaches to the solution. That includes random car placement and random host selection (if needed), and excludes any dependence on Door #1/Door #3 that you think is included. Her shell game analogy did not number the shells, yet she said it was the same problem. As you yourself point out, the cup experiment does not mention picking cup #1 or lifting cup #3. You can't treat that part as an error; it, too, has to be part of any "exact match to the rules she sets up." So Morgan's interpretation does not qualify.
- In the literature, there are two camps that do not agree with each other, and that never reconcile their differences (well; actually, MvS does - she said Morgan's is a different problem). Those that follow Morgan's "conditional problem" and those who agree with Seymann, that "the host is to be viewed as nothing more than an agent of chance who always opens a losing door, reveals a goat, and offers the contestant the opportunity to switch to the remaining, unselected door." Insisting that only the former group is correct is POV, and that is what I mean when I say you inject your POV. I will only stop saying it, when you stop doing it. Morgan's POV can be handled in the article; but it clearly is not the problem MvS intended. That fact is acknowledged in literature. So the article needs to first address the problem she said was intended, and then add the second opinion in as a variant, and clearly label it as a variant. That is the NPOV approach.
- Martin, I firmly believe that Morgan never intended to introduce theirs as a "variant" problem, or to avoid making assumptions like the one about car placememt. All they ever say, is that one of the assumptions MvS made is not necessary to answer the question "should we switch?" They misspoke when they said "assumes no additional information," they quite clearly meant "assumes no unnecessary information," since they did make assumptions. And their answer to the Monty Hall Problem does not use the 1/(1+q) result, it only shows that no specific value is needed. It was an intellectual excersize only, and is not intended to be the MHP. So yes, it would be inconsistent to give an answer that includes q but not P(C1), P(C2), and P(C3). My point here is that they don't - but they also fail to make it clear to their readers that they don't. JeffJor (talk) 17:14, 4 January 2010 (UTC)
- Jeff, I agree, the Morgan paper is not entirely without interest or value. They show how, in a more general case, the host behaviour is important, firstly in never showing a car, then in choosing which door to open, but the player can never do worse by swapping. The problem is that they make such a bad job of what they do that it is hard to work out exactly what their main point is, except to criticise others.
- My main point is that there is no justification for saying that in the MHP, with standard rules, the action of the host is important. There are only two logical and consistent ways to look at Whitaker's question. Take it as a real world question about the actual probabilities on hypothetical TV show, in which case it depends on so many factors that the answer is indeterminate, or take it as a mathematical puzzle, in which the normal mathematical puzzle assumptions are made, undefined distributions are taken as random etc. Anything else is, as you say, just an intellectual exercise. Martin Hogbin (talk) 18:11, 4 January 2010 (UTC)
I disagree on 1 point, Martin. "Suppose you're on a game show..." means the car placement and host choice, as far as the contestant is concerned, are random. This is true whether it's a hypothetical game show, or a mathematical puzzle. Because that is the host/contestant relationship on a game show. And it's every bit as much a premise of this math puzzle as '1 car and 2 goats' which is clearly stated. Because 'Suppose you're on a game show...' has also been clearly stated. Glkanter (talk) 18:44, 4 January 2010 (UTC)
- It depends on the presumed state of knowledge of a contestant on the hypothetical show. Maybe the contestant has watched the show and discovered that the car is most often behind door 1. I do agree that a natural assumption would be that a contestant would have no knowledge of how the car might be placed or the host would choose but in real life this might not be the case. I am not seriously pushing the real life option, just stating that it is the only logical alternative to the 'puzzle' option. Martin Hogbin (talk) 20:01, 4 January 2010 (UTC)
- Martin, this is not about "real life." It is a thought puzzle, only. No biases, or unrandom occurrences, can be assumed unless specifically stated. But this has nothing (directly) to do with "Suppose you are on a game show..." It has to do with its not being mentioned. You must assume any unmentioned options have to be random between the possibilities (and so this is another reason the conditional solution can't be used.) JeffJor (talk) 20:32, 4 January 2010 (UTC)
- Jeff you are the hardest person in the world to agree with sometimes. As I said to Glkanter, the real world scenario is the other consistent option, but I agree that the real one in the 'puzzle' option. Martin Hogbin (talk) 20:40, 4 January 2010 (UTC)
- Martin, you have a problem because you don't acknowledge the fact that it is your argument, not your conclusion, that I disagree with. There is no supportable definition for what "suppose you are on a game show" means, so it is pointless to try to argue for a meaning behind it. There also is no justification for trying to place the problem in a real-world setting, which is one of the problems with Morgan. (Just like "What is the probability that the woman I met yesterday, who has two children, has two boys?" The real-world answer is 100%, because she does have two boys. The intent of the question, as a puzzle, is 25%; and the real-world scenario is completely irrelevant.) So I view any arguments based on eithar as being counter-productive. JeffJor (talk) 17:31, 6 January 2010 (UTC)
- Martin, this is not about "real life." It is a thought puzzle, only. No biases, or unrandom occurrences, can be assumed unless specifically stated. But this has nothing (directly) to do with "Suppose you are on a game show..." It has to do with its not being mentioned. You must assume any unmentioned options have to be random between the possibilities (and so this is another reason the conditional solution can't be used.) JeffJor (talk) 20:32, 4 January 2010 (UTC)
- It depends on the presumed state of knowledge of a contestant on the hypothetical show. Maybe the contestant has watched the show and discovered that the car is most often behind door 1. I do agree that a natural assumption would be that a contestant would have no knowledge of how the car might be placed or the host would choose but in real life this might not be the case. I am not seriously pushing the real life option, just stating that it is the only logical alternative to the 'puzzle' option. Martin Hogbin (talk) 20:01, 4 January 2010 (UTC)
- Jeff - I'm merely suggesting where Morgan et al. got the "odd" scenario from. MvS explicitly said lots of things, but never (as far as I know) explicitly said anything about how the host chooses when given the chance (at least not before the Morgan et al. paper was published). This is NOT "my" POV, but Morgan's POV (that I seem to be the only one representing here). I'm FINE with treating this specific scenario as a variant (it's already been moved to the variant section), but I don't think it means that a conditional solution must also be deferred to a variant or that a discussion of whether the question refers to the unconditional probability or the conditional probability must be deferred to a variant. The unconditional and conditional probability for the fully explicit version (including "host picks randomly if given the chance") are the same - we all agree about this - but I think clarifying that these are different questions and which approach addresses which question should be part of the initial "Solution" section. -- Rick Block (talk) 19:12, 4 January 2010 (UTC)
- Rick - I'm not saying that your agreeing with Morgan, per se, is POV. I'm saying that disagreeing with MvS on the same subject, where she disagrees with Morgan, is POV. You said "[Morgan's interpretation] exactly matches the rules [Marilyn vos Savant] sets up for the experiment she describes in her 3rd column." It does not - why do you ignore that part of what I said? It exactly matches what Morgan said her set of rules was, a set she explicitly denied was her intent. There are two parts to that difference, and it requires both of them to make Morgan's interpretation applicable to anything: There is the question about whether Door numbers are important (they aren't, and she explicitly made that clear because she does not use the numbers in her two analogies), and there is the question about how the host treats door numbers if they are important (which MvS doesn't mention, but doesn't need to: as per Seymann, the host is only an AGENT OF CHANCE and so everything he does must be goverened by chance alone. That is the statement you are missing. But even if that is not accepted, since the door numbers are not important, no such bias can be used. You yourself pointed that out, because her experiment was "explicitly addressing the unconditional probability of winning by switching rather than the conditional probability." Those are what her rules are, according to her.)
- To say there is a match requires that you ignore MvS's comments and accept only Morgan's. We simply have to beleive MvS's statements of what her intent was, over Morgan's. That's what makes their treatment a variant, and what makes disagreeing with her "set of rules" POV even if you find sources for support. Plus, you misinterpret Morgan (I still must be speaking in cat whan I say this): They do not present a solution to the MHP that includes q. They show that if the MHP is CHANGED (they don't call it a change, but MvS does, so we have to treat it as one to be NPOV), then you get the same answer ("switch") no matter what q is. But they ignore placement bias the exact same way MvS does, because it does affect the answer, and is part of being an "agnet of chance."
- There is no justification for saying the host's preferences for opening one door over another matters, because the MHP is about the strategy, not the doors. Any mention of it belongs with the variants, and even then we have to say that you can't use it unless you know it. JeffJor (talk) 20:25, 4 January 2010 (UTC)
- Jeff - How do the rules of Morgan's vos Savant scenario not match the experiment she described? As far as I can tell, the difference is whether you're ignoring the conditional question and intending to answer the unconditional question (like vos Savant does) or whether you understand there may be a difference and are intending to answer the conditional question (like Morgan et al. says is what the question asks). Which specific comments of MvS's (from her columns) did they ignore? It sounds like you're claiming the quote "Anything else is a different problem" applies to anything you'd like it to apply to. Here's the full quote:
- Rick - I can keep repeating this as often as you ignore it: Morgan assumes that the door numbers were intended to be used as part of the problem, and not meant as examples. MvS does not. That is, Morgan addresses what you call "the conditional proeblem," which is part of what MvS calls "a different problem." As I said twice yesterday, this is clear because the two analogies she uses do not use those door numbers in the solution, not even the one (and only one) that includes them in the description. So how can you ask this question? Her experiment that does not use door numbers, and that you said "exactly matches" a treatment that does. So I'm not "ignoring" the conditional problem, I'm saying (1) It isn't there to be ignored, (2) teh conclusion that it is comes from a misreading of the problem (K&W say the door numbers are sematically just examples) (3) MvS has denied it is intended, which allows us to ignore it and still be NPOV (4) Seymann acknowledges it is not how the problem shoud be read, and (5) The vast majority of sources that address what they call the MHP, especially in popular literature where the controversy exists, do not consider it at all. Selvin, Savant, Gardner, Tierney, Delvin, Mlodinow, and others all either ignore door numbers, or uses them as examples and treat them as though they represent "without loss of generality" selections. JeffJor (talk) 17:25, 5 January 2010 (UTC)
- Morgan et al. assume the door numbers are persistent and the problem pertains to a player who has initially selected a specific door and has seen the host open another specific door. They use the example case of door 1 and door 3 as representative of any other (equivalent) case. Selvin (2nd letter), Morgan, Gillman, Grinstead and Snell, Chun, Falk, and others all approach the problem this way using this case "without loss of generality". -- Rick Block (talk) 19:48, 5 January 2010 (UTC)
- Only partially right, Rick. Not all use the phrase "without loss of generality." Those that do, are acknowledging that they are using single cases to represent all of the symmetric cases. Those that never consider "host strategies" other than random selection - like Selvin in what he calls an alternate solution - are implicitly doing the same thing. It is the insistence that the problem [i]must[/i] be solved with conditional problility AND that this solution must consider p!=q that we object to. Once you remove that second possibility, there is no need to use specific doors. You can change "Door #1" to "the chosen door," "Door #3" to "the opened door," and "Door #3" to "the remaining door." There is no way apply the parameters p and q this way. Doing so removes a level of complication that obfuscates the MHP to the casual reader, and so makes the article more readable. JeffJor (talk) 17:22, 6 January 2010 (UTC)
- So let's look at it again, remembering that the original answer defines certain conditions, the most significant of which is that the host always opens a losing door on purpose. (There's no way he can always open a losing door by chance!) Anything else is a different question.
- You're telling me you're absolutely sure "anything else" here does not refer specifically to the host opening a losing door on purpose but to all the "certain conditions" that she hasn't enumerated (and, so, anything she might need to justify her solution is included). Furthermore, you're telling me that in spite of explicitly labeling the cups she never meant the labels to mean anything (!?), and in spite of explicitly randomizing the initial distribution of the penny under the cups as well as the contestant's initial choice but saying nothing about the host's choice when the host has a choice she clearly meant the host to choose randomly as well (?).
- What is "anything else" comparing? Questions. What is the subject of the paragraph? Conditions that are defined for her question, by her answers. "Anything else" means anything other than the conditions she describes - those stated in her original problem OR implied by her solution. And it is quite clear in "opens a losing door" is just one example of a condition she means, and yes it is quite clear there are others. So absolutely, positively, can't mean just that one specific condition. JeffJor (talk) 17:25, 5 January 2010 (UTC)
- I think we both know she simply wasn't thinking of the conditional case at all - which is precisely what Morgan et al. criticize her for since in their view the question is clearly what is the chance of winning by switching in a specific case (such as the player has picked door 1 and the host has opened door 3). The chance is the same as the unconditional chance only in "certain conditions" one of which is that the host choose randomly when able to choose - or, less realistically, you can't tell the difference between the doors - or, more pedantically, you're restricting the solution to the player's SoK and assuming the player has no way to know about a host preference (essentially a sophisticated way of saying you can't tell the difference between the doors). You're saying MvS clearly understood this and clearly meant this to be one of the conditions of the problem. Fine. You're certainly entitled to your opinion, but since she never explicitly mentioned it in her columns (or, BTW, even in her rejoinder to Morgan et al.) it seems like there's a pretty good argument that she overlooked this condition. It was one of the first things Selvin mentioned in his second letter about the problem in response to the letters he got. It's not a "throwaway" detail. What I'm saying is that this article should not treat it as a throwaway detail either.
- As far as my POV, what I'm saying is that there is a difference between a solution that inherently addresses only the unconditional probability and a solution that is able to address the probability in a specific case. I would like BOTH to be presented, as equally valid solutions to the fully specified, symmetric, problem (where the answer is the same). Based on how violently you're objecting to this are you arguing that a conditional solution is somehow wrong? -- Rick Block (talk) 01:07, 5 January 2010 (UTC)
- There is no "specific case." It is an example. Here's another one: A woman I met at random yesterday has two children. What is the probability she has two boys? Answer, for the specific case: 100%. Answer clearly intended by the question: 1/4. It's a puzzle that uses an example to describe the random process. The example is not intended, and never is in such questions unless specifically included. And it has to be so, because probabilities do not apply to specific cases. They only apply to random processes. JeffJor (talk) 17:25, 5 January 2010 (UTC)
- Since I can enumerate the specific cases (player picks door 1 and host opens door 2, player picks door 1 and host opens door 3, etc.) your claim that there is no specific case seems rather curious. Bottom line, you're saying "yes" a conditional solution is wrong? -- Rick Block (talk) 19:48, 5 January 2010 (UTC)
- And once again, you insist on ignoring the point so that you can justify including Morgan. The numbers are not a part of the problem, they are only examples used to illustrate the problem. Stop me if you've heard this before: "Say," when used as an adverb like that, means "for example." The problem statement without examples is "Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door ... and the host, who knows what's behind the doors, opens another door ... which has a goat. ... Is it to your advantage to switch your choice of doors?" The parts I removed are simply not part of the question that is being asked. Period. We also know MvS intended it that way, because the statement as I just worded it is the only statement that is equivalent to Marilyn's two experiments. The only source that addresses why we should, or shouldn't, use the so-called "conditional problem" is K&W. All others merely assume one or the other (some implicitly and some explicitly, but still an assumption) without justifying why. And when they present that argument, K&W say the door numbers are not semantically part of the question. So by the Wikipedia guidelines, the only problem we can consider "the" MHP is the unconditional one. Morgan's treatment of it does not address her problem, because they changed the wording in the quote to insert the door numbers into the actual problem. So we cannot use Morgan as a source that addresses our main problem, and any solution they depends on specific door numbers must be relegated to the section on the variant they created. (Note that Selvin's conditionl solution does not depend on the door numbers, as it sets all probabilities by the assumption of uniformity. I've always agreed it can be used as a solution, but not with q != 1/2. And it doesn't help explain the controversy.) JeffJor (talk) 16:41, 7 January 2010 (UTC)
- Let's continue this during mediation if you don't mind. -- Rick Block (talk) 19:41, 7 January 2010 (UTC)
JeffJor, I'm with you on this 100%. So, living with the requirement that since they're published, Morgan and its ilk must be included in the article, how would you apply your argument to the article? Bear in mind, imho, the conclusion that 'Morgan's paper does not address the MHP' is, unfortunately, OR. Unless you have a source? Seymann just couldn't quite say it. Glkanter (talk) 18:03, 7 January 2010 (UTC)
Formal mediation
The informal mediator bailed, so the next step is formal mediation. The guts of the request are the sections I've put up at User:Rick Block/DraftMed. Before filing the request, I'd suggest anyone who's interested take a look. There are separate sections for issues the "filer" is asking to be mediated as opposed to issues other parties want mediated (feel free to add whatever you'd like). I don't care who actually files it (it really shouldn't matter), but have filled this out as if I'll be the one doing the filing. If anyone strongly objects to this we can rearrange things so someone else will be listed as the "filer" (but then whoever this is will have to actually file it at Wikipedia:Requests_for_mediation/File, and notify the involved parties). -- Rick Block (talk) 02:48, 5 January 2010 (UTC)
- Unlike informal mediation, formal mediation will be rejected unless all parties named as involved parties agree to mediation. Please indicate here whether you consider yourself to be an involved party, and whether you're willing to participate in mediation. For more on what is involved, please see Wikipedia:Mediation Committee/Policy. If you consider yourself to be an involved party and refuse to participate, the mediation committee will refuse to take the case. At a minimum, I consider myself, Glkanter, Martin Hogbin, and JeffJor to be "involved parties", although Nijdam, and Kmhkmh have been fairly involved in the past as well. Naming more, particularly anyone who is not willing to participate in the process is extremely counterproductive. -- Rick Block (talk) 05:22, 5 January 2010 (UTC)
- Well, it's a 'chicken or the egg' kind of thing. Just because you listed someone doesn't mean he will participate/accept the results. I tried to do a 'pre-agree' and after a few days passed, you said that wasn't how it works. The people I added have been participants in the last month or two, and should be offered the chance to be involved. Almost all of them were either part of the consensus or agreed to take part in the formal mediation when I asked.
- But I agree, we don't want a named person bringing the formal mediation to a halt. Has Nijdam indicated his willingness to participate to you? Glkanter (talk) 13:08, 5 January 2010 (UTC)
- Nijdam has not indicated anything to me. Any pre-agreement is not official. I've trimmed the list back to those who previously indicated a willingness to participate, plus Nijdam and Kmhkmh (I'll ask them both directly). If neither of them indicate they're willing we should talk about how to proceed. -- Rick Block (talk) 14:37, 5 January 2010 (UTC)
- I have posted an edit at Seems selective regarding the invitee list. Glkanter (talk) 16:21, 6 January 2010 (UTC)
- I cannot claim to be involved. Although I wanted to participate, lately I have found myself so busy with other things that I have not had the time to dedicate to trying to help with this thorny problem. Good luck and good grace to all, though.--Father Goose (talk) 08:08, 5 January 2010 (UTC)
- I'm not sure (and it perhaps depends on the mediator and/or the case), but I think even if you're not listed as an involved party you can generally participate. -- Rick Block (talk) 14:37, 5 January 2010 (UTC)
- I would like to participate. Heptalogos (talk) 20:03, 5 January 2010 (UTC)
Do we need it?
Despite continuing discussion and some disagreement, I think there has been a move towards increased understanding between all the parties involved. We should also all congratulate ourselves on keeping the discussions civil and avoiding edit warring.
Now is the time to move on and start improving the article. Some of us have written down our objectives for the article for the mediator and they do not look that far apart. With goodwill and some concessions on both sides we should be able to make considerable progress. We all need to give a little and be prepared to drop some of the finer points of our arguments. For example, I believe that there is no rule telling us that the K&W formulation must be treated conditionally (and I am happy to continue this discussion on the 'arguments' page) however, I am prepared to accept a statement along the lines of, 'strictly speaking this problem is one of conditional probability', if appropriately placed.
I appreciate that I am not a neutral party in all this but I am going to start a new section below, which I will call 'Self-mediation', just to see if we can find things that we agree on. Martin Hogbin (talk) 09:59, 5 January 2010 (UTC)
Self-mediation
This is a section where I hope to find as many things that we can all agree on. Can editors please indicate their agreement or otherwise below. I will add just two items to start, which I hope there will be fairly general agreement on.
Make better use of the two talk pages
This is just a matter of the mechanics of our discussion. This talk page should be reserved for discussion of proposed changes to the article: the general approach and format, what to say, where to put things, what diagrams to have etc. Discussion of the rights and wrongs of different sources, what is conditional and what is not, and other philosophical and mathematical issue around the subject should take place on the arguments page. It is important that this division is voluntary with only the gentlest of reminders to other editors to take their points elsewhere, if needed. Martin Hogbin (talk) 10:16, 5 January 2010 (UTC)
Agree
- Martin Hogbin (talk) 10:16, 5 January 2010 (UTC)
- Except I don't see why we need to be particularly gentle about it. Rick Block (talk) 16:10, 5 January 2010 (UTC)
- Demanding people take their discussions elsewhere does not work and creates ill feeling. Here is the deal as I see it. Everyone agrees to use the two pages properly and engage in discussion on both pages. This means that those who want to keep the article as it is must engage in discussion about the underlying issues on the arguments page, otherwise those that do want change will bring those arguments here. Martin Hogbin (talk) 16:58, 5 January 2010 (UTC)
- Except both pages are too long, and I find them hard to work with becaseu of that. JeffJor (talk) 16:38, 5 January 2010 (UTC)
- The answer to this is archiving, if somebody knows how to do that. Martin Hogbin (talk) 16:58, 5 January 2010 (UTC)
Disagree
Have essentially two sections
I think nearly everyone agrees that we should have two sections. In one section we should have the simple non-conditional solution (I think that non-conditional is a useful term in this discussion, which I use to mean not specifically mentioning or discussing conditionality, but not necessarily because the problem is agreed to be unconditional. Maybe the issue is being initially glossed over, in the interests of simplicity and clarity). In the second section we can mention the conditional nature of the problem and other variants and complications.
I am not yet talking about exactly what should go in each section, or what they should be called. I am just trying to get agreement to having a starting section that treats the problem simply, as we now have. Martin Hogbin (talk) 10:16, 5 January 2010 (UTC)
Agree
Martin Hogbin (talk) 10:16, 5 January 2010 (UTC)
The "conditional problem" - i.e., the one that uses Door #1/Door #3 - is a variant that definitely does not belong in the main section. Not only becasue it does not address the actual MHP, but because it does not help the uneducated reader to understand why the unintitive answer is correct. It only confuses him. It is NPOV to do it this way, because it separates the sources that treat different problems into sections that handle their own problems, rather than assuming the sources that handle the originally-intended problem are somehow wrong about what the problem they presented is. JeffJor (talk) 16:38, 5 January 2010 (UTC)
Disagree
I agree the initial section should focus on the fully symmetric problem, but strongly disagree about deferring a conditional probability solution to a subsequent section. IMO, this would not be NPOV. -- Rick Block (talk) 16:13, 5 January 2010 (UTC)
- This is roughly chronological order of the sources. None of the sources before Morgan mentions conditional probability, why should we? My main aim (and that of may others I believe) in the first section is to keep it simple. I personally would not object to a footnote stating things might be a bit more complicated. Martin Hogbin (talk) 17:03, 5 January 2010 (UTC)
- "None of the sources before Morgan mentions conditional probability"? This is absolutely false. Selvin's second letter [8] has a solution using conditional probability. The MHP was a well-known conditional probability problem in academia years before vos Savant's column. -- Rick Block (talk) 19:16, 5 January 2010 (UTC)
- Yes, you are quite right, I was persuaded by the arrogance of the Morgan paper that they had discovered conditional probability. Selvin's second letter does indeed consider the probability with which the host chooses a given door. He, naturally, takes this to be 1/2 (as has already stated that the host will choose randomly when he has a legal choice) and proceeds, without fuss to solve the problem. In the light of this it is hard to see what the Morgan paper adds to the story.
- "None of the sources before Morgan mentions conditional probability"? This is absolutely false. Selvin's second letter [8] has a solution using conditional probability. The MHP was a well-known conditional probability problem in academia years before vos Savant's column. -- Rick Block (talk) 19:16, 5 January 2010 (UTC)
- Even so, this is not what I would like to see at the start of this article. I think we need to have a balance between what some see as mathematical correctness and simplicity, so that we can fulfill the basic function of WP of informing our readers. Let me make a proposal below, it is similar to that on Nijdam's development page. Martin Hogbin (talk) 11:34, 6 January 2010 (UTC)
Somewhere in between
Too soon the article is getting complex. The Whitaker question however seems to be most famous, but other well known examples from reliable sources are welcome. My main suggestion would be to use the information which is already in the article, but reshape it:
1. Introduction. (Until "When the above statement". Move the last two paragraphs to the "History" chapter.)
2. Popular solution
3. Conditional solution
4. Aids to understanding
5. History (general)
---similar problems
---Monty Hall
---American Statistician
---Parade
6. Arguments and methods (detailed)
---Conditional or not
---Variants
---Bayesian analysis
7. Links and references
Heptalogos (talk) 21:47, 5 January 2010 (UTC)
Discussion and proposals aimed at reaching a compromise on this subject
This section is intended for discussion of what editors would like to see as the first solution.
I would like to see this as the first solution in the article, with pretty pictures, of course. I also would accept a footnote of some kind to indicate that some people regard this solution as incomplete, exact wording to be negotiated.
You choose a goat | You choose a goat | You choose a car | |||
The host opens a door to reveal a goat | The host opens a door to reveal a goat | The host opens a door to reveal a goat [1] | |||
You Stick | You Swap | You Stick | You Swap | You Stick | You Swap |
You get a Goat | You get a Car | You get a Goat | You get a Car | You get a Car | You get a Goat |
Rick, could you accept this? Nijdam? Martin Hogbin (talk) 11:47, 6 January 2010 (UTC)
- I'm not Rick or Nijdam. I have stated at least twice that the only appropriate way to 'major edit' the article is to start with the current version. That way all adds/deletes/modifies are clearly discernible with a true audit trail. Is this obtuse to anyone? Do I need to rephrase that sentence for clarity? Do you catch my drift? Glkanter (talk) 12:11, 6 January 2010 (UTC)
- My suggestion above is not suitable for immediate inclusion in the article because it needs pretty pictures. Everything done on WP is recorded so any agreements made here can be later transferred to the article with an 'audit trail' as you have put it the past. Do you like the suggestion above (with your choice of footnote, including none)? Martin Hogbin (talk) 13:51, 6 January 2010 (UTC)
- And the footnote infers that 'Probability' is the only discipline available to solve the puzzle. That is false, and I will not support it's inclusion with the 'probability/logic solutions'. Unless you want to go to 4 distinct solutions in chrono order: The one above from 1975 (with footnote), Selvin's indifferent conditional from 1975, vos Savants probability/logic solution from 1990, and Morgan's non-solution from 1991. Glkanter (talk) 12:46, 6 January 2010 (UTC)
- Would you be happy with a different footnote? If so what? Would you like to see the above solution with no footnote? Martin Hogbin (talk) 13:51, 6 January 2010 (UTC)
- As an alternative, I suggest again a unified solution section, more or less like #Proposed unified solution section. The first part of this is (I think) pretty much exactly like what you're suggesting. But then instead of a footnote it continues with a conditional solution of the symmetric problem presented as an alternative. Maybe this will be the meat of the mediation, but I don't see how deferring a conditional solution to a later section rather than including one at this point is anything other than POV favoritism. The difference is only one screenful of text and figures, basically one paragraph. -- Rick Block (talk) 14:51, 6 January 2010 (UTC)
- Here's another word Rick and I disagree on the meaning of: 'compromise'. Rick thinks (as per his edit summary) that going from 2 separate solution sections (as the article has today) to 1, is a compromise with the guy(s) who want 3 or 4. Another thing we disagree on is that chronological order has a POV. Maybe we should somehow have all the solutions typed on top of each other? To be fair, of course. Glkanter (talk) 15:13, 6 January 2010 (UTC)
- If that's going to be your attitude then I see no point in continuing this discussion until we have a mediator. -- Rick Block (talk) 15:21, 6 January 2010 (UTC)
Rick, as you say, my proposed diagram (which was taken from 'The Curious Incident of the Dog...') is quite similar to yours but it does not have door numbers which, according to K&W, only confuse people. The footnote has the advantage of allowing us to present a clear diagram, which most people can actually understand, but still be correct. I think you have forgotten how difficult this problem is for most people when they first see it. We need to do all that we can to make the problem and solution simple, at least to start with. The main point of the problem is that the answer is 2/3 and not 1/2. We must get this across first.
I do not think that starting simply and then going into more detail can be regarded as POV. It is how most good text books work. Can we leave what happens after this diagram for the moment. Would you accept the diagram, with an appropriate footnote?
Glkanter, do you like my proposed diagram at all? Would you be happy for the article to start with this?
Finally, I do not think that formal mediation will achieve anything more than we are doing here. Everyone needs to compromise a little. Martin Hogbin (talk) 16:40, 6 January 2010 (UTC)
- If the diagram represents the sources accurately, what else is there to say?
- OK. Lets compromise. 1st question: Is the contestant ever aware of a host bias? How? Why? 2nd question: How many Solution sections? In what order? 3rd: What about the SoK problem in the Variants - Slightly Modified Problems section? 4th: Other than as part of Morgan's solution, will the probability/logic solutions be describes as 'false'? Will there be a statement that Morgan's view is not universal, more likely a minority opinion?
- I think we need the Formal Mediation so that we can go to arbitration on ownership and filibuster issues. Otherwise, the consensus will continue to be improperly restricted from improving the article. Glkanter (talk) 16:55, 6 January 2010 (UTC)
- As far as I can see my simple diagram meets all your requirements. Can you confirm that you like it and would be happy for the article to start with it? If not, I am wasting my time with it. Martin Hogbin (talk) 17:03, 6 January 2010 (UTC)
- If you feel this would improve the existing article, I encourage you to make such an edit. If you are asking my opinion, I cannot render it out of the context of the existing article. It's a Featured Article. Why would anybody start over at ground zero, rather than add/delete/modify the existing article, or a copy? I've said this countless times, and you guys all just set up sandboxes all over the place. I take no responsibility for how anybody spends their time here, other than for myself. Glkanter (talk) 17:16, 6 January 2010 (UTC)
- You thanked me for my support below, how about some from you now. Before I spend my time creating pictures and uploading them, I want to be reasonably confident that I am not wasting my time. I am not starting at ground zero, most of what is currently in the article can stay as far as I am concerned. I just want to start the article with a simple, convincing solution that shows that the player has a 2/3 chance of winning by swapping. This is what is missing, in my view. If I added such a diagram and solution, would you accept it? Martin Hogbin (talk) 17:52, 6 January 2010 (UTC)
- Did I?
- If I've learned anything from Wikipedia, it's that nothing can be 'assumed'. Until your mods are in the article, or a copy, I can only assume where you're putting it, what else you're changing, etc. And if I do that, then I haven't learned anything after all. Oh, and I don't agree with a footnote, or any other disclaimer. Until Morgan's direct criticism in Morgan's solution section. Glkanter (talk) 20:06, 6 January 2010 (UTC)
- I guess I have to leave you to do things your way then. I was trying to reach some kind of consensus here but if you want to try, mediation, arbitration, edit warring, or whatever then go ahead. 86.132.191.65 (talk) 20:23, 6 January 2010 (UTC)
I think your table is quite clear, but I don't think it's easier to understand than the simple pictures in the Popular solution section. Actually I guess it's about the same as the first big picture. What's really different? Heptalogos (talk) 22:25, 7 January 2010 (UTC)
- Interesting division in Wikipedia policy: Text - no OR; Images - OR is OK. Glkanter (talk) 22:49, 7 January 2010 (UTC)
Maybe The Host Has A Bias Towards Pretty Women?
So, by watching, you realize he nods his head at where the car is. Now the female contestant has a 100% likelihood of selecting the car.
This is equivalent to Morgan's argument about a left-most door bias. It's published, but pretty darn stupid.
There is no contestant, or viewer, awareness of a host bias on a game show. And the puzzle begins, 'Suppose you're on a game show...' Glkanter (talk) 17:20, 5 January 2010 (UTC)
- Equivalent, except your scenario is NOT published in a well known peer reviewed statistics journal. And, in the extreme, if it were published (and there were multiple confirming sources) we should include it.
- These little sections you keep adding that suggest no specific change to the article at best belong at /Arguments. Please stop posting them here. -- Rick Block (talk) 19:23, 5 January 2010 (UTC)
- Glkanter's point, which I agree might be better in the argument page, is that even in a real-life scenario the player is unlikely to know much about the host's door choice and would therefore be reasonably expected to treat it as random. As this is, in fact, a mathematical puzzle, as the article makes clear at the start, it would be perfectly normal to take an unknown initial distribution as uniform. This makes the conditional answer exactly equal to the unconditional answer and the issue of conditionality somewhat irrelevant to explaining to a typical reader of this article why the answer is 2/3 and not 1/2. This is the improvement to the article that many of us here want. Martin Hogbin (talk) 23:51, 5 January 2010 (UTC)
- Thanks for the support? My point is that this Pretty Woman variant is just as likely as the left-most door variant that Morgan (or Rick?, I've lost track and interest) uses to stigmatize the probability/logic solutions. They're both made from whole cloth. Mine is just less opaque in it's ridiculousness. I would not strike the keys of my keyboard to discuss the initial distribution. Although, I have spent 15 months arguing over a host bias. Glkanter (talk) 06:17, 6 January 2010 (UTC)
- ^ We take it that it is unimportant which of the two possible doors that would reveal a goat the host opens. In the case that the host makes this choice randomly it turns out that this is correct, but nevertheless the problem is strictly one of conditional probability (ref Morgan), the condition being the door that the host opens. This, together with the variation that the host is known to choose non-randomly, is discussed in more detail below.
What is the argument in support of reporting Morgan's solution prior to the others'?
Why should Morgan's solution 'jump the line' over the other solutions, which were published earlier? It's no 'better' than any other. That would be a NPOV violation. Glkanter (talk) 16:37, 6 January 2010 (UTC)
- Where are you getting the notion that anyone is arguing Morgan's solution should be reported prior to any others? Is there a change to the article you're suggesting here, or is this actually a response to something else? -- Rick Block (talk) 19:48, 7 January 2010 (UTC)
Recent changes and discussions (early Jamuary 2010)
changes
I noticed that the article has changed quite a bit and not all for good.
good
- the current introduction looks good to me it thankfully stays away from Kraus&Wang and the unconditional vs conditional issue. But just state the problem in the parade version which made it famous
not good
- However, this is not the only mathematically explicit version of the problem. Were the names of the doors (the numbers 1, 2, and 3) fixed in advance (painted as huge numerals on each door), or are we naming the doors retrospectively: you choose a door and we call that door 1; then Monty Hall opens a door and we call that door 3; we then give the remaining door the name door 2? This latter appears to have been the intent of Marilyn vos Savant herself. <--- What's that supposed to be? Please no personal speculation of what the problem might be. And in the same manner no (unsourced) speculation of what vos Savant, Morgan or whoever might have had in mind. Stick to summarize what they've actually written.
- Source: MvS site. Part of the actual problem statement: "the host opens another door, say #3". "Say" reasonably meaning to give it a random name out of three. This enables the host to say, in the particular event: "Do you want to pick door #2?", also quoted from the statement.
- The first similar example described by MvS uses three shells. They are not numbered, reasonably because the chosen one is identified by a finger on top of it, and the other two similarly need no identification other than 'empty or not'.
- The second example presents all six possibilities, including openings of door 2, which are counted as valid outcomes. This is probably the most explicitly convincing one.
- The third example uses three playing cards in repeated experiment. The cards can't be numbered at all, because the numbers would reveal their value after a few times. This is probably the most implicitly convincing example.
- The last example is the experiment actually performed. This one is trying to cover randomness by throwing dices. Three cups are numbered, in reference to the only valid outcomes of the dices. Again, both cups no. 2 and 3 may be lifted and are counted as valid outcomes.
- So, it can be reasonably understood that the intent of MvS was not to fix numbers to doors, but rather to identify any possible situation at a certain moment. Heptalogos (talk) 13:11, 9 January 2010 (UTC)
- Marilyn herself raised the fact that she was misquoted by Morgan: [9], creating early misimpressions. She accuses Morgan of purposely focussing on semantic issues. She further writes that no additional stated conditions appeared important to a general comprehension of the problem because circumstances in default are reasonably considered random. And finally she states that 'we' (herself and Whitaker?) published no significant reason to view the host as anything more than an agent of chance who always opens a losing door.
- Morgan answered that they consider Whitaker's question as an original question, which makes any comment by MvS irrelevant. I don't know if MvS replied to that again, but she ends her letter with the phrase "I have given up on getting the facts across properly and have decided simply to sit back and amuse myself with the reading of it all".
- They both make sense and I think this is another example of our need to take distance from opinion, taste and ethics, and present the issues as they arise, anywhere relevant and reliable. Heptalogos (talk) 22:21, 9 January 2010 (UTC)
- Popular solution - 1975. <---Excuse me? None of the sources/descriptions in this section is from 1975 but they are from the 90s onward. Furthermore Selvin (which is the supposed 1975 reference) published a unconditional (=popular) and a conditional solution in 1975. Conclusion this section header complete nonsense.
- Probabilistic solution - 1991. <--- Similarly off as the other section title. All solutions are "probabilistic" if they compute probabilities and use probability theory. That's the case for Gardner, Selvin and later treatments (including that partially that of vos Savant herself). There is also no unconditional vs. conditional difference between 1975 and 1991 if the section header is supposed to allude to that. The only thing that was "new" in 1991 was a generalization of the conditional solution to model different host behaviours.
discussion
Much of the discussion still evolves around "What the real MHP is (according to us)", "What the appropriate or true solution has to look like (according to us)", "What vos Savant thinks the problem means (according to us)", "What Morgan thinks the problem means (according to us)", etc.. While this can be an interesting discussion in its own merit it is largely pointless for the article. For the article we have to provide an accurate/representative summary of he how the problem was defined/solved/treated in reputable literature and that's it. It's not up to us to "decide" whether Morgan or vos Savant or whoever was ultimately "right" or did solve the "real" MHP while the rest was doing something else. If all participants would stick to summarizing all reputable literature in a representative and readable fashion as a goal and stay away from cherry picking sources and pushing their personal view of the problem much of disagreements would vanish.--Kmhkmh (talk) 01:47, 8 January 2010 (UTC)
- So, what should the Solutions heading read for the first source that calls the Selvin/vos Savant solutions 'false'? Who gets the credit? What year did that happen? Some long-standing editors of the article seem to think that's a significant point in the history of the puzzle. Do you agree? Glkanter (talk) 22:47, 8 January 2010 (UTC)
- You need to decide whether you want to have the chapters being organized by content or chronology. If want to order them by content, then as explained above the current section headers make no sense. If you want to order them chronologically then the headers should be something like 1975 _ Selvin when Selvin posed & solved the problem and coined the term MHP. And 1990 - Parade/Whitaker/vos Savant when the problem became widely known and the "controversy" started. I don't see any particular importance of Morgan in the time line here. He was just the possibly first of string of academic and math publications that followed after the parade affair. The problem with chronological sections however is that you cannot separate the conditional from the unconditional solution (since both are around in 1975). Furthermore we have a chronological overview in the history section anyhow. So if we organize by content the section headers could be something like simple/popular/unconditional solution (essentially with the current content) and conditional solution/detailed mathematical analysis (partially with the current content (conditional solution)m but possibly also the bayesian section and the variants. The header detailed mathematical analysis might also indicate to readers, that people just looking for simple and sufficient explanation do not have to bother, however people interested in various other perspectives or a more "advanced" treatment of the problem might read on.--Kmhkmh (talk) 23:25, 8 January 2010 (UTC)
- Is the following structure in compliance with the requirements?:
- 1. Introduction. (Until "When the above statement". Move the last two paragraphs to the "History" chapter.)
- 2. Popular solution
- 3. Conditional solution
- 4. Aids to understanding
- 5. History (general & chronological)
- Similar problems
- Monty Hall
- American Statistician
- Parade
- Similar problems
- 6. Arguments and methods (detailed)
- Conditional or not
- Variants
- Bayesian analysis
- Conditional or not
- 7. Links and references
Heptalogos (talk) 11:47, 9 January 2010 (UTC)- That looks like feasible approach to me, alternatively i'd like to suggest te following structure maybe slightly better suited for a compromise:
- 1. Introduction (as is)
- 2.Problem (as is)
- 3.Solution (unconditional as is)
- 4.Aids to understanding (as is)
- 4.1 Why the probability is not 1/2
- 4.2 Increasing the number of doors
- 4.3 Chance of Picking Goat With the Assumption of Switching
- 4.4 Simulation
- 5.Detailed Mathematical analysis (contains all "advanced"/more complicated mathematical treatments)
- 5.1 conditional solution (basically the old "Probabilistic solution - 1991" as is)
- 5.2 Variants - Slightly Modified Problems (as is)
- 5.2.1 Other host behaviors
- 5.2.2 N doors
- 5.2.3 Quantum version
- 5.3 Bayesian analysis
- 5.x other math aspects
- 6. Psychological analysis (here Krauss & Wang, Mueser,Granberg and others could be treated in greater detail)
- 7. Sources of confusion (can treat math and psychological aspects together or alternatively moved in subchapters of 5 and 6
- 8. History of the Problem (as is)
- 9. See also
- 10. References
- 11. External links
- From my perspective either suggestion might be a starting point for the mediation.--Kmhkmh (talk) 13:01, 9 January 2010 (UTC)
- That looks like feasible approach to me, alternatively i'd like to suggest te following structure maybe slightly better suited for a compromise:
- 1. Introduction. (Until "When the above statement". Move the last two paragraphs to the "History" chapter.)
featured status
Given the recent changes in particular, but also the constant quarrel and maybe latent edit warring. I think it is time to review the featured status. This doesn't have to done right now and might be combined with the mediation procedure (or afterwards) but it should be done. Because the current or future article might be somewhat to significantly different from what was reviewed in 2005. Aside from some of problems listed further of the current article stability is also a criteria for a featured article.--Kmhkmh (talk) 01:47, 8 January 2010 (UTC)
- I agree to literally all you say. But an accurate/representative summary of how the problem was treated in reputable literature cannot at all be created without a (our) perception of it. So some of your points are literally useless. Let's do it your way, fully objective, consequently, and replace the article by a list of all reliable sources. Not even a choice of quotes of course. You think that's the essence of the featured status? Heptalogos (talk) 12:01, 8 January 2010 (UTC)
- Or could it be that you are suffering from the bad symptoms of our inspiration? The same inspiration that created the featured article, while the bad symptoms are almost all in the talk pages. I like your 'not good' stuff, but the 'discussion' paragraph is largely pointless for the article. Heptalogos (talk) 12:11, 8 January 2010 (UTC)
- The spirit that created the originally featured article has nothing to do with the endless and ultimately rather boring discussion that followed the 5 years after (in particular the last year, when I paid some attention to it). People are still arguing about changing the article in a way that essentially makes vos Savant or Morgan look "more right", it is just done in more subtle ways. As in "who is mentioned first", "Who is not solving the real MHP", "who should be considered a variant", "what should be moved to separate article", "should both approaches be described in a combined fashion" or for the latest the odd section titles described above. Quite often in the discussion people seem to willfully ignoring or misrepresenting sources not fitting their POV as well as the statements of other participants. If you look at the edit history of some of the involved participants, it also makes you wonder.... Imho the whole thing is as petty and pointless as the original squabble between vos Savant and Morgan, in fact this seems to the wikipedia extension of it.
- If you summarize the reputable literature, it is rather obvious that the article needs to contain both, an unconditional and a conditional treatment. Yet here we are, having a year long struggle of how to implement/realize the obvious and having "proxy battles" about marginal differences (section titles, who goes first, etc.). And for intermission we also doubt the obvious by inserting our own WP:OR and giving our own personal version of the real MHP and judge which reputable literature is wrong and which is right according to it. This is Wikipedia at its worst as far as constructive collaboration is concerned. It might be different though if you are here for the show or for sociological research or other reasons.--Kmhkmh (talk) 13:14, 8 January 2010 (UTC)
- I agree to most of what you say, but the irony is that everything you write after the 'good' and 'not good' is about the same kind of drama. Try to look at it this way: the article really has improved over the last year. Apart from that, people are learning on the talk pages. At least recreating. Do you maybe know where suggestions can be assigned for a background discussion forum? Heptalogos (talk) 14:39, 8 January 2010 (UTC)
- My point is that the article has not significantly (or at all) improved over the last 6 (or even 12) months. The only positive outcome the quarrel has produced are some additional sources, which is arguably good but not that important since the article due to wealth publication tends to rather oversourced than undersourced anyhow. Neglecting some the latest changes it has barely managed not to get worse (probably mostly due due to Rick Block constantly editing out the biggest nonsense). I would agree however that it still has significantly improved over the 2005/2006 version that became a featured article (see [10]). Looking at the early version however I wonder why it got featured at all, presumably the criteria and selection were still somewhat less strict back then. I understand that people learn on discussion pages and that a learning phase might be required in some discussions. However the main or strictly speaking sole purpose of this page is just (constructive) collaboration discussion to improve the article (which i barely see for last 9 months now, though the formal mediation might be bring some difference here). If people just want to discuss/argue their views of the problem, they can do that here Talk:Monty_Hall_problem/Arguments (see template at the top of the page as well), at another internal page created for such a purpose or outside Wikipedia (web fori, usenet, irc, real life), but ideally not on this discussion page. Anyhow just my observation or 2 cents if you will. I do not intend to join the neverending debate for long, aside from maybe helping in the mediation as this would be a more promising constructive attempt.--Kmhkmh (talk) 15:52, 8 January 2010 (UTC)
- Wikipedia is intended to work on a consensus basis, recognizing reliable published sources. You have chosen to not be part of the consensus. And the will of the consensus continues to be rebuffed.
- I may be the only active editor who came here simply as a reader of the article. In October, 2008, it was horrible. It is orders of magnitude more useful now. And could still be a lot better. FA or not. Glkanter (talk) 15:28, 8 January 2010 (UTC)
- I agree with Glkanter. You have chosen not to take part in most of the discussion yet you presume to tell us how the article should be edited. There is strong feeling amongst many editors that the article does not explain the basic puzzle and solution very well. This is a major failing in an encyclopedia for the general public that needs be be addressed.
- You refer to 'latent edit warring'. This is how WP is meant to work. Editors should discuss issues to reach a consensus then edit the article appropriately. The fact that you have intentionally absented yourself from this process does not give you any special rights here. Martin Hogbin (talk) 15:57, 8 January 2010 (UTC)
- I'm neither telling anybody how the article should be edited nor I'm claiming any "special rights", but I was outlining what follows directly from the wikipedia guidelines (and common sense actually). There is no problem with editors wanting to have a simple explanation. In fact we have one prominently featured early in the article (though under a questionable section header). There is however a problem with editors wanting to edit out everything other than "their" simple unconditional solution and who in doubt even do not mind to resort to unsourced material and apparently want the article to create the impression that the simple unconditional solution is all that there is to MHP.
- The fact that Wikipedia for the most part primarily targets the general public, does not mean we write a Wikipedia for Dummies and it does not mean WP only contains material "that everybody can understand". WP collects the knowledge of the world and that means a comprehensive treatment of topics. The important thing here is that articles are properly structured, i.e. information/content requiring a different level of background knowledge is in different chapters, with more complicated treatments and extensions towards the end of the article (which we kinda have here as well).
- 'latent edit warring' is not how WP is supposed to work. It is supposed to work by constructive collaboration not by never ending quarrels over essentially the same things and editing things back and forth. It is supposed to work by achieving a reasonable compromise/result which adheres to WP guidelines and then edit the article in agreement. Maybe the mediation will achieve that, we'll see.
- And finally regarding Glkanter's point. Wikipedia is not just some arbitrary consensus by currently active editors somehow using some reliable sources. Wikipedia is a consensus within the WP guidelines and representing the (available) sources appropriately. Or to put it this way there is no such thing in WP as consensus outside the guidelines or that misrepresent sources.
- This is only partly true. It may be that 90% of WP is actually arbitrary consensus, while 90% of the editors sees no issues in it and leaves it unquestioned. As long as people agree with it, most people won't bother to request resources. Check this site: Conditional probability. Where do all text and examples come from? What about the definition of the main subject in de second line: "the possible outcomes of the experiment are reduced to B". It's simply an editor's opinion, but most editors simply agree with it. That's WP also. Heptalogos (talk) 21:20, 8 January 2010 (UTC)
- I do however agree that if the current problems are fixed this version can be seen as an improvement over October 2008 and having the unconditional and the conditional solution clearly separated is a plus ("different background knowledge in different chapters"). However this does not mean the criticism of the unconditional can simply be ignored, but it can be discussed in a separate section comparing both approaches or in the section of the conditional solution.
- Anyhow I'm off until the mediation assuming we get one anytime soon)--Kmhkmh (talk) 17:20, 8 January 2010 (UTC)
- Ugh! More Straw Men. Enough!
- We call them Aunt Sallies in the UK. Knhkmh has made up a list of mad things he claims we all want to do just to show how bad we are. Martin Hogbin (talk) 10:44, 9 January 2010 (UTC)
- Ugh! More Straw Men. Enough!
- Where does this presumption that the 'consensus' editors respect Wikipedia guidelines less than you, and Rick, and Nijdam (oh, please) come from? I presume everybody is operating in good faith, until they demonstrate, usually via some hipocracy, otherwise. Your interpretation of Wikipedia's guidelines has merit, just like anyone else's. To present it as a fact that some violation has or will occur is hardly appropriate or supported by the writings of any editor, including Glkanter. And it ignores the prevailing Morgan POV of the current article. Glkanter (talk) 17:40, 8 January 2010 (UTC)
What EXACTLY was the original Parade problem statement?
It seems there is some variations in what the actual problem said. RussAbbott had recently changed the wording of the problem in a minor way, apparently to make it read better to him. Since it is supposed to be a quote, I went off to the "external link" for MvS's web site, as listed at the bottom of the article, and cut-and-pasted what she lists as the original question. Here that is, verbatim:
- Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?
I did post that change with it a snide comment, referring to how Morgan had significantly changed the meaning of the problem by rewording it. I guess Rick took offense, and "changed" it back to what he says appears in a "copy of the column as it was published." That consisted of some insiginicant changes: two in puncuation (the first period becomes a colon, and the first comma in the second sentence becoems a semicolon), and using "No." instead of "#" to mean "number." But two minor wording changes were also included, inserting the words "then" in "He then says to you," and removing "of doors" from the end.
The problem is, I don't have a copy of the column (it would be helpful to post it, Rick) but I have seen numerous variations in the alleged "quotations" of this problem in literature. And what I consider to be the best source amid conflicting ones, MvS quoting herself, agrees with my version:
- MvS's book The Power of Logical Thinking is exactly as I listed it, except it uses "number" instead of "#". I consider that to be just a change in editorial style, maybe even done without her knowledge.
- Rosenhaus quotes that exactly; but does attribute it to the book, and not the column.
- Grinstead and Snell substitute "Monty Hall's Let's make a Deal!" for "a game show," and "Monty" for "he" later on. They also move "You're given a choice of three doors" to the second sentence, separating it from what I listed with a comma. And they don't use any form of "#", just listing the bare number.
- Krauss and Wang use a semicolon where Rick did, and capitalized "Number" wherever it appears in place of "#". But they also added Rick's "then" and changed "pick door #2" to "switch to Door Number 2".
- Morgan takes significant liberties with the quote, which are discussed elsewhere. They include using "No." and Rick's "then."
- Are there any others?
My question is, is there an internet source to verify the actual (not quoted) version? And if not, shouldn't we trust MvS's own quote of herself over all others, since there seems to be a inexplicible tendency to misquote it? JeffJor (talk) 17:54, 8 January 2010 (UTC)
- Oh for Christ's sake, you think I'm LYING? I have a printout from a microfilm reader of the Sept 9, 1990 edition of Parade Magazine (as published with the Knoxville News Sentinel). The quote as it has been in the article for quite a long time is exact (including punctuation and capitalization). I am NOT going to scan the copy I have and post it. If you don't believe me, you can verify this for yourself (go find a library that has a microfilm copy of a newspaper that included the supplements). -- Rick Block (talk) 03:00, 9 January 2010 (UTC)
- No, I don't think you are lying, Rick. I said no such thing, and had no intent to imnply it. But I had actually cut-and-pasted my version from a direct source, so I felt about the same way you do about the edit you made. And I didn't go on to accuse you of calling me a liar. The fact that I didn't put it back to what that source says, and asked for verification and discussion, shows that I am exhibiting a great deal more integrity than you are blaming me for.
- And, I do think (1) That microfilm of a newspaper can be mis-read when it comes to the difference between a comma and a semi-colon; (2) That sometimes we see what we want to see even if it isn't there, and (3) that it just might be possible that some editors - like, say, those in Knoxville - "corrected" what they thought was incorrect usage. We actually have evidence of these sorts of thing, since Morgan grossly misquoted it and the internet reference and the on-line version of MvS' book contain at least one such change. Again, I'm not accusing you of any of these things, I'm saying we can't know unless we can all see your source.
- The point is, I don't know what was originally published; but we have an easily-verifiable source that anybody can see, and that doesn't require looking at microfilm and/or wondering if some phantom editor changed something. And since you were the one who took it upon yourself to make those insignificant changes (unless you think there is a significance?), yes I do think the onus is on you to prove it is correct when MvS herself quotes it differently. We can change the reference to her book, which is about the column, if you desire, and remove all these difficulties. Or keep it as is, and reference Parade Magazine, as published in the Knoxville News Sentinel.
- And this digression kinda echos the entire problem with this article. It seems there are multiple versions of "truth," and the discussion tends to be more centered on "Why can't all you idiots see the truth the way I do?" rather than handling the different "truths" for what they are - truths that are based on different sets of assumptions. Morgan, et al, did solve an actual ptoblem; but it clearly (sources, including MvS, say so, Rick; and the only significant difference in their interpretation is whether door numbers are important, unless you know of anoither. Everything else they criticze is what they say MvS assumed, which is not a difference in interpretation) isn't the one Marilyn intended, or what her problem (which she formatted, not Craig Whitaker) sematically says. JeffJor (talk) 17:15, 11 January 2010 (UTC)
- I agree that it is important that we get this right. Just out of interest, does anyone know if Craig Whitaker was a real person? If so has he ever made any comment about the subject? Martin Hogbin (talk) 18:22, 8 January 2010 (UTC)
- Per above, if you think it's not right feel free to verify it yourself. Craig Whitaker is (well, at least was) a real person. My understanding from a reliable source (that I will not divulge, which means it's not a "fact" as far as Wikipedia is concerned) is that some reporter (surprisingly, not John Tierney from the NY Times) tracked him down and that what appeared in Parade is not exactly what he wrote (if that's where you're going) but omitted that he thought the answer could be 1/2 or 1/3 depending on what was assumed. -- Rick Block (talk) 03:00, 9 January 2010 (UTC)
- Since I, too, live in Columbia Maryland, I looked him up in the phone book. I found no Craigs. And I didn't want to bother the ten-or-so Whitaker families to ask, since it is likely he graduated from one of our High Schools about fifteen years ago. And if we know it isn't exactly what Craig wrote (as per Rick Block's recollection), we need to remove his name from the reference list and make it refer to MvS. Quote her, quoting Craig, and don't imply Craig is an expert at anything. JeffJor (talk) 17:15, 11 January 2010 (UTC)
- I was asking just as a matter of interest, I do not doubt any of what you say. I just thought that, if he could be contacted, it would have been interesting to ask him what exactly he had in mind. It did just occur to me that he might have been someone made up by the editor or vS. It is good to know that he is real. I wonder what the makes of the furore that his question created.
- Similarly I do not think that Jeff was accusing you or anyone else of anything. Because the exact question is important in statistics, both Jeff and I think that it is necessary to get the original question exactly right. Thank you for confirming this. The matter is of particular importance to some of us because of the way that, in our opinion, Morgan morphed the question into a different one. Martin Hogbin (talk) 10:34, 9 January 2010 (UTC)
What is the best internet source available? Is it the MvS site? Is the Parade edition available in a public library? Do we have arguments about difference between both? If so, what is the WP policy about reliability of non-internet sources? Heptalogos (talk) 14:11, 9 January 2010 (UTC)
- Not every citation needs an online source although I agree this is desirable. The article currently cites Whitaker's letter in Parade as this is the definitive source. Martin Hogbin (talk) 15:52, 9 January 2010 (UTC)
- The policy regarding sources is Wikipedia:Verifiability with more detailed guidelines at Wikipedia:Reliable sources, and you basically have your question backwards. From Wikipedia:Reliable sources#Quotations: Quotations should be cited to the original source if possible. Parade is a magazine supplement published with hundreds of newspapers (in the U.S.). Microfilm copies of newspapers are available in many public and university libraries. For a long time this quote was indirectly referenced ("as cited by") to an article by Bohl et al. in the 1995 edition of Journal of Recreational Mathematics. As part of the last featured article review I did a large amount of referencing work (one of the concerns was "Huge sections of unsourced content") and as part of this chased down an actual copy of the Sept 9, 1990 edition of Parade (as I mention above).
- This entire thread is grossly insulting. I reverted Jeff's change because it was wrong, not because I took offense to his snide comment about Morgan. I really couldn't care less what he thinks about Morgan (or, at this point, pretty much anything else). What I said in my edit summary was "revert to the actual quote - I'm looking at a copy of the column as it was published". In this thread he's both attributing a motivation for my change that he simply made up (which is bad enough) and calling me a liar. Martin echoed Jeff's insult "I agree that it is important that we get this right" (what about "I'm looking at a copy of the column as it was published" is not clear???). And you (Heptalogos) are questioning what the sourcing policy is (as if any internet source could possibly be more reliable than the actual magazine).
- I'm trying very hard not to say something grossly insulting in return here. Let's just say that I think all three of you should apologize. -- Rick Block (talk) 16:25, 9 January 2010 (UTC)
- I can see why Rick has taken exception. His edits, his research, and his very integrity have, in his mind, been attacked, without provocation or fact-based support.
- Yes, there's an RfC open on your behavior. Your post here is yet another example of your bad behavior. I invite anyone reading this to add their opinions to Wikipedia:Requests for comment/Glkanter. -- Rick Block (talk) 17:28, 9 January 2010 (UTC)
- Rick, you are taking this far too personally. I, and I suspect Jeff, just thought that there was some doubt as to the exact wording of the problem, looking at the various sources on the subject. You have confirmed that the current wording is exactly as it is in Parade magazine, some good work on your part. That is the end of the matter as far as I am concerned.
- Heptalogos was, I think, just suggesting that some online references would be good (if there are any for the exact quote) so that readers and future editors can verify the problem statement for themselves. Martin Hogbin (talk) 17:48, 9 January 2010 (UTC)
- Since the reference is to the printed magazine, it can't be verified definitively against any online source, and so the question of what online source is the best to use to verify the quote is correct is meaningless. At this point, Wikipedia is as good as any other online source - actually, IMO, better since anyone is free to verify it with their own eyes (as I have done) and then correct what it says. In addition, it's a featured article meaning many editors have closely looked at the entire article so the chances that someone has actually verified the quote here is correct are quite high (this applies to any featured article, not just this one).
- And, as far as taking it personally, I think I'm taking it in the spirit in which it was intended. -- Rick Block (talk) 18:54, 9 January 2010 (UTC)
Let's stay with the facts and strip them from complex emotions. This is what I understand of it:
1. The Whitaker quote in this article is different from the quote on the MvS site, which is claiming to be an original Parade publishment.
2. A WP-editor, senior Admin, claims to have seen the original Parade article which is exactly the same as in the article.
3. Quotations should be cited to the original source if possible.
4. Any internet source is less reliable than the actual magazine.
5. We have two internet sources quoting the original source: senior Admin and MvS.com.
6. Senior Admin is more explicit in the claim that his quote is very exact.
7. In absence of written policy, do we have any decisional law (jurisdiction) on this? Heptalogos (talk) 20:03, 9 January 2010 (UTC)
It seems to me now, from this source, that Marilyn is about just as explicit in her claim of excellent quoting. An interesting sentence in that, which may even give a hint towards the cause of apparent disparity, is: "Here are both (question and reply), as they first stood". Indeed, MvS site introduction states: "This material in this article was originally published in PARADE magazine in 1990 and 1991". Rick, is the Sept 9, 1990 edition of Parade Magazine, the first one about the problem? Heptalogos (talk) 22:45, 9 January 2010 (UTC)
- '...then perhaps we would' and the page stops! What?! We would 'what'?!? I gotta know!
- These guys are sorry excuses for whatever it is they are. And we've got the 5th Beatle right here on Wikipedia. Glkanter (talk) 23:05, 9 January 2010 (UTC)
- Hard to find the complete PDF, but here's another interesting source: [11]. Heptalogos (talk) 23:45, 9 January 2010 (UTC)
- The initial column was Sept 9, 1990. Well, at least, that's what I'm claiming. How about if you verify what the magazine actually says yourself? Just go to a library and look it up. There's a list of newspapers that currently carry it here. Call your local public library and ask if they have microfilm of any of these newspapers (from 1990), including the Sunday supplement sections (specifically Parade Magazine). And, yes, I'm saying in her reply to Morgan et al. vos Savant (like Morgan et al.) trivially misquoted her column. She used a comma rather than semicolon following "Behind one is a car", used "#" rather than "No.", and dropped "then" and added "of doors" in the last sentence.
- JSTOR makes only the first page of the references it carries available for free. I have the next page of this one as well. I could tell you what it says, but given how absurd this thread has become I suggest you go to a library (probably a university library for this one) and look it up yourself. -- Rick Block (talk) 01:27, 10 January 2010 (UTC)
- On my first visit to the US I might take a look in Parade. In that case I should sign the source witness list, because when I leave here, another editor may ask the same question. I am trying to understand how WP works. But you are convincing and the change seems to be a cosmetic improvement. It makes sense. Marilyn is a proud lady. Heptalogos (talk) 10:18, 10 January 2010 (UTC)
- You could try to obtain a reprint of the Parade column as well, see [12]. However, I'm not sure if a reprint obtained this way would necessarily be identical to what was originally published. Another idea would be to find a library (possibly a university library) that might have microfilm of a major US newspaper that carries the magazine. Possible examples would be the Chicago Tribune or the Los Angeles Times. If you're a student or faculty member at a university I'm sure your university library could obtain a copy for you, and if you're not at a university then the reference librarian at any decent public library should be able to help. -- Rick Block (talk) 17:07, 10 January 2010 (UTC)
- Rick, I have to take issue with one point you make above. You say, '...vos Savant (like Morgan et al.) trivially misquoted her column...'. I agree that vos Savant's misquotation was trivial but Morgan's was not. It was part of a subtle process that changed the question into something different.
- Anyone who has studied the MHP, either here or elsewhere, must know that the exact question is important. Things which at first sight might seem irrelevant turn out to make a critical difference, for example, most people are surprised to be told that it makes a difference whether Monty knows where the car is.
- The Morgan misquotation starts to move the problem statement from saying that the host opens a door with and explanation of what that might mean, to the host opens door 3, where it is clearly the intention of the question to specify which door the host opened. These things are important in probability problems and Morgan's misquotation is quite inexcusable. Martin Hogbin (talk) 10:45, 10 January 2010 (UTC)
http://www.huffingtonpost.com/2010/01/08/our-little-genius-kiddie-_n_416440.html
To argue that the host indicating to the contestant where the car is located is consistent with the statement "Suppose you're on a game show..." is contradicted by the facts. Glkanter (talk) 18:32, 8 January 2010 (UTC)
- Can you be more explicit in what your exact proposal or criticism is about? Heptalogos (talk) 20:54, 8 January 2010 (UTC)
- Sure. Morgan, and others call the simple solutions 'false'. Then they claim to 'prove' this by concocting a host bias, that presumably the contestant is aware of.
- Issue #1 is that this is not a valid method of dis-proving a solution, this criticizing problem 'B' as a means to discredit problem 'A'.
- Issue #2 is that there can be no such transfer of knowledge, in any way, shape or form from the host/producer to the contestant. As this article demonstrates. I've posted 2 Wikipedia articles on this subject previously. Symmetry prevails by definition.
- Morgan and others published it. It goes in the article. Thoughtful editors may choose not to over-emphasize Morgan's critique in the article. Glkanter (talk) 21:18, 8 January 2010 (UTC)
- OK, can you please move this to the arguments page first? Heptalogos (talk) 21:28, 8 January 2010 (UTC)
I'm done with it. Besides, it's about how to edit the article. That's what this page is for. Glkanter (talk) 21:37, 8 January 2010 (UTC)
How Goes The Formal Mediation Filing?
I tried to sign that I'm willing to go along, or whatever. Is there a place to do this?
Has Nijdam responded? Glkanter (talk) 13:21, 9 January 2010 (UTC)
- The request has not been officially filed yet since Nijdam has not responded. I've tried to contact him via email. He hasn't made any edits since Jan 2. I suspect he may be on holiday. I'd like to give him another week or so to respond, but if you're unwilling to wait that long you can certainly file the mediation request yourself (you'd have to rearrange the draft I created a bit). -- Rick Block (talk) 18:07, 9 January 2010 (UTC)
- If I filed it without listing Nijdam, is it still valid? Would Nijdam or any other editor be able to claim I had filed a 'biased' request? Glkanter (talk) 18:14, 9 January 2010 (UTC)
- I think you are maybe misunderstanding the idea of mediation. The mediator will not try and force an agreement on us, they will just try to help us to work together. So, if Nijdam is happy with the mediation process, it will continue. If anybody says that will have nothing to do with it but they will continue to push for their wishes then it cannot work. I do not have much faith in the process now, after failing to get even one side of the argument to agree, but I am happy to give it a go. Martin Hogbin (talk) 18:30, 9 January 2010 (UTC)
- If I filed it without listing Nijdam, is it still valid? Would Nijdam or any other editor be able to claim I had filed a 'biased' request? Glkanter (talk) 18:14, 9 January 2010 (UTC)
So, it's intractable, then. What official Wikipedia steps remain available to us? Or do we just keep arguing and over-editing each other in the article in a non-3RR manner? Glkanter (talk) 18:39, 9 January 2010 (UTC)
I have done some editing
Having tried in vain to reach any form of consensus I have made a few edits to see what the general reaction is.
I have deleted the unsourced comment about door numbers and replaced it with a comment (citing Seymann) that Morgan address their interpretation of the problem, in that specific doors are identified in the problem statement.
I have added what should be an uncontroversial explanation of the existing diagram. Martin Hogbin (talk) 14:35, 9 January 2010 (UTC)
- Kmhkmh's criticism about the unsourced comment is in the 'Recent changes' chapter on this page. I added the source and explanation to that, so I guess we'd better check or discuss it before deleting the entire thing. Heptalogos (talk) 18:44, 9 January 2010 (UTC)
- The point is that the section I removed seemed to be a view that is pretty well unique to you, and certainly not one I have seen mentioned in any reliable source. I think the fact that the numbers of the doors may not be intended to be important is better covered by Seymann's comment on the Morgan paper, even though he does not specifically state this.
- I think that you are making much the same point but in a different way. Martin Hogbin (talk) 18:53, 9 January 2010 (UTC)
- "Unique to you": do you know that Rick added this view? Heptalogos (talk) 20:10, 9 January 2010 (UTC)
- Sorry, I though it was you who was promoting this possibility. Martin Hogbin (talk) 21:18, 9 January 2010 (UTC)
- What view did I add? -- Rick Block (talk) 01:29, 10 January 2010 (UTC)
- I checked the history and thought I saw that you were the one adding the paragraph in the Problem section about 'fixed door numbers'. Isn't that true? Heptalogos (talk) 22:18, 10 January 2010 (UTC)
- I found another source. Please check under 'Recent changes'. I may understand your question about Whitaker being a real (third) person. Heptalogos (talk) 22:57, 9 January 2010 (UTC)
With the above changes I am not that unhappy with the article. I would still like to.
1) Rename 'Popular solution - 1975' maybe 'Simple solution' are just 'Popular solution' again.
- The year really has no meaning here.
2) Rename 'Probabilistic solution' 'Conditional solution'.
- All solutions are probabilistic, this is the conditional solution.
3) Move the 'Aids to understanding' section to be immediately after 'Popular solution'.
- If you read this section you will see that none of it relates to the conditional problem/solution
Martin Hogbin (talk) 16:05, 9 January 2010 (UTC)
- I've gone ahead and deleted the dates from the headings. It seems clear there's no support for including these dates. -- Rick Block (talk) 16:39, 9 January 2010 (UTC)
How about we do away with the 'History' section, and present the Solution sources chronologically? With headings and sub-headings for clarity? Then, dates would be very useful. Glkanter (talk) 17:25, 9 January 2010 (UTC)
- Thanks. Do you see my point about the 'Aids to understanding' section? This is aimed at the general reader, who may not accept or understand the solutions presented. Later sections are for more advanced readers. Martin Hogbin (talk) 17:35, 9 January 2010 (UTC)
- I still think we only need one Solution section including both an unconditional style and conditional style of solution. In my opinion, the more you separate the "popular solution" and the "conditional solution" the less NPOV you make it. Please think carefully about your reason for wanting to avoid a conditional solution up front. Is it really because it's a more complicated approach, or is it because it doesn't fit your POV that the problem should be approached unconditionally? -- Rick Block (talk) 17:51, 9 January 2010 (UTC)
- My main reason for wanting to change this article has always been that it fails to provide a simple and convincing solution and explanation for the general reader. I think you have lost sight of just how hard it is for most people to understand and accept the simple solution to the non-conditional problem. The last sentence in the lead says, 'Even when given a completely unambiguous statement of the Monty Hall problem, explanations, simulations, and formal mathematical proofs, many people still meet the correct answer with disbelief'. If we just state the facts without making them understood by our readers, no matter how well they are supported by reliable sources, we fail in our job of producing a good encyclopedia. The main point of the MHP is that you can tell people the answer, and still they do not believe you.
- My POV is that the issue of conditionality is not that important at the start of the article, although it should be discussed later as it was raised in a published source. That is why I say that I want to treat the problem non-conditionally (meaning just not dealing with that particular issue) rather than unconditionally, to start with. I am happy to continue to discuss the subject of conditional probability on the arguments page with anyone who is interested. Martin Hogbin (talk) 18:22, 9 January 2010 (UTC)
- I've mentioned this before (perhaps I should look for a published source supporting this view), but IMO at least one reason many people disbelieve the typical unconditional answer is because it does not have the same form as how people generally interpret the problem - in particular, the fact that the host has opened a particular door completely vanishes. Most people (the K&W study does support this) internalize the question as asking what is the probability in a specific case, e.g. given the player picks door 1 and the host opens door 3. The unconditional solution isn't restricted to this case. I think a solution that addresses BOTH the unconditional situation and the conditional situation is likely to be far more convincing than only an unconditional solution. -- Rick Block (talk) 19:50, 9 January 2010 (UTC)
- This really is OR. K&W showed that door numbers generally confused people and that people generally did better when they were not involved. My take on this is that the problem is more easily explained and understood without door numbers. Just as we do now to start with.
- I am sure that nobody finds the problem hard just because they think it might matter which door the host opens. Most people miss this point completely, and assume that it cannot possibly make any difference (whereas in truth it could possibly make a difference but does not actually do so, with consistent assumptions). I certainly did not imagine that the door opened by the host could matter, and it is not even mentioned in the 'Three prisoners problem'. More to the point there is no source that I am aware of that claims that this possibility is what makes the problem difficult. Martin Hogbin (talk) 20:14, 9 January 2010 (UTC)
- Though I kinda agree with you here, I do on the other hand not mind having 2 different solution sections (unconditional/simple and conditional/detailed mathematical analysis) - in particular if this provides compromise everybody might be live with. If one takes a step back from what he might consider the "optimal" version of article from his perspective but rather thinks of an acceptable or sufficient version (not being optimal to oneself but acceptable to all involved editors and readers in general), then this should be the way to go imho.
- Also separating "easy" from the "hard" is definitely good idea. This is a general organizing principle for a well written article/book/whatever anyhow. The fact that this approach might be convenient for Martin potential POV ("it has to be solved unconditional") is irrelevant, since the advantage of that approach are real and having nothing to with Martin. The German probability book, I've mentioned occasionally (Henze) for instance pursues exactly that approach. It mentions MHP in the introduction as an example for probability theory or problems in the public domain and then later gives the unconditional solution. Much later after having laid some theoretical groundwork and having introduced conditional probabilities he revisits the problem for a more detailed analysis and a conditional solution. I see no reason why the our article can't do the same.--Kmhkmh (talk) 20:42, 9 January 2010 (UTC)
- Maybe we are moving towards some kind of acceptable compromise here. I wonder what others think? Martin Hogbin (talk) 21:15, 9 January 2010 (UTC)
- I agree to most suggestions, especially to the terms 'Simple solution' and 'Conditional solution'. We must remember that the 'Monty Hall problem' is basically a paradox simply solved by the popular solution, and that only specific problem statements, like in Parade, or specific interpretations of them, need conditional solution. This is another argument for starting with the simple solution. Also indeed, 'Sources of confusion' should really be (far) below 'Aids to understanding'. But I am getting into trouble as to where chapters should be, and how they should be structured. It tends to become rather POV or endlessly arguable if we don't use any objective structure. As Glkanter proposes: use chronology. I think it may be good if the whole article is made up that way, which will naturally present the entire scope as it grows in complexity and perspectives. I do understand that most 'normal' subjects are better off with a simple definition and explanation first, but if there's one thing this issue has proven, it is the fact that there is no single truth here. There's lots of different questions with lots of different answers and lots of different methods. The introduction may spend some words on it, to explain the choice for chronology. This should really reduce our disagreements! What is important in this scenario, is to have an exhausting 'contents' tree, from which one can easily jump to the section of choice. The contents section should give enough description to have a fair idea of what it's about. Heptalogos (talk) 22:01, 9 January 2010 (UTC)
- Chronological, but none of those 'pointy, POV dates' will be allowed. Glkanter (talk) 22:15, 9 January 2010 (UTC)
- Chronology contradicts my suggestion and what Martin seems to be indicating as well. So you cannot support the above (organizing by content and difficulty level) and wanting a chronological organization for the sections, it is either or here. Again I'm getting the impression we are moving 1 step ahead and 2 back. As soon as there seems to be some reasonable common ground, another issue is raised or something rather inconsistent statement is put forward as well. --Kmhkmh (talk) 23:03, 9 January 2010 (UTC)
- As a matter of fact I can support changes to the current structure as well as favoring another structure. It's an expression of flexibility which enables reinforcement of common ground, while at the same time offers an opportunity to improve significantly in the long term. Although I am very aware of the extra energy it takes for the time being. If we all keep getting tired, I think we indeed keep getting tired, for a much longer time. Heptalogos (talk) 23:21, 9 January 2010 (UTC)
Why Didn't Whitaker Write To Morgan, et al, Instead?
Because they don't have a general interest column read by millions every week published as a supplement to hundreds of newspapers across America.
In her letter to The American Statistician vos Savant directly tells Morgan her interpretation. She tells them *her* interpretation of Whitaker's letter. But these guys claim to know these things better than vos Savant herself. She had helped to educate over 1,000 PhDs on this paradox, she says. But not this Professor Morgan and his 3 assistants. They are different. Or, having spent 15 months on this article, maybe not so different.
And if Whitaker is fictional? Then vos Savant and her publisher made him up. And they know what they meant to ask.
So, back to the article. How will the article cover the aspect of Morgan calling all simple solutions 'false'? That means Selvin, vos Savant, Adams, Devlin, etc., etc. all are wrong. Sad state of affairs in academia these day. Hasn't been the same since 1991, really. All because Morgan claims a game show host can tell a contestant where the car is.
Will it lead off the solution section? Go last, but bold? Mentioned after each of the other solutions? I want to know. Where else will this revelation be placed in the article? How many times? Because Selvin already did the conditional solution, in 1975. What is it that Morgan's paper is noteworthy for then? The only thing left is calling the simple solutions 'false'. I guess countless Professors, etc. haven't gotten the word on this paper, yet, because they still teach it.
Morgan says this: "...(the producer)...is free to consider a variety of factors in determining how the game will be run." That's correct. Including all applicable laws. Game show hosts and producers do not tell contestants where the car is. Glkanter (talk) 13:55, 10 January 2010 (UTC)
- What is YOUR suggestion? -- Rick Block (talk) 14:50, 10 January 2010 (UTC)
- What I understand, and why, I have made abundantly clear. I have, and will continue to share my views with the consensus of editors.
- But, you're the long-standing defender of Morgan's paper. What do you think are the new issues the paper brought forth in 1991? Why is it significant? Glkanter (talk) 15:39, 10 January 2010 (UTC)
- I have just added a section [13] to my Morgan criticism page, which shows the question that Morgan have actually answered. I would be very interested to hear from both of you whether you agree that Morgan's paper actually addresses the question that I have stated. This may stop some pointless argument. Please leave your comments on the associated talk page. The footnotes are simply to address Morgans claim to have given a solution based only on information given in the problem statement. For the moment, do you both agree that the Morgan paper is a fair answer to my stated question? Martin Hogbin (talk) 15:08, 10 January 2010 (UTC)
- I agree that your problem statement is one wording of what Morgan et al. calls the "vos Savant" scenario. You might note that Gillman addresses this same problem as well. -- Rick Block (talk) 16:30, 10 January 2010 (UTC)
- It looks as if we can agree on the exact question that Morgan actually answer. Martin Hogbin (talk) 17:09, 10 January 2010 (UTC)
- I agree that your problem statement is one wording of what Morgan et al. calls the "vos Savant" scenario. You might note that Gillman addresses this same problem as well. -- Rick Block (talk) 16:30, 10 January 2010 (UTC)
- You are being rather kind to Morgan though. The fail to mention in their paper that: the car is originally randomly placed, the player chooses randomly, the host can never open the player's originally chosen door, and the host must always offer the swap. Perhaps they take these rules to be somehow transcluded from vos Savant's analysis or elsewhere. Martin Hogbin (talk) 17:43, 10 January 2010 (UTC)
- Falk, also. It's one of the versions Krauss and Wang address, too. -- Rick Block (talk) 16:34, 10 January 2010 (UTC)
- Glkanter, do you agree that Morgan answer my stated question correctly? Martin Hogbin (talk) 17:11, 10 January 2010 (UTC)
- You can find the question that they answered here [14]. Note that I do not claim that this is the MHP. In fact I assert that it is not. Martin Hogbin (talk) 23:53, 10 January 2010 (UTC)
This New Archiving Taking Place
I just took a look at archive #12. I think sections are being moved, but not in the order they were created. So the archive does not preserve the original discussions as they took place.
Would anyone mind confirming this? Is this the way archiving should work? Glkanter (talk) 13:05, 11 January 2010 (UTC)
- I've asked the bot owner. See user talk:Misza13#Archive order. -- Rick Block (talk) 19:48, 11 January 2010 (UTC)
- Thanks for doing that, Rick. I just read his response. He says that's the way it's supposed to work. I'm only experienced in 'data' archiving, not 'conversation' archiving. I really don't know, but it seems like it defeats the purpose. It never occurred to me that MHP archives 1 - 11 were built that way. Glkanter (talk) 00:39, 12 January 2010 (UTC)
So, What Are The Significant Events, And Why, Of The Monty Hall Problem Paradox
Some people certainly didn't like the chronology I posted on the this talk page. Heck, it was vandalized, then they put up an RfC/U on me because of it.
And when I added the year to the Solution sections, an edit war damn near broke out. And editors turned on editors.
So, who, what, where, when, and why? But especially this Morgan paper. It seems its only contribution is to use an unimaginably wide paint brush to call the unconditional solutions false. Glkanter (talk) 13:32, 11 January 2010 (UTC)
Glkanter, if you would answer whether you think Morgan have answered this question correctly, it might throw some light on the subject. Martin Hogbin (talk) 13:47, 11 January 2010 (UTC)
- I looked at it earlier this morning. First, I would need to know why it even matters. Otherwise, I just can't bend my brain to comprehend that stuff. 6 footnotes? Sorry, it's just not an area I'm strong in, or that I have much interest in. Nor do I think my opinion on that OR is relevant to editing the article Glkanter (talk) 13:59, 11 January 2010 (UTC)
- Ignore the footnotes for the moment, just read the question. I am trying to reach some kind of resolution over your, often repeated point, 'Suppose you're on a game show'. Note that my question does not suggest that perspective, it simply asks you to solve a problem based only on the information given in the problem statement. Do you agree that Morgan have answered that question correctly? Martin Hogbin (talk) 14:19, 11 January 2010 (UTC)
- What's the point? Who are you trying to influence? And why? Just look at Nijdam's newest contribution to the discussion. But they won't give a straight answer to 'Is The Contestant Aware?' or 'How Can Huckleberry Do Better By Knowing The Equal Goat Door Constraint?'. I prefer to expose intellectual dishonesty, rather than enable it. Glkanter (talk) 17:12, 11 January 2010 (UTC)
- See Monty Hall problem#History of the problem. Morgan et al. is one of the "Over 40 papers have been published about this problem in academic journals and the popular press". Although the history section doesn't say this, it is (to my knowledge) the first paper specifically addressing the problem published in an academic peer reviewed statistics journal. -- Rick Block (talk) 19:55, 11 January 2010 (UTC)
- I guess it's also the only published paper on the issue that vos Savant publicly replied to. Heptalogos (talk) 20:47, 11 January 2010 (UTC)
- I'm quite sure that my obvious answer to you will be of no benefit, so I'll leave it here. Heptalogos (talk) 22:05, 11 January 2010 (UTC)
- Yes, I was anticipating your response, Rick. But you chose not to answer this back then. So, tell me now, how and why is Morgan the Uber-Monty-Hall-Problem-Wikipedia-article-source? Glkanter (talk) 21:31, 11 January 2010 (UTC)
- Glkanter, I hate the Morgan paper as much as you do, but unfortunately we are stuck with it to some degree, for the reasons given above. The important thing to me is to see it for what it is, a solution to a somewhat contrived and restrictive formulation of the problem that does not represent the MHP as most people understand it. Martin Hogbin (talk) 23:01, 11 January 2010 (UTC)
- Yes, I was anticipating your response, Rick. But you chose not to answer this back then. So, tell me now, how and why is Morgan the Uber-Monty-Hall-Problem-Wikipedia-article-source? Glkanter (talk) 21:31, 11 January 2010 (UTC)
- Of course. It's published. I said that just yesterday. It has no new point to make (except 'as we contrive it, all unconditional solutions are false'), and with all the article's flaws there's no rationale for it being the focus of, and the 800 lb gorilla looming over every aspect of the article. Martin, we're unnecessarily just arguing with ourselves. By now you know that I understand that Morgan is published, and that gives Rick the ability to cling to it. We're being stifled from our legitimate ability to edit the article as the editorial consensus. What's left to say on the various talk pages, by either 'side'? Nobody is budging, clearly, and the article remains confusing and cluttered to the Wikipedia readers. Glkanter (talk) 23:18, 11 January 2010 (UTC)
(outindent) How is this paper "the focus of, and the 800 lb gorilla looming over every aspect of the article"? I said before (toward the end of #What If Morgan Had Used A Different Variant?) "As far as I can tell the only mention of this POV is in the "Probabilistic solution" section, the 4th paragraph in "Sources of confusion" and a paragraph in "Variants"." and you said "It's in every word except the intro and the Simple solution section." Perhaps we should go through the article paragraph by paragraph starting with "Sources of confusion". Here's a list (based on this version - current as I'm typing).
- Sources of confusion, paragraph 1: no Morgan et al.
- Sources of confusion, paragraph 2: no Morgan et al.
- Sources of confusion, paragraph 3: no Morgan et al.
- Sources of confusion, paragraph 4: a mention of Morgan et al.
- Why the probability is not 1/2: no Morgan et al.
- Increasing the number of doors, paragraph 1: no Morgan et al.
- Increasing the number of doors, paragraph 2: no Morgan et al.
- Increasing the number of doors, paragraph 3: no Morgan et al.
- Chance of picking goat with the assumption of switching: no Morgan et al.
- Simulation, paragraph 1: no Morgan et al.
- Simulation, paragraph 2: no Morgan et al.
- Simulation, paragraph 3: no Morgan et al.
- Simulation, paragraph 4: no Morgan et al.
- Simulation, paragraph 5: no Morgan et al.
- Other host behaviors, paragraph 1:no Morgan et al.
- Other host behaviors, paragraph 2:no Morgan et al.
- Other host behaviors, paragraph 3:a mention of Morgan et al.
- Other host behaviors, table:one of nine cases mentions Morgan et al.
- N doors, paragraph 1: no Morgan et al.
- N doors, paragraph 2: no Morgan et al.
- Quantum version: no Morgan et al.
- History of the problem, paragraph 1: no Morgan et al.
- History of the problem, paragraph 2: no Morgan et al.
- History of the problem, paragraph 3: no Morgan et al.
- History of the problem, paragraph 4: no Morgan et al.
- History of the problem, paragraph 5: no Morgan et al.
- History of the problem, paragraph 6: no Morgan et al.
- History of the problem, paragraph 7: no Morgan et al.
- History of the problem, paragraph 8: no Morgan et al.
- History of the problem, paragraph 9: no Morgan et al.
- Bayesian analysis, paragraph 1: no Morgan et al.
- Bayesian analysis, paragraph 2: no Morgan et al.
- Bayesian analysis, paragraph 3: no Morgan et al.
- Bayesian analysis, paragraph 4: no Morgan et al.
- Bayesian analysis, paragraph 5: no Morgan et al.
- Bayesian analysis, paragraph 6: no Morgan et al.
- Bayesian analysis, paragraph 7: no Morgan et al.
- Bayesian analysis, paragraph 8: no Morgan et al.
- Bayesian analysis, paragraph 9: no Morgan et al.
- Bayesian analysis, paragraph 10: no Morgan et al.
- Bayesian analysis, paragraph 11: no Morgan et al.
- Bayesian analysis, paragraph 12: no Morgan et al.
Let's count, shall we? I come up with 42 paragraphs (starting with "Sources of confusion") and 3 references to Morgan et al. Does this make it "the focus of, and the 800 lb gorilla looming over every aspect of the article" and "It's in every word except the intro and the Simple solution section"? If you're not suggesting eliminating Morgan et al. completely from the article (which you keep claiming is NOT what you're suggesting), then what are you suggesting? -- Rick Block (talk) 15:03, 12 January 2010 (UTC)
- Last February I wrote that 5% of the article added value, the other 95% was waste. I understand Wikipedia's policy's better now, so I'm willing to double it. Make it 10%.
- The very existence (and certainly the content) of Aids and Sources shows a pre-disposition to Morgan's claim that the simple solutions are all false and/or inadequate. The existence of all the variants, except the Forgetful (Random) Monty sprout from Morgan. Morgan gave license to these other contrivances you call variants that are more appropriate for a shell game than a game show. Only the forgetful Monty informs the contestant at the same time as the observer, by revealing the car. All the others rely on collusion or ESP.
- So Selvin came up with simple and conditional in 1975. vos Savant came up with random (essentially Deal or No Deal) in 1990. Morgan contrives his stuff to claim the simple solutions are false in 1991. Bayesian? I have no comment. History? Shows that a poor job was done earlier in the article. Just call everything after the Solutions sections 'Diversions'. Glkanter (talk) 16:19, 12 January 2010 (UTC)
- So, you're suggesting deleting the entire article following the Solution section? Is this the change you think there's a consensus for that you keep complaining you're being prevented from making? -- Rick Block (talk) 19:31, 12 January 2010 (UTC)
- Nope. Not at all. The consensus of the editors agreed on the various benefits of the 3 proposals. I'll be one voice of that consensus that makes changes, eventually. Even Dicklyon made note in his comments of Wikipedia violations of UNDUE in the article.
- So, why is Morgan significant? The paper strikes me like Paris Hilton. She's celebrated for being a celebrity. Morgan's paper is, in your estimation, anyways, important for where it was published. Not many of us share that POV. I don't believe Wikipedia's policies support that either. Glkanter (talk) 21:32, 12 January 2010 (UTC)
- You apparently have a fundamental misunderstanding of Wikipedia:Consensus. Consensus applies to edits, not editors. It is definitely NOT the case that the article will be "open for editing" ONLY to some set of "consensus" editors. If this is what you're looking for you will never get it, by any process at Wikipedia. I keep asking you about specific changes, because that is the ONLY thing consensus applies to. This will perhaps become more clear to you if/when we get to formal mediation. -- Rick Block (talk) 15:32, 13 January 2010 (UTC)
- I'm just trying to understand what you're talking about. You say the article is rife with a pro-Morgan POV and at least imply you think everything after Solutions might as well be deleted. I asked if this is indeed what you are suggesting. Your reply says you're not talking about deleting everything but that you'll be part of a consensus that makes changes, making it sound like some "consensus of editors" will have carte blanche to make whatever changes they collectively want. What I'm saying is that this is not how it works and you're at least implying you know that. OK. So please say what specific changes you are suggesting. Take any one (or more) section I've listed above. Say "I, Glkanter, would like the pro-Morgan POV in section blah blah blah to be eliminated by changing <something> to <something else>". Thank you. -- Rick Block (talk) 18:08, 13 January 2010 (UTC)
Door numbers matter
Consider the following game. Roll a (fair) dice, call the outcome X. Put as many white balls as the outcome in an urn and complete with black balls till 6 balls all together. Then before you draw a ball from the urn, predict its colour. Then draw a ball; if your prediction was right you win a car, otherwise a goat. What will be your prediction? Nijdam (talk) 17:04, 11 January 2010 (UTC)
- I suggest that we continue this on the arguments page, I have copied your question there. Martin Hogbin (talk) 17:43, 11 January 2010 (UTC)