Talk:Monty Hall problem

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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)

How do both of your answers correspond to either advancing MHP Paradox knowledge, or following Wikipedia principals? I don't see them doing much of either. Glkanter (talk) 21:20, 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)

Not if you don't post it. I have no idea whatsoever you might reply. Glkanter (talk) 22:49, 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)

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)

Another Straw Man, aka Aunt Sallie. Don't you have anything better to do with your time? Glkanter (talk) 16:08, 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)

Sure. Right after you answer the subject of this section's heading, 'So, What Are The Significant Events, And Why, Of The Monty Hall Problem Paradox', with special emphasis on Morgan, who did not come up with the conditional argument. Or 'Is the Contestant Aware of a Host Bias?'. Or tell me how your 'FCC guy' answer relates to 'How Would The Equal Goat Door Constraint Benefit Huckleberry?', and how it's relevant to the MHP Paradox article. It seems you're more eager to write what you think I think, rather than what you think. Glkanter (talk) 18:25, 13 January 2010 (UTC)

Significant events:

1) Selvin publishes the problem in a letter to American Statistician., giving an unconditional solution. In a second letter Selvin responds to mail he's received suggesting he's wrong, clarifying his assumptions (the host knows where the prize is and chooses randomly if it comes up) and includes a conditional solution.

2) Others publish the same problem. It becomes a standard example of conditional probability in statistics textbooks.

3) vos Savant publishes a version in Parade, and defends her answer with an unconditional solution and then, in response to continued criticism, clarifies all of her assumptions except that the host chooses randomly if given the chance (and says nothing about conditional vs. unconditional)

4) Morgan et al. publish what is apparently the first peer reviewed paper on the problem, in American Statistician (no one has ever claimed this as far as I know, but I suspect it's not a coincidence that this is the same journal Selvin's problem originally appeared in), making the point that the problem is inherently conditional and criticizing vos Savant's solution/clarification as well as other unconditional solutions, noting that "The distinction between the conditional and unconditional situations here seems to confound many". The paper is fairly lighthearted since the problem has been well known in academia for years and is mathematically rather trivial. Since vos Savant ignored (or simply missed) the effect a potential host preference has on the player's chance of winning, the paper explores this specific aspect of the problem concluding that the player should switch regardless of any host preference (since the probability of winning by switching is between .5 and 1 even assuming a host preference). The conditional probability is 2/3 which is the same as the unconditional probability assuming the host picks randomly between two goats.

5) Gillman publishes a note (presumably without knowing about the Morgan paper) that says essentially the same thing as the Morgan paper

6) popular sources continue to publish unconditional solutions, saying the probability is 2/3 and (following vos Savant's lead) ignoring the issue of host preference and the distinction between the conditional and unconditional situations (which seems to continue to confound many).

7) numerous academic papers examine all aspects of the problem, ranging from what assumptions people make and how they understand it to esoteric variations (e.g. the quantum version)

8) the problem continues to be a standard example of conditional probability in many statistics textbooks. At least some (such as Grinstead and Snell) say the unconditional solution doesn't exactly answer the problem that is asked.

NOW will you please say how you're suggesting the "pro-Morgan POV" might be eliminated from any of the sections that you're complaining about? -- Rick Block (talk) 02:28, 14 January 2010 (UTC)

Imho you need to reverse 2) and 3) though, to my knowledge it mostly became a standard example in probability textbooks at large after the parade affair.--Kmhkmh (talk) 05:11, 14 January 2010 (UTC)

I think 2) should probably come after 4). Text books seem to use Morgan terminology. Are there any textbooks, making a big issue of the conditionality, dating before the Morgan paper? Martin Hogbin (talk) 10:05, 14 January 2010 (UTC)

What about 'Is The Contestant Aware Of A Host Bias' and 'How Can Huckleberry Do Better From Knowing The Equal Goat Door Constraint'? Glkanter (talk) 06:58, 14 January 2010 (UTC)

I thought you said "or". I'm getting the distinct impression you either don't have a suggestion or don't want to say what it is. Mediation will presumably help this. -- Rick Block (talk) 14:50, 14 January 2010 (UTC)

Once again, you spend more time on what you think I'm thinking than what you, yourself have to contribute. So, what about 'Is The Contestant Aware Of A Host Bias' and 'How Can Huckleberry Do Better From Knowing The Equal Goat Door Constraint'? These both demonstrate that Morgan, despite being published in the same journal as Selvin, has contributed nothing of value. But it's published, so it goes in. Glkanter (talk) 15:41, 14 January 2010 (UTC)

It's already in, in a way I'm comfortable with. I'm not the one saying the article needs to be changed in this regard. So are we OK here, or would you like to see it in, in some other way? And, if so, how? Again, I'm getting the distinct impression you either don't have a suggestion or don't want to say what it is. -- Rick Block (talk) 15:51, 14 January 2010 (UTC)

Well, what can I say, Rick? You put up an RfC/U on me for not editing the article often enough. Then I very lightly edit the article for clarity, and I touch off a near-edit war, and am accused of violating NPOV with the 2 'POINTy' dates I added to the headers. Then I tell you I look forward to being one of the consensus of editors, and you jump all over me for that. I've updated my concerns on the mediation request. They're consistent with my statements of 15 months now.

So, tell me how you intellectually justify Morgan's claims in light of 'Is The Contestant Aware Of A Host Bias?' and 'How Can Huckleberry Do Better By Knowing The Equal Goat Door Constraint?'? Glkanter (talk) 16:01, 14 January 2010 (UTC)

So your suggestions are at Wikipedia:Requests for mediation/Monty Hall problem#Additional issues to be mediated. Thank you. That's all I was asking for. These are concrete things we can work on (except for the "blather" comment). Regarding your questions - there is no reason whatsoever that I or anyone else should have to justify Morgan's claims to your satisfaction. They're published in a VERY reliable source backed up by similar (if not identical) claims made in other reliable sources. You apparently do not personally agree with them. Fine. Nobody says you have to. -- Rick Block (talk) 16:47, 14 January 2010 (UTC)

I have moved the 'Aids to understanding section'

I have moved the 'Aids to understanding section' to immediately follow the 'Popular solution section to see how it looks.

In my opinion it makes more sense there for the following reasons:

The section mentions nothing about conditional probability or the specific door that the host opens.

The article now has a logical sequence from simple to complicated.

The move does not affect Morgan's claim in the 'Probabilistic solution' section that conditional probability must be used.

Nobody who does not understand the basic problem is going to make and sense of the Morgan paper and subsequent discussions of conditional probability.

No doubt not everyone will like this but there seemed be be something of a consensus forming that it might be a better way to organise things. Note that I have not changed any wording, just moved a section, although I do notice many uncited claims and statements in the 'Aids to understanding section. This should be addressed regardless of the position of the section. Martin Hogbin (talk) 20:23, 13 January 2010 (UTC)

I strongly object to this, and as I've said think that it is not in keeping with WP:NPOV. We're going to talk about this in formal mediation (Nijdam has agreed). I'll file the request today. -- Rick Block (talk) 02:33, 14 January 2010 (UTC)

Why do you object? Have you read the article as it is now? It makes perfect sense and does not discredit the conditional solution in any way. If you could accept this change, which is just of order and not content, I do not think we would need mediation. I think most editors would be happy with the major content and structure of the article. I cannot see how moving the sections into a logical order without any other changes can be described as POV.

I do not know if you filed your formal mediation request in response to my action or not. I have agreed to it anyway. Remember that the mediator does not attempt to impose a view on us. My action was rather bold but it was a genuine attempt to bring this dispute to a close. The change it would be a major step in the right direction, if you could accept it. If not please give reasons. Martin Hogbin (talk) 09:58, 14 January 2010 (UTC)

I note that my change has been reverted with no reason other than to 'wait for mediation'. There is much support for this cgange and no logical reason against it has been proposed. These changes should be discussed here, as I am trying to do. Martin Hogbin (talk) 15:00, 14 January 2010 (UTC)

I said above I object on the basis of WP:NPOV. In more detail, this change presents the problem and then presents an unconditional solution as "the solution". The alternative conditional solution and the POV (of the sources we all know by now) that the unconditional solution does not exactly address the problem as they see it are buried in the article. This creates an "anti-Morgan" POV in the structure of the article. Unlike the Bayesian analysis, which is highly technical and arguably of little interest to a general readership, the conditional solution is well within the grasp of a general readership. Rather than bury this in the article I think we should actually go the other way and have a SINGLE solution section that presents both an unconditional and conditional solution. Per Wikipedia:Make technical articles accessible the marginally simpler unconditional solution should be presented first, but to comply with WP:NPOV I think an alternative conditional solution should immediately follow.

Regarding mediation - my impression is Glkanter wants mediation with or without this change. I filed the request in response to his continued insistence about this, not specifically because of this change. -- Rick Block (talk) 15:32, 14 January 2010 (UTC)

Formal mediation request filed

I've filed the request for formal mediation and informed all the users listed as involved parties, see Wikipedia:Requests for mediation/Monty Hall problem. If you are listed as an involved party please go to the request page and indicate your official agreement to participate in this process. If you think there are other issues to be mediated, please add them to the "Additional issues to be mediated" section. Wikipedia:Requests for mediation/Common reasons for rejection has some helpful comments that may be relevant. Thank you. -- Rick Block (talk) 03:01, 14 January 2010 (UTC)

Has Nijdam Invoked A 'Freeze' On MHP Article Editing?

His only comment I could find anywhere supporting his revert of Martin's edit was his edit comment 'Wait for mediation'. Does he have this unilateral power as an editor? Mediation might not be accepted, and we probably won't know for over a week. Interesting that his action immediately follows his agreement to take part in the Formal Mediation. Glkanter (talk) 14:18, 14 January 2010 (UTC)

He has as much and as little "power" as Martin or anyone else. The "normal" editing cycle is described at WP:BRD. If anyone makes a change that someone else reverts, the revert is a direct indication that this change does NOT have consensus. Making a change to "test the waters" is fine. Reverting such a change is fine. Reverting a revert is NOT fine. -- Rick Block (talk) 15:00, 14 January 2010 (UTC)

Consensus does not mean unanimity. There is, I think, much support for the change. I do not want to start edit warring but if there are no logical arguments presented as to why the change is wrong, I think it would be quite reasonable of me to revert again. I cannot see how changing the order of sections to make the article read better can be described as POV. I have given full reasons above as to why the change is an improvement. Why do you think it is not? Martin Hogbin (talk) 15:35, 14 January 2010 (UTC)

See above. -- Rick Block (talk) 15:39, 14 January 2010 (UTC)

If one doesn't want to wait for the mediation, than I can say I'm strongly against the proposed change. I want the "simple solution" directly being followed by the correct one, together with the critical notes about the simple solution, so anyone reading the article will guaranteed (I hope) see this. Nijdam (talk) 16:33, 14 January 2010 (UTC)

I would say that until the reader has understood the simple solution they have little chance of understanding the issues involved in conditional probability. In fact many readers will not be able to understand this anyway and many will not be interested. We cannot make our readers read something that they do not want to. The essence of the MHP is its simplicity. Martin Hogbin (talk) 17:00, 14 January 2010 (UTC)

Well, it's clear that keeping it the way it had been would be supporting Nijdam's pro-Morgan POV. We can't have that. Tell me Nijdam, how do you intellectually justify Morgan's claims in light of 'Is The Contestant Aware Of A Host Bias?' and 'How Can Huckleberry Do Better By Knowing The Equal Goat Door Constraint?'?

What we can't have is POV is either direction. You apparently think the current article has a pro-Morgan POV. The fix for this cannot be to make it POV in the other direction. A mediator might help us reach a better solution. -- Rick Block (talk) 18:18, 14 January 2010 (UTC)

Nijdam wrote, "I want the "simple solution" directly being followed by the correct one..." He gives no other reasons. I think it's correct to classify his justification solely as pro-Morgan. You've made similar POV conclusions based on a whole lot less. Glkanter (talk) 18:35, 14 January 2010 (UTC)

And maybe you could clarify for us exactly what you and Boris resolved before he left the discussion? Glkanter (talk) 17:12, 14 January 2010 (UTC)

I would also ask Nijdam to prove on the arguments page that any method of solving the symmetrical problem that does not involve conditional probability must be wrong. Martin Hogbin (talk) 17:21, 14 January 2010 (UTC)

IMO, these questions are asking for WP:OR. It doesn't matter what Boris and Nijdam resolved or whether or not Nijdam can prove anything. The only thing that matters is what reliable sources say. You and Glkanter seem to be having a great deal of trouble with the basic concept of saying what reliable sources say without injecting your own POV. This is another issue a mediator might help with. -- Rick Block (talk) 18:27, 14 January 2010 (UTC)

Please, Rick. You've posted on these pages for what, 6 years now? I've read plenty of your personal interpretations and offers to try various game simulations and slightly different problems. There's a difference between Wikipedia policies and intellectual honesty. And there's a difference between how an article is edited and what good faith editors discuss on talk pages. Glkanter (talk) 18:35, 14 January 2010 (UTC)

What there's a difference between is trying to help people understand what the article says (which I've done plenty of) and suggesting edits based on whether or not an editor can "prove" something (which is what you and Martin seem to be doing). The latter has no place in Wikipedia. Edits are based on what reliable sources say, not on what editors can prove. Again, I think this is an issue a mediator might help with. -- Rick Block (talk) 18:50, 14 January 2010 (UTC)

Nijdam and you are the ones asserting that an unconditional solution must always be wrong, you must therefore prove this. There is one reliable source which makes a similar claim (for their interpretation of the question) but as I say below, there is another source which says that they might have misunderstood the question. Martin Hogbin (talk) 18:57, 14 January 2010 (UTC)

My interpretation is that Nijdam is asserting his personal view here, and on the /Arguments page, that the problem is inherently conditional. Although his view matches that of some of the sources, I don't think he's insisting the article take this as its POV. It's not supposed to matter but you are aware that he's a professor of mathematics (right?). I'm asserting there are multiple reliable sources that say the unconditional solutions don't exactly address the Parade version of the problem statement and want (insist) the article fairly represent this POV. -- Rick Block (talk) 21:37, 14 January 2010 (UTC)

Face the facts, Rick. In a peer reviewed journal, Morgan claims all the simple solutions are false. That's the entirety of Morgan's contribution to the literature. Morgan's paper has many flaws, including misquotes and math errors (you could ask Nijdam about this aspect), has a disclaimer from Seymann attached to it, and his claim has not been acclaimed by the professional community in the 19 ensuing years. Plus, it's inconsistent with the problem statement, 'Suppose you're on a game show...' What's left? The peer-reviewed journal part? How is the fact that it is in a peer reviewed journal of any utility or interest to the Wikipedia reader? It's published. It goes in. Why not chronologically? Glkanter (talk) 18:47, 14 January 2010 (UTC)

Rick, this seems to be the line that you resort to when you are beaten in logical argument. As you well know, there are plenty of reliable sources that treat the problem unconditionally, there are also reliable sources which treat it conditionally. There is one source (Morgan) that suggests that the unconditional treatment is incorrect, there is one, equally reliable, source (Seymann) that suggests that Morgan may have misunderstood the question. All these sources are, quite rightly, reflected in the article. Our job as editors is to decide the best way to do this, for the benefit of our readers. No source tells us how to do that. Martin Hogbin (talk) 18:51, 14 January 2010 (UTC)

This is the line I resort to when it becomes obvious further logical argument is pointless. The fact is many sources, not just Morgan, say the problem as it appeared in Parade asks a conditional probability question and that unconditional solutions are not directly responsive to this question. Our job as editors is to insure the article is written (per WP:NPOV) "from a neutral point of view, representing fairly, proportionately, and as far as possible without bias, all significant views". No source tells us how to do that, but since you personally "hate" (your quote above, and your "criticism" page, etc.) the Morgan et al. paper you have a clear bias. Of course this doesn't necessarily mean you can't edit in an unbiased fashion, but you do seem to be having trouble with this. Yet again, I think this is an issue a mediator might help with.-- Rick Block (talk) 19:18, 14 January 2010 (UTC)

Nijdam reverted Martin's 'Aids to Understanding' placement edit solely because of his pro-Morgan POV and bias. He wants the allegedly only 'correct' answer encountered by the reader as soon as possible in the article. That violates NPOV. Badly. Glkanter (talk) 19:06, 14 January 2010 (UTC)

Is it his statement (that this is the only correct answer) that you are saying violates NPOV, or his edit? As long as he doesn't edit the article to say or imply the conditional solution is the only correct solution he's free to think whatever he wants (as are you). We all have our own personal POV. What NPOV says is we have to make sure the article doesn't have a POV. This is something else I think a mediator might help with.-- Rick Block (talk) 19:41, 14 January 2010 (UTC)

I am happy to try mediation but please remember that it is not the mediator's job to enforce rules or decide who is right. They can only help us to agree amongst ourselves. Martin Hogbin (talk) 20:23, 14 January 2010 (UTC)

Right. A mediator would also help us communicate (would, in fact, probably lay down some pretty strict rules about how we should say things and interact with each other) and presumably would be happy to explain relevant policies and guidelines. My impression is Glkanter is not happy with my attempts at explaining policies. -- Rick Block (talk) 21:37, 14 January 2010 (UTC)

Glkanter will express again that you should spend less time thinking and conjecturing and writing about what I'm thinking. I'm not shy about letting you all know what I'm thinking. Glkanter (talk) 22:21, 14 January 2010 (UTC)

I'm sorry, but I find getting a straight answer out of you for what seem to me to be simple direct questions (for example, per the thread somewhat above where you insisted I answer certain questions before you would deign to respond) to be nearly impossible. I sincerely hope a mediator can help us communicate. -- Rick Block (talk) 23:27, 14 January 2010 (UTC)

Martin made an edit and explained his non-POV reasons why. Nijdam reversed it giving ONLY his Morgan-POV reason shy. Seems clear to me where NPOV is being violated. Strange that you can't see that, Rick. Glkanter (talk) 20:53, 14 January 2010 (UTC)

We all know you're being snide here. Please stop it. This is something else a mediator could help with. -- Rick Block (talk) 23:27, 14 January 2010 (UTC)

Yes, Rick. Quoting people in context is 'snide'. And dates are 'POINTy'. And we continue to disagree on nearly ever topic broached in the last 15 months on these talk pages. Glkanter (talk) 02:03, 15 January 2010 (UTC)

To clarify, what I'm referring to as snide is "Strange that you can't see that, Rick". -- Rick Block (talk) 03:28, 15 January 2010 (UTC)

Let us give it a try, but remember, both sides will have to give something to reach a consensus, mediator or not. Martin Hogbin (talk) 23:37, 14 January 2010 (UTC)

Logical and scientifical solutions

The popular solution is actually a logical solution, while the probabilistic solution is a scientific solution. It is quite clear that within science several sources explicitly state the problem to be conditional and therefore the unconditional to be wrong, while no scientific source states the opposite (including Seymann). Some scientific sources used logic in fact, and were corrected by their colleagues. They did not respond to that, nor did any of them criticize the conditional approach.

The 'conditional' scientists all corrected their colleagues, because the latter were not practicing pure science, even though they may have been fully right logically. The problem with the last option is that logic just can't be proven. although logical sources have certain reliability too. I don't see any other possible interpretation.

This is yet another attempt to agree on this matter, because as I said before, we are all right basically. There is no wrong or right, but within disciplines. Logic can only be beaten logically, while science can only be beaten scientifically. They are both as well superior as inferior to another, and this very extreme paradox cannot simply be positioned objectively. Heptalogos (talk) 22:20, 14 January 2010 (UTC)

What does "The problem with the last option is that logic just can't be proven." mean? It sounds like it contradicts the academic teachings of 'logical proofs'. Glkanter (talk) 22:26, 14 January 2010 (UTC)

When so called logic is proven, it becomes law. Any logic outside the law is unproven. As a result, no proven logic exists. Suppose you use logic to create and claim law. The law should be defined as generally true (provable), to some degree. However, when the law is proven, there is no automatic proof for the logic behind it. Unless you use the same logic to create several proven laws. "The same logic" must then be clearly defined and becomes law.

Some principles may be or seem so clearly logical, but not existing as law (yet). On the other hand, as they don't exist as law, are they really that logical? At the moment, there is no law on practicing conditional solution other than to use any exact given information as a condition. Even if all intelligent people in the world would agree on a specific exception, as logically true, it's not scientifically true. These people could maybe create and claim a new general law, which may be proven, after which even the most consequent scientists will agree on this 'logic', but only because it has become law. And they are right, not beaten, because they're still being as consequent as they should, scientifically. So these are really different dimensions. Heptalogos (talk) 08:49, 15 January 2010 (UTC)

The discussion between Morgan and Marilyn didn't make any sense. They were in different dimensions. Their argue is one of historic value! Most famous paradox, most intelligent person, and most consequent scientists ever. It may be a pitty that the original paradox has been kidnapped for this, but on the other hand this 'new issue' is what it made even bigger. So I think both issues really deserve a prominent position in the article, as long as we can clearly separate them, starting of course with the original paradox. Heptalogos (talk) 09:40, 15 January 2010 (UTC)

Something else to add to the mediation list

One of the points of the three fundamental content policies (WP:V, WP:OR, and WP:NPOV) is to avoid the sorts of arguments that have been raging on this page and the /Arguments page. In total, what these policies mean is all we need to agree on is what sources are reliable (WP:RS makes that pretty easy), whether these sources say what we respectively are claiming they say (nothing too difficult there), and how much weight to give what each source say. That's it. The first two should be easy as pie (alright, maybe easy as pi) since whether we agree with what any of the reliable sources say shouldn't even come up. We simply need to agree whether the article is representing what the sources say "fairly, proportionately, and as far as possible without bias" (this quote is from WP:NPOV). The last one (weight) is trickier and sometimes leads to heated arguments. On this page, we've mostly been arguing about the stuff that is supposed to be EASY. I hope mediation will help this, although I fear that the discussion will then morph into a heated discussion about how much weight we should give specific sources. Assuming mediation happens, while we're there we should make sure we explicitly address the weight issue as well. -- Rick Block (talk) 04:13, 15 January 2010 (UTC)

Rather than wikilawyering we should concern ourselves with the overriding objective to improve the article, for the intended readership, which is a wide section of the general public. Although we are entitled to WP:ignore the rules to improve the article, I am not necessarily suggesting that we do that, just that the rules should not in any way direct us to write the article in a way that will not be useful to most of the general public. Martin Hogbin (talk) 10:48, 17 January 2010 (UTC)

Martin - you are at least implying the article can violate WP:NPOV if doing so would make it more useful to most of the general public. IMO, you are completely wrong about this. I would say if we can't write an article that is both useful to the general public and complies with NPOV we are simply incompetent editors, but if we have to pick one we have to pick NPOV. From WP:NPOV: The principles upon which these policies [NPOV, OR, and V] are based cannot be superseded by other policies or guidelines, or by editors' consensus. -- Rick Block (talk) 17:50, 17 January 2010 (UTC)

N, I am not suggesting we abandon NPOV, although, in the end, everything on WP, including policies, is decided by consensus. The question we have to address is the exact interpretation of those policies and my suggestion is that we do this in order to meet the fundamental purpose of WP which is to inform. Martin Hogbin (talk) 11:24, 18 January 2010 (UTC)

"Wikilawyering", I love it. They've seen it all, haven't they? In my opinion, some good things came out of the discussions of the last couple days:

1. The real items for mediation have made themselves known. How much weight to give Morgan, and is the 'host bias' conditional problem statement consistent with the information given in either Selvin's or Whitaker/vos Savant's MHP?

2. All those "let's start fresh and try to make a compromise article" attempts were destined to fail.

3. Rick's threats of the consensus "violating NPOV" were bluster, unsupported by Wikipedia policy.

Glkanter (talk) 13:20, 17 January 2010 (UTC)

Your point #1 is half right. One of the issues is how much weight to give the viewpoint expressed by Morgan et al. However the second half of this ("is the 'host bias' conditional problem statement consistent ...") is not for us to decide. Sources have decided this. And, for many many sources the decision is clear. I have no idea what you're talking about with your second point. Regarding your third point: I have made no threats, "the consensus" is not a group of people, and this point is yet more low level harrassment directed my way entirely consistent with the objectionable behaviors described here. Please just stop it. -- Rick Block (talk) 17:50, 17 January 2010 (UTC)

The problem of NPOV in MHP

It is NOPV to chronologically:

• Start with the original paradox and explain it fully, logically as well as mathematically.
• Describe the complex issues within science to solve particular statements.

It is also NPOV to:

• Start with the most famous particular statement of the MHP, which is from Parade.
• Explain this one logically as well as mathematically.
• Which will at the mathematical solution already start the complex issues, because of the particular Parade statement.

So, how do we solve this? Heptalogos (talk) 10:11, 15 January 2010 (UTC)

For practical reason, I think we all agree on the Parade statement to start with? A problem description is initially needed, reflecting a mundane situation, instead of basic formula. Any mundane problem description should come from reliable sources and be significant. So without an alternative, there's not much of a discussion here. Heptalogos (talk) 10:31, 15 January 2010 (UTC)

As the Parade statement is particular in a certain way, the mathematical solution has an enormous tail of complexities. What if we do it like this: logical explanation of the paradox first, as it is now. Next the mathematical solutions, again the same, but this section not being significantly bigger (in text) than the logical section. Because that would not be NPOV either. This section should preferably describe the unconditional and conditional scientific solutions, and mention the issue of scientific consensus in favour of the conditional method. Then it may direct to a lower section in the article to give a full description of all scientific sources and arguments. This seems to me as a reasonable compromise, because Nijdam is justly being served at the issue of common readers being aware of the essential criticism on the unconditional method, while Martin and Glkanter are being served at the issue of having the "aids to understand" section above all detailed sources and arguments of science. Heptalogos (talk) 10:51, 15 January 2010 (UTC)

This is exactly the structure I favor. Further, I don't see any particular reason to separate the "logical" solution and the "scientific" solution into different sections. It doesn't seem to me to be that difficult to do both in one section per the proposal above (see #Proposed unified solution section). For a time, I thought there was good progress being made on that solution. -- Rick Block (talk) 15:19, 15 January 2010 (UTC)

This does not address the problem that will be raised for the vast majority of our readers. As Glkanter has pointed out, none of the thousands of letters that vos Savant received was concerned with which door the host opened, they were all about why the answer is not 1/2. This is the question that we must answer first. Martin Hogbin (talk) 15:33, 15 January 2010 (UTC)

Rick, I don't like the unified solution because it spends only a few lines on the paradox that confuses so many people and is then already starting to mention Morgan et al. The scientific solution is very complex, which is the reason to describe it separately. Heptalogos (talk) 19:53, 15 January 2010 (UTC)

Martin - even in the unconditional case you have to talk about the door the player picked, the door the host opened, and the other door. You might as well call these "door 1", "door 3", and "door 2" (respectively). Do you want to add "for example" before every door number to clarify that the initial unconditional solution is only using door numbers as examples (which, btw, is true of the conditional solution as well)? Player picks door 1, host opens door 3, and player can now switch to door 2 is used as the representative example of the general case in nearly all sources. You seem to be saying that using the door numbers in this solution means we're limiting the response to only those players who have picked door 1 and have seen the host open door 3. This is not what ANY of the sources who discuss the solution in terms of these door numbers are intending. Your refusal to see this point mystifies me.

Your table (#Discussion and proposals aimed at reaching a compromise on this subject) is incredibly confusing. Why are there two identical columns for "you choose a goat"? You don't KNOW what you chose. What you initially know is the car is behind your door, or one of the other two doors, and then you find out the car is NOT behind the door the host opens. So, you start with 3 doors (not car, goat, goat - for example, what if rather than goats the losing doors are simply empty?). This model (door centric, rather than car/goat/goat centric) is exactly the basis of vos Savant's case analysis solution (go look at it). This approach matches the example in the problem description (yours doesn't).

I am incredibly tired of arguing about this. How about if we start talking exclusively about how sources handle this. My claim is the unconditional case is handled by vos Savant and nearly all other popular sources using the door numbers given as examples in the problem statement. If you'd like to argue whether this is what popular sources do or not, we can presumably put together a list and go look. Would you like to offer up a source that does not use door numbers? -- Rick Block (talk) 18:29, 15 January 2010 (UTC)

I am not sure exactly what Heptalogos is suggesting. The order I want is U/C solution, Aids to understanding, C solution, Causes of confusion, Variants etc. Martin Hogbin (talk) 15:36, 15 January 2010 (UTC)

I am suggesting about the same as you do, while also offering a comment, within the U/C section, about scientific consensus in favor of the conditional approach. Apart from that I think it's better to replace U/C by L/S meaning logic/scientific. Heptalogos (talk) 19:53, 15 January 2010 (UTC)

No, Martin is disagreeing with you. He wants an initial section that only discusses what you're calling the Logical solution with no mention of conditionality and no mention of scientific consensus, and then an entire section on "aids to understanding" this "logical" solution, and only then a section on the "scientific" solution. -- Rick Block (talk) 21:27, 15 January 2010 (UTC)

It's about the same. He also doesn't want the Morgan battle in between the logical solution and the aids to understand it better. But then again I agree with you that the conditional (scientific) approach should really be mentioned and explained basically right after the logical approach (NPOV), but with a reference to another section to go in detail. Heptalogos (talk) 21:45, 15 January 2010 (UTC)

Unfortunately, we cannot have it both ways, or everyone would be happy and there would be no argument. I want the 'Aids to understanding' section to follow the simple solution section. This is not to intentionally downgrade the conditional solution but simply because most of the reader's difficulties in understanding will be with the simple solution, just as it was for vos Savant. People insisted that the answer was 1/2, not that the host goat door choice was important. Martin Hogbin (talk) 00:12, 16 January 2010 (UTC)

Please read through the current 'Aids to understanding' you will see that it refers only to the issue that I have described above. There is no mention of the host's choice of goat door. Martin Hogbin (talk) 00:15, 16 January 2010 (UTC)

I know what you want, but others also have justified wishes. So what is specifically wrong with the compromise that I propose? Heptalogos (talk) 09:52, 16 January 2010 (UTC)

Summary

See Talk:Monty Hall problem/Arguments. Nijdam (talk) 14:34, 15 January 2010 (UTC)

Let's Discuss, As Per Rick's Comment On Wikipedia Policy, How Much 'Weight' Each Source Receives In The Article

I love this topic! It's what I've been arguing about for months. Sure, Morgan is published in a peer reviewed journal. So what? What do they have to say that's new? The article has at least three gaping errors. Only with the errors can they make the contrived claim that all simple solutions are false. I wonder, even ones that hadn't been published yet?

Here's Rick's diff.

How about Selvin? I added '1975' to the Popular solution section. Someone pointed out that Selvin's 1975 paper isn't even referenced there. And his conditional paper of the same year isn't in the Probabilistic solution section.

This is what real life editors do! They 'weigh' the newsworthiness of various articles from various sources. And they discuss it. With ideas, and logic, and facts.

So, perhaps more Selvin, and less Morgan POV? Glkanter (talk) 18:50, 15 January 2010 (UTC)

I would suggest we wait until mediation for this because it is going to immediately get into Wikipedia policy issues that I think would be best explained by someone you would perceive as a neutral party (i.e. not me). I will say that this is an area that takes editorial judgment, which means it's not nearly as black and white as "is X a reliable source". -- Rick Block (talk) 19:29, 15 January 2010 (UTC)

You may choose to participate or not, Rick. I'm sure there are others like me who will get bored waiting another week or so. Now that you agree 'weight' does not equate to 'biased POV', we can have this discussion again.

So, we can talk about how to edit the article, and actually edit it, or we can offer more urns puzzles, formal probability notations, and narratives on 'truth'. Or, some may wait for the mediator. Glkanter (talk) 19:50, 15 January 2010 (UTC)

Morgan's weight:

• Very explicit arguments against many (unconditional) methods used so far. (are they the first?)
• Very explicit arguments against the method Vos Savant is using. (are they the first?)
• Very first attempt to solve the problem without making assumptions. (are they the only ones?)
• 'Scientifically' published and peer reviewed. (did any unconditional solution?)
• The only scientists to have an extensive debate with Vos Savant.

About the errors: that's POV and it doesn't even really matter in the weight issue.

Please suggest something about Selvin 1975. Do you have his letters? Heptalogos (talk) 20:11, 15 January 2010 (UTC)

The errors are not POV. They are real. The paper discredits itself. Misquotes, math errors found by Martin and Nijdam, treating the host and the car placer differently, assigning 'bias' and transferring this to the contestant on a game show... Much like 'the Earth is flat' doesn't get much emphasis, neither should 'the host, but not the car placer, has a bias known by the contestant' Glkanter (talk) 14:44, 16 January 2010 (UTC)

I think before anyone can talk about this, they need to go read WP:WEIGHT. Weight and POV are different, but related, issues. -- Rick Block (talk) 21:24, 15 January 2010 (UTC)

Certainly. Why your comment? Heptalogos (talk) 21:49, 15 January 2010 (UTC)

Because we all need to able to talk about the issues that will come up using the same terminology. The appropriate weight relates to (from the link I provided) "prevalence in reliable sources, not its prevalence among Wikipedia editors or the general public" (emphasis in the original). Weight also relates to viewpoints as opposed to specific sources. These are things I'm fully expecting to argue with Glkanter and Martin about, but I want there to be a shared understanding of what weight means before we start arguing specific issues (or else I will certainly be accused of "filibustering" or trying to subvert the will of the masses or whatever).

I would truly prefer to defer this discussion until we're in mediation so that I don't have to both help people understand Wikipedia policy and also argue one "side" of this. Doing this apparently tends to make the policy clarifications sound like partisan arguing. -- Rick Block (talk) 23:59, 15 January 2010 (UTC)

OK, I understand. I was basically trying to convince Glkanter personally about the significance of Morgan, nothing more. But I can define the same arguments a little bit different to present the viewpoints in proportion to the prominence of each in the sources. First of all I think Morgan is mentioned a lot in other reliable sources, while it's already most reliable because it was published within scientific media. Then they spent much energy in disproving others, including Vos Savant, on which they published another article. Another (the) prominent part in their main article is their solution without assumptions. The only aspects now missing are Morgan being the first ones, which is probably indeed not relevant, except for Glkanter and myself personally.

I don't see the problem of you switching roles as soon as mediation starts. Is there any Admin responsibility conflicting with that? Heptalogos (talk) 10:55, 16 January 2010 (UTC)

There is no actual conflict. Anyone is free to try to clarify Wikipedia policies during a discussion (although this sometimes leads to Wikipedia:Wikilawyering). As far as content disagreements are concerned admins have no more and no less authority than anyone else - specifically admins can't take administrative actions to enforce their preference of content, i.e. cannot block users or protect a favored version of an article. What I would like is for policy issues to be brought up by an acknowledged neutral party, and thus keep the "meta" (policy) issues distinct from any conflict we might have about content. -- Rick Block (talk) 18:07, 16 January 2010 (UTC)

Regarding a couple of questions raised above. Glkanter asks about Selvin's letters. These letters to the editor (they're not "papers") are referenced in the "History" section. They're not referenced elsewhere since they are being treated here as primary sources. Please see the first bullet at Wikipedia:Reliable sources# Scholarship.

Heptalogos asks

• Is Morgan et al. the first to explicitly argue against many (unconditional) methods used so far? I believe so.
• Is Morgan et al. the first to explicitly argue against the method vos Savant uses? I'd say they were technically the first, although Gillman is effectively simultaneous (the Morgan et al. paper was published in November of 1991, Gillman's in January of 1992).
• Is Morgan et al. the only attempt to solve the problem without making assumptions? Specifically meaning not making assumptions about the host preference between goats, Gillman approaches the problem in exactly the same way. This same approach is one of several presented by Krauss and Wang. At least several others (all presumably assuming initial random car placement) show the same result, i.e. the chance of winning by switching given a host preference p for the door that has been opened (meaning the host opens this door with probability p if there are two goats to pick from) is 1/(1+p).
• Are any unconditional solutions "scientifically" published and peer reviewed? I'm not sure exactly what you mean by this, but assuming you mean a source that acknowledges the difference between the unconditional and conditional questions and then uses an "unconditional" solution to rigorously address the conditional case (like Gill's WP:OR, below) I don't know of such a source.
• Are they the only scientists to have an extensive debate with Vos Savant? I believe the sense of what you're saying is true, although "extensive debate" seems like an overstatement. They published a paper. She wrote a letter to the editor in response. They published a rejoinder.

As mentioned in the "History" section, Barbeau's book contains a survey of the academic literature through about 2000. He published an earlier version of this survey in 1993, it's the Barbeau 1993 reference in the article. -- Rick Block (talk) 20:31, 16 January 2010 (UTC)

Thanks for the answers Rick. Those are the exact answers I was aiming for. Heptalogos (talk) 22:40, 16 January 2010 (UTC)

Do any of these sources criticize the simple solutions based only on the information given in the various forms of the problem statement? Or do they all contrive a 'host preference', then criticize the simple solution for not solving this different problem? Do any of them also contrive a 'car placer preference'? Glkanter (talk) 21:37, 16 January 2010 (UTC)

I've provided quotes from Morgan et al., Gillman, and Grinstead and Snell before (see above #What If Morgan Had Used A Different Variant?) - what they're all saying is that the "simple" (unconditional) solutions don't address the problem as they interpret it, so yes they are criticizing these solutions based on the problem statement. The fully conditional solution in Grinstead and Snell accommodates a 'car placer preference' although they (explicitly) assume the initial car placement is random. These sources don't "contrive" a host preference but say the probability of winning depends on it, and if you don't know what this preference is or don't assume a value for it you can't exactly say what your probability is of winning by switching. Krauss and Wang include a discussion of this as well. They consider the "two door scenario", where the door the player picks as well as the door the host opens are given, to be the "standard version" and say "... one has to make assumptions about what Monty Hall would do in A1 [the case where the player has initially picked door 1 and the car is behind door 1] and estimate the probability that Monty Hall would open Door 3 rather than Door 2." -- Rick Block (talk) 23:55, 16 January 2010 (UTC)

Do any of these sources reason why they do assume e.g. random car placement (etc.), but not random host behavior? Heptalogos (talk) 11:23, 17 January 2010 (UTC)

Morgan et al. are clearly trying to match vos Savant's assumptions - they even call it the "vos Savant scenario". She (and they) assume initial random car placement, that the host always opens a door revealing a goat, and that the host always makes the offer to switch. They mention at the end of the paper that it would also be possible to consider non-uniform probabilities of car placement. Assuming they overlooked or ignored this case is (IMO) simply willful misinterpretation of what they wrote.

Gillman restates the problem (does not quote the Parade version) and explicitly states the initial car location is to be taken as random.

Grinstead and Snell quote the Parade version as reprinted in vos Savant's book "Ask Marilyn" (I haven't checked whether their quote matches this version - what they quote is definitely not the same as the original Parade version) and explicitly say (without rationale) they're assuming the car is initially located randomly.

One might surmise that making this assumption focuses the problem on the effect of the host opening a door. By making this assumption, the probability before the host opens a door is clearly 1/3 and the question becomes what is the probability after the host opens a door. For editing purposes, my claim is we don't care why this assumption is made. The fact is this assumption is made. If the sources don't say why, then the article can't say why and, unless there are other sources that explain or question this assumption, the article can't either. -- Rick Block (talk) 21:04, 17 January 2010 (UTC)

Rick, very clear, thank you. It's quite a mess. However, it triggers me to find the exact relations between sources. Heptalogos (talk) 11:58, 18 January 2010 (UTC)

How Richard Gill110951 sees things now

I distinguish three Monty Hall problems, and I think they are all legitimate problems to discuss; they have all been discussed in the literature of brain-teasers, mathematics, psychology...

There is no law saying that exactly one of these three is "the" Monty Hall problem

Here they are:

0: Marilyn vos Savant's question "would you switch?"

1: A mathematician's question "what is the unconditional probability that switching gives the car?"

2: A mathematician's question "what is the conditional probability that switching gives the car?"

Please specify the sources for 1 and 2, otherwise there's no use in discussing it here. Heptalogos (talk) 11:31, 16 January 2010 (UTC)

Whitaker. Sorry, misread the comment above. I agree that Whitaker should be interpreted as 0. Martin Hogbin (talk) 15:02, 16 January 2010 (UTC)

The following analysis gives the right answers to the questions 1 and 2 under the assumptions conventionally thought to be appropriate. I use Boris Tsirelson's beautiful trick ("symmetry") to deduce the answer to question 2 from the answer to question 1. I finally make some comments on question 0.

My set-up:

The quizteam hides the car, the player chooses a door, the quizmaster opens a door.

Three random variables taking values in {1,2,3}.

I don't care what your interpretation of probability is (subjective or frequentist or ...).

I don't care (for the time being) whose probabilities we are talking about at which stage of the game.

Notation:

C = door where Car is hidden

P = door first chosen by Player

Q = door opened by Quizmaster

Assumptions: with certainty,

Q unequal to P

Q unequal to C

Because of the first assumption we may define

S = door which follows by Switching = unique door different from P and Q

1) Short solution to problem 1:

If Prob(P=C)=1/3 then Prob(S=C)=2/3, since the two events are complementary.

2) Short solution to problem 2:

In this problem, the door chosen by the player is fixed, P= x, say.

We are to compute Prob(S=C|Q=y) for a further specific value y unequal to x. Let y' denote the remaining door number, besides x and y.

Assume that (given the chosen value of P), C is uniform, and the distribution of Q given C is uniform.

So by assumption Prob(P=C)=1/3 and therefore as in Problem 1, Prob(S=C)=2/3.

The latter probability is the weighted average of the two probabilities Prob(S=C|Q=y) and Prob(S=C|Q=y'), weighted by the probabilities Prob(Q=y) and Prob(Q=y').

Since the distribution of C gives equal probabilities to y and y' and since Prob(Q=y|C=x)=Prob(Q=y'|C=x)=1/2, nothing is changed by exchanging y for y' and vice-versa.

Thus the two conditional probabilities Prob(S=C|Q=y) and Prob(S=C|Q=y') are equal, and equal to their (weighted) average 2/3.

0) Short solution to problem 0.

I don't know what strategy the quizteam and quizmaster use, so naturally I had chosen my door uniformly at random, independently of the car's actual location.

Since I know game theory I know that "always switching" is the minimax strategy. It guarantees me a 2/3 (unconditional) chance of winning the car.

I don't care a damn what my conditional probability of winning is, given my specific initial choice: say door 1, and the quizmasters' specific choice: say door 3.

I don't know this probability anyway, since I don't know the strategy used by quiz-team and quiz-master.

I only know that Monty Hall always opens a door revealing a goat.

Gill110951 (talk) 11:03, 16 January 2010 (UTC)

This belongs on the arguments page. Heptalogos (talk) 14:12, 16 January 2010 (UTC)

Gill, I have shown much the same thing on my analysis page using fixed door numbers.

What is your opinion on the order of sections within the article? I would like to see the 'Aids to understanding' immediately follow the 'Popular solution' section as this is the section that most people cannot understand. Martin Hogbin (talk) 15:08, 16 January 2010 (UTC)

@Gill: as I wrote in my mail, once the problem is well stated, the soluton is (must be) obvious. Now your points 0, 1 and 2 do not clearly state the problem. I will ask you to go to the arguments page under Talk:Monty Hall problem/Arguments#Summary, where I also tried to formulate the different views on the MHP, and specify your versions 0, 1 and 2, and if possible relate them to my versions A, B and C.Nijdam (talk) 11:29, 17 January 2010 (UTC)

@Nijdam, @Martin, @Heptalogus, thanks for your comments. Let me explain what I am trying to do here. Problem 0) is supposed to be the question exactly as posed by Marilyn vos Savant, i.e., in her own words. Problems 1) and 2) are what people around here call the unconditional and the conditional problems respectively. They are two different ways in which mathematically inclined people have converted Marilyn's verbal problem into a formal mathematical problem. For each of the three problems the solution is immediate. I wrote it down in explicit, respectable mathematics (especially after I cut a lot of crap out of my first try at problem 2). The point I am trying to make is that Marilyn vos Savant asked whether or not one should switch doors, NOT what some probability was. Her problem was not a maths problem. It is only if you decide that the right way to make your final choice of door is by computing a probability, that you arrive at problems 1) and 2) - the "conventional" unconditional and conditional variants of the problem, which people spend their time here fighting about. I think that, if you think Marilyn is asking you for a probability, then whether 1) or 2) is closer to the question she is asking is a question of interpretation of American-English idiom. Did she refer to the actual door numbers painted on the doors in advance of the show, or did she mean that we decide to call the door which the player chooses "door 1"? Anyway my point is, she did not ask for a probability, she asked for a strategy. The usual *solution* to the conditional problem, problem 2, requires one to make all kinds of assumptions which are not justified by the problem as originally phrased. However, lots of people have talked about it, so it should be on wikipedia, since we are not supposed to be presenting *our* opinion as to what the MH problem ought to be, but merely writing down what it has been to various people. My own humble opinion is that the most satisfactory treatment of the problem uses the language of game-theory, i.e., we explicitly take account of any strategy which might be used by the quiz-team & quiz-master. We don't make unwarranted assumptions about it, whose raison d'etre is merely that they are designed to deliver the answer 2/3. Gill110951 (talk) 20:52, 17 January 2010 (UTC)

Gill, I completely disagree with you about game theory, this, in my opinion adds further complicationto an already difficult problem. If you want to discuss that further, I suggest that we do so on the arguments page. Martin Hogbin (talk) 21:01, 17 January 2010 (UTC)

Gill, I completely agree with you about game theory. If you want to discuss that further, I suggest that we do so on the arguments page, until you come up with a reliable source suggesting this method to solve the MHP. Heptalogos (talk) 22:26, 17 January 2010 (UTC)

I spoke to some game theorists who said they "know" the game theoretic solution to the Monty Hall problem. And any way, it really *is* an easy exercise after you have done Game Theory 101! Probably there exists a published discussion somewhere. Sometime I will post a combination, and condensation of my notes so far (see references Gill 2009a, 2009b, and since an hour ago also **** 2010 ****) to arXiv.org and submit to a light-weight but respectable peer-reviewed journal (more respectable than the American Statistician). BTW I think that Morgan et al. is a very poor paper. It is solving a Statistics 101 problem in a pompous and arrogant way, as well as being definitely un-scholarly in being dogmatic about their version being "the" version; for which purpose they even misquote earlier works. In the meantime my job is to go on looking for reliable sources, creating reliable sources if necessary, and learning from what people say here. In particular I must check what wikipedia already has on game theory. Game-theory ought to be more accessible and more well-known. Personally, I find a game-theoretic approach illuminating. In fact I find it essential since it is the only way as far as I know to give a decent argument for always switching, whatever the conditional probabilities..., without making articial assumptions about the quizmaster. @Martin and @Heptalogus, I am happy to discuss this with anyone, anywhere they like, I have tried to provide information which anyone can use to figure it out for themselves. Anyone who wants to erase anything I put on wikipedia can go ahead. No problem. Gill110951 (talk) 18:01, 18 January 2010 (UTC)

Game Theory 101: von Neumann's (1928) minimax theorem. References:

The minimax theorem (von Neumann, 1928), http://en.wikipedia.org/wiki/Minimax_theorem#Minimax_theorem

Von Neumann's seminal contributions to game theory: http://en.wikipedia.org/wiki/John_von_Neumann#Economics_and_game_theory

What game theory is nowadays: http://en.wikipedia.org/wiki/Game_theory

Honestly, I don't think that the minimax theorem complicates matters. It simplifies matters because we know there is a minimax solution and once we have guessed it, it is easy to check that we were right. And the two party's minimax strategies are exactly the player's and the quiz-master's "symmetric" probability distributions, used to randomize their choices. The probability distributions which turn up all over the place on these pages.

Gill110951 (talk) 19:45, 18 January 2010 (UTC)

I have no objection at all to a section about game theory and the MHP. What I would not want to see is game theory replacing the simple solution of the problem in which the car placement, the player's initial choice, and the host's legal door choice are all assumed to be uniform at random, as is the standard in mathematical puzzles.

After the discussion about the affect of a known or suspected host door opening policy (per Morgan), a section on game theory would be most welcome in my opinion, especially as it shows that, if both the player and the host take the game seriously and competitively, the chances of winning by switching are back to 2/3 again. This puts Morgan's 1-q twaddle back in its proper place. Martin Hogbin (talk) 22:24, 18 January 2010 (UTC)

In some ways the Morgan paper can be seen as a dismally failed attempt to discuss game theory. Martin Hogbin (talk) 22:26, 18 January 2010 (UTC)

A question for the 'conditionalists'.

The MHP is notoriously difficult for most people to understand and many peope do not accept the solution even when it is clearly explained to them. In the lead we state, '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'.

Does anyone here think that the above statement does not apply to the unconditionally stated problem, such as that given by Morgan, You will be offered the choice of three doors, and after you chose the host will open a different door, revealing a goat. What is the probability that you win if your strategy is to switch. ?

In other words, does anyone here think that the unconditional problem is not the MHP because it is too easy? Martin Hogbin (talk) 11:00, 17 January 2010 (UTC)

Nijdam, would you care to comment? You say above, 'once the problem is well stated, the soluton is (must be) obvious'. Martin Hogbin (talk) 15:54, 17 January 2010 (UTC)

What is there to be commented?Nijdam (talk) 16:59, 17 January 2010 (UTC)

I take it that you do believe that the solution to the unconditional problem is obvious. Is that right? Martin Hogbin (talk) 20:18, 17 January 2010 (UTC)

Martin - this is the "no-door" version discussed by Krauss and Wang. They consider it an easier version of the MHP, but not the "standard" version. Perhaps it should be discussed in the "Variants" or "Aids to understanding" section. -- Rick Block (talk) 18:03, 17 January 2010 (UTC)

Rick, I understand what the problem is and there is no ulterior motive behind my question. I am just trying to understand your POV. Do you think that the solution to the unconditional problem is obvious? Martin Hogbin (talk) 20:18, 17 January 2010 (UTC)

Why do you care what I think? For editing purposes we actually have a highly reliable source (in fact, a psychology source) that discusses this very question. They say it's an easier version. They say it's not the standard version. My (irrelevant) opinion is that they're right although I doubt that it is a significantly easier version. IMO (more irrelevance) most people would still try to solve this version by thinking about a specific example case - e.g. hmmm, let's say I pick Door 1 and the host opens Door 3, then there would only be two doors left but I still wouldn't know where the car is, so after the host opens a door there's one car and two doors and the chances must be 50/50. -- Rick Block (talk) 20:38, 17 January 2010 (UTC)

I care what you think because I want to resolve this dispute. I am trying to find out why a bunch of, I guess, reasonably intelligent people cannot agree. Both sides seem to keep saying the same thing over and over again but making no impression on the other side. What are we missing? Why do we continue to talk past one another? That is what I am trying to find out.

Nijdam, it would seem, does think that a solution to the well-defined unconditional problem statement is obvious. I have continued this point on the arguments page as I think this may be an important cause of disagreement. Martin Hogbin (talk) 20:56, 17 January 2010 (UTC)

Richard Gill also thinks that the solution to the well-defined unconditional problem is obvious. If the door you first chose has the car behind it with probability 1/3, then switching gives you the car with the complementary probability 2/3, since switchers get the car whenever stayers don't get the car, and vice-versa. He is furthermore really pleased with Boris Tsirelson's proposal to solve the conditional problem under the supplementary condition of *symmetry*, by using *symmetry*. Thus: if after you have chosen door 1 the probability the car is behind each other door is the same, therefore 1/3, and if the quizmaster opens a door by tossing a fair coin when he has a choice, then doors 2 and 3 are exchangeable. Therefore Prob(car is behind 2|player chose 1, quizmaster opened 3)=Prob(car is behind 3|player chose 1, quizmaster opened 2) and both are equal to the unconditional probability Prob(car is behind the remaining closed door|player chose 1, quizmaster opened a door)=2/3. All of this verbal maths argument can be converted into formulas, as I did yesterday. Gill110951 (talk) 21:16, 17 January 2010 (UTC)

When you say obvious, do you mean that most people would be able to spot this solution? 86.132.191.65 (talk) 22:50, 17 January 2010 (UTC)

Yes, spot the solution, or at the very least, quickly understand and accept it when it is presented to them. Martin Hogbin (talk) 11:16, 18 January 2010 (UTC)

Lots of people *do* spot the solution to the unconditional problem. And certainly most people accept it once they have heard it. The exception being lawyers, as was discovered by a survey at the University of Nijmegen. Everyone initially gives the wrong answer (including lawyers), afterwards everyone agrees with the right answer (except lawyers). Gill110951 (talk) 17:28, 18 January 2010 (UTC)

Do you have any more information on what proportion spot the solution without help, and are these results published anywhere. Martin Hogbin (talk) 00:36, 19 January 2010 (UTC)

Explicit assumptions

The 'probabilistic solution' in the article states that "This analysis depends on the constraint in the explicit problem statement that the host chooses randomly which door to open after the player has initially selected the car." Why not mention the other assumptions implicitly made?

This section starts with "Morgan state that many popular solutions are incomplete, because they do not explicitly address their interpretation of the question". And it ends with a solution which does the same. It doesn't make sense. Heptalogos (talk) 12:15, 17 January 2010 (UTC)

This is probably the result of arguments being played out on the page. The best way to correct the problem depends on who you are. I would like to change the first quote to, "Conditional analysis of the problem is only required if it is known that the host may not choose randomly which door to open after the player has initially selected the car." Others may not agree. Martin Hogbin (talk) 12:59, 17 January 2010 (UTC)

Yes (others may not agree). IMO, the article is at this point kind of a mess directly as a result of the senseless bickering on this page. Rather than actually improve the article by presenting a peacefully coexisting (per Boris) pair of unconditional and conditional solutions to the fully symmetric problem, these are being presented (and discussed here) as incompatible solutions. For the symmetric problem these solutions are essentially two sides of the same coin. The unconditional solution says the average probability of winning by switching is 2/3. The conditional solution says the conditional probability of winning by switching in any example case is also 2/3 (which of course means the average must also be 2/3). There is NO conflict between these approaches. You may not think the conditional approach is necessary, but it is certainly not wrong. You may not think the unconditional approach exactly answers the question, but it certainly says what the average probability is. These are both useful solutions, and it is also useful to understand the difference. And, unsurprisingly, this is exactly how most sources (at least most academic sources) treat the problem. -- Rick Block (talk) 18:30, 17 January 2010 (UTC)

I see you both react from your political programs. But I am not at all trying to continue the same discussion from yet another angle. Within the conditional section, which is a valid one as we all agree, I noticed some defects. How can we repair? Heptalogos (talk) 19:35, 17 January 2010 (UTC)

Mo comment was not intended to be political or to be supporting my view. I was just pointing out that, unless we can reach agreement on how to structure the article there is unlikely to be a solution to the problem that you have found. Maybe mediation will help. Martin Hogbin (talk) 20:15, 17 January 2010 (UTC)

An actual suggestion for an improvement rather than continued bickering? WHAT ARE YOU THINKING?  :)

I'd suggest revising the section to be more or less like the relevant paragraph out of the proposed unified solution section, above, which at least is intended to present the conditional approach as an alternative solution without introducing the POV that the unconditional solution is wrong. I think others may disagree that it accomplishes this goal, but I think it's an improvement over what's there now and by working on it together we can make incremental changes toward this goal. -- Rick Block (talk) 20:23, 17 January 2010 (UTC)

As a temporary solution, until we can reach a consensus on how to move forward, I am not fussed. Martin Hogbin (talk) 20:38, 17 January 2010 (UTC)

I like a lot of what Rick Block says above. I would like to underline that there is now a fantastic opportunity to simplify the whole article by presenting the answer to the conditional problem (with the supplementary assumption of symmetry between remaining two doors given player's first choice) as a corollary to the answer of the unconditional problem (with the assumption only that the first choice has probability 1/3 of being correct). My POV is furthermore that since both conditional and unconditional problems are interesting in their own rights and frequently discussed in the literature, both by "amateurs" and by "professionals", both problems need to be treated on the encyclopaedia page. Finally I want to repeat again that Marilyn vos Savant's simply asked "would you switch", she didn't ask for a probability, let alone specifying a conditional or unconditional probability. Personally I think that her problem is a sensible problem and as far as it looks like a math problem, it looks to me more like a problem of game theory than a problem of probability theory, since what the player ought to do depends on what the player believes the quizmaster is doing. Perhaps a sensible player prefers not to follow the advice of a calculation based on a specific assumption about the behaviour of the quizmaster. Perhaps the player would be happy just to discover that by starting with a uniform random choice of door and thereafter always switching (s)he gets the car 2/3 of the time, whatever the quizmaster does, and that this is the best one can hope for. It's amusing that the quizmaster's minimax strategy is actually the symmetric stategy, which makes the conditional probabilities equal to the unconditional ones. Gill110951 (talk) 21:35, 17 January 2010 (UTC)

If only it were that simple. There's still the issue of Morgan saying 'All simple solutions are false', then 'proving' it by contriving a host bias, which leads to the conditional non-symmetrical non-solution which is clamoring for equal time as well. Glkanter (talk) 21:46, 17 January 2010 (UTC)

Now also Gill is presenting his game theory in this section. We should really bring more structure in our discussions. Let's separate possible article restructuring (?) from improving the current structure, which will also be very helpfull in case of restructuring. If you state there cannot be such improvement without agreement on restructuring, then you're actually saying that you don't want to participate, which is really not necessary to mention. So I understand Rick is working out a proposal, which is quite welcome to me. Heptalogos (talk) 22:44, 17 January 2010 (UTC)

I'm not sure what proposal you're talking about. I suggested a unified solution section above. I thought progress was being made, but it was definitely derailed and is somewhat moribund at this point. I think what's actually going to happen is the mediation committee will decide fairly soon whether to accept the mediation request. Although I think it makes sense to wait on structural changes until after we know their decision, there's certainly no harm in making improvements we can all live with in the interim. -- Rick Block (talk) 04:17, 18 January 2010 (UTC)

No structural changes indeed, but (y)our suggestion to revise the section. I don't think that 'the unconditionalists' would mind too much about changes in the conditional section anyway. There are obvious defects that may be repaired relatively easy. Heptalogos (talk) 08:23, 18 January 2010 (UTC)

I now see that you already did change, thanks for that. Heptalogos (talk) 08:47, 18 January 2010 (UTC)

"Suppose you're on a game show..."

"You're" is a contraction for "you are." This means YOU. The reader. Do you know of any host bias? Can you assign anything but a uniform distribution to the host's 2 goat door choice? Glkanter (talk) 18:26, 17 January 2010 (UTC)

If you're not talking about what sources say, this thread belongs on the /Arguments page. Please move it there (or simply delete it since we've talked about this ad nauseam already). Thank you. -- Rick Block (talk) 18:40, 17 January 2010 (UTC)

No, Rick, the arguments page is for this:

"Please place discussions on the underlying mathematical issues on the Arguments page."

I'm not discussing the underlying mathematical issues. I'm discussing how much weight Morgan's paper should receive in the article. That's an article editor's responsibility. Which is the purpose of this talk page.

And we have NOT discussed this before. We have discussed that the MHP needs to be solved from the contestant's SoK. We have certainly not discussed that I, or you, or anyone, as the contestant, have no knowledge of any host bias. Glkanter (talk) 18:55, 17 January 2010 (UTC)

That's really nonsense Glkanter. You are not at all correctly weighing any viewpoint by personally judging the content of a source. Then you found a gap in the reference to the arguments page, misusing that. So we should change the reference from an including to an excluding one: "please post any underlying discussion not directly adressing changes in the article to the arguments page." I suggest that these new sections, including the one from Gill110951, all get removed directly by the admin. Heptalogos (talk) 19:52, 17 January 2010 (UTC)

Hepatlogos: you want to move the new section I wrote to the arguments page. Fine by me, but I put it here because it leads to proposals of how the article could be organised: it can be much shorter and needs much less maths, now we have a short integrated solution to the two main variants which people like to formulate. Gill110951 (talk) 21:45, 17 January 2010 (UTC)

Your arguments cannot lead to acceptable proposals because they are fully POV if there's no reliable source telling us what the MHP is(, but you). Heptalogos (talk) 21:55, 17 January 2010 (UTC)

Glkanter, you are saying that if the player doesn't know anything about the host behaviour, then the player *has* to assign a uniform distribution. I disagree on this. I think that the player has to take account of all possible host behaviours. Gill110951 (talk) 21:45, 17 January 2010 (UTC)

I didn't see this argument coming. Other than, 'Here I am on this game show, heck, I don't know where they put the car', what else is there for me to account for? Glkanter (talk) 21:52, 17 January 2010 (UTC)

Though Laplace promoted it, the argument that ignorance should be represented with uniform probabilities is not much believed these days. Especially when it is not difficult to account for the fact that you don't know the strategy of the quiz-team and quizmaster. It's called game theory. More or less invented by von Neuman, one of the greatest scientists of the 20th century. Nowadays *everyone" knows about it, and it is used and abused all over science and economics and politics. I know of a lot of disasters in applications of statistics where people plugged in uniform probabilities when they didn't know what to plug in, not realizing that this choice can actually produce a very biased/unrealistic answer. EG the legal case of the suspected Dutch serial killer nurse, Lucia de Berk. Gill110951 (talk) 18:23, 18 January 2010 (UTC)

I'm not so good with 'subtle'. And you're raising an issue, 'can we assume a uniform distribution?' that I thought had long been settled. So, if you will, please offer your comments of the Huckleberry section. Glkanter (talk) 18:40, 18 January 2010 (UTC)

Despite the very long discussion we have had, so far all have managed to remain civil, which is to our credit. Demands to move discussions and threats to call admins do nothing to cool tempers here. The only way to move forward is to all try to understand the other side's point of view. That may require still more discussion. That may be tiresome for those that believe the article is right as it is but better to discuss that edit war. I suggested earlier that we all made the effort to use the two discussion pages effectively but said that should be done gently. As the original point was essentially about an underlying philosophical issue I would ask the original poster to consider moving this to the discussions page. Martin Hogbin (talk) 20:35, 17 January 2010 (UTC)

I don't agree that we should account tempers and practice gentleness, or whatever emotions that give the discussion other dimensions than plain reasoning. We'd better also not explicitly imagine such emotions. Heptalogos (talk) 22:15, 17 January 2010 (UTC)

We should always be civil here and it seemed to me that the above conversation was heading in a direction where it could have become uncivil. Martin Hogbin (talk) 11:14, 18 January 2010 (UTC)

Revised version of probabilistic solution

I've edited the content of the Probabilistic section, attempting to make it more NPOV (similar to the proposal above). If anyone violently objects to this feel free to revert, although I hope it is viewed as an improvement. -- Rick Block (talk) 05:41, 18 January 2010 (UTC)

Good change. This is also better explaining how the conditional approach adresses the very specific, although it might not seem to matter in this case. Heptalogos (talk) 08:52, 18 January 2010 (UTC)

Nijdam has reverted this change saying "it was no improvement". More specific comments would be helpful. Another idea is to incrementally edit, rather than revert wholesale. Here's what I changed it to. It was intended to address at least most of JeffJor's comments as well, on the version now archived at /Archive_13#Proposed unified solution section. -- Rick Block (talk) 03:16, 19 January 2010 (UTC)

---

Tree showing the probability of every possible outcome if the player initially picks Door 1

Another way to analyze the problem is to determine the conditional 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 probabilities in all cases where the player has initially picked Door 1 can be determined by referring to the figure below (note the case where the car is behind Door 1 is the middle column) or to an equivalent decision tree as shown to the right (Chun 1991; Grinstead and Snell 2006:137-138 presents an expanded tree showing all initial player picks). Given the player has picked Door 1, the player has a 1/3 chance of having selected the car. Referring to either the figure or the tree, if the host then opens Door 3, switching wins with probability 1/3 if the car is behind Door 2 but loses only with probability 1/6 if the car is behind Door 1. The sum of these probabilities is 1/2, meaning the host opens Door 3 only 1/2 of the time. The conditional probability of winning by switching for players who pick Door 1 and see the host open Door 3 is computed by dividing the total probability (1/3) by the probability of the case of interest (host opens Door 3), therefore this probability is (1/3)/(1/2)=2/3. Although this is the same as the average probability of winning by switching for the unambiguous problem statement as presented above, in some variations 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.

Car hidden behind Door 3	Car hidden behind Door 1		Car hidden behind Door 2
Player initially picks Door 1
Host must open Door 2	Host randomly opens either goat door		Host must open Door 3
Probability 1/3	Probability 1/6	Probability 1/6	Probability 1/3
Switching wins	Switching loses	Switching loses	Switching wins
If the host has opened Door 2, switching wins twice as often as staying		If the host has opened Door 3, switching wins twice as often as staying

---

The MHP - relations between sources

Selvin described a problem which he called the MHP. Savant Vos described another problem. Several sources reacted to Savant Vos (Morgan, Gillman, Grinstead) but did not mention Selvin. Who connected Selvin to Vos Savant? Or even more interesting (at least to me): can we create a graphical presentation of the links between all sources?

The reason why this could be interesting to all, is IMO that the question "what is the MHP" can only be answered by such a graphic. Where is the centre of gravity and how are sources connected? If any sources are outside (not connected), they should not me mentioded as the MHP. Heptalogos (talk) 12:46, 18 January 2010 (UTC)

Similar problems.

Can we judge other problems to be similar, if not related by sources? Heptalogos (talk) 12:08, 18 January 2010 (UTC)

Prominence.

I don't think 'weighing sources' is a formal Wiki-term. Only viewpoints in proportion to the prominence of each in the sources is mentioned. What is prominence? Apart from the position of a viewpoint within a source, how about the amount of sources in which a viewpoint exists? How about the amount of readers of a source (and thus the viewpoint)?

If many secondary sources write about a primary source, should the viewpoints of all secondary sources together be more prominent in the article than the viewpoints of the primary source? Would that be strange? Heptalogos (talk) 12:37, 18 January 2010 (UTC)

Any 'source' which attributes to the contestant some knowledge of how the host opens doors is not describing a story problem which begins, 'Suppose you're on a game show...' Which, as I understand it, is how most (all?) popular versions of the MHP begin. Glkanter (talk) 14:24, 18 January 2010 (UTC)

That's your opinion, and you're welcome to it. But since there are not just one or two, but many sources that don't agree with you we have to go with what the sources say. We can't exclude them because of something you or anyone else thinks about them. If it helps you understand their viewpoint any better, just imagine (for yourself) they're saying "Suppose you're on a game show and you knew ...". Furthermore, we can't even say (in the article) anything like "these sources violate the premise that you're on a game show" unless there's some published source we can attribute this to. The bottom line is what you or anyone else thinks about what reliable sources have to say is irrelevant. If they've made egregious errors, there would presumably be other reliable sources that call them on it. -- Rick Block (talk) 15:13, 18 January 2010 (UTC)

No Rick, my paragraph above is not an opinion. It is a logical conclusion. Glkanter (talk) 15:43, 18 January 2010 (UTC)

Rick, you wrote this, above:

"just imagine (for yourself) they're saying "Suppose you're on a game show and you knew ...". "

That contradicts the very essence of a game show. And the problem begins, "Suppose you're on a game show..."

And it's not in an any problem statement. That the host will always reveal a goat, and always offer the switch, have over time become 'accepted' premises. The contestant either colluding or mind reading with the host has not. Glkanter (talk) 16:04, 18 January 2010 (UTC)

@Heptalogos: the notion of prominence (it is also called weight) is primarily discussed in non-scientific articles, e.g. biographies. One example - during the recent US presidential election there was a continuous debate at talk:Barack Obama over how much prominence (if any) to give to Obama's relationships with William Ayers and Tony Rezko. These were stories that Fox News was broadcasting constantly, but mostly ignored by the mainstream media. The point is that accurately reflecting the prominence of a viewpoint within the complete set of reliable sources is an integral part of being NPOV. The readership of a viewpoint is not the issue, but rather the prominence of a viewpoint within reliable sources. One of the goals of this policy is to prevent Wikipedia from being used to promote "fringe" theories or partisan causes (this is policy as well, see WP:NOT#Wikipedia is not a soapbox or means of promotion). Prominence within secondary sources, not primary sources, is exactly what is meant. -- Rick Block (talk) 15:13, 18 January 2010 (UTC)

I have to wonder how prominent a source should be when it is accompanied by a commentary such as Seymann's. Is that common in peer-reviewed professional journals? Glkanter (talk) 15:43, 18 January 2010 (UTC)

@Glkanter, The American Statistician is a peer-reviewed journal for professional statisticians and professional teachers of statistics but not what within professional statistics would be called a research journal. It contains discussion and gossip and a teacher's corner and the like... I am not being disparaging, I am just trying to say that from a professional research-oriented statistician's point of view the journal does not carry a lot of weight and that particular article certainly doesn't contain much work. People who do important novel work publish it in big journals and maybe later do some advertising in The American Statistician. The Morgan et al paper exists and makes an important point which people like to refer to (distinguish conditional from unconditional) so it became a standard reference. At some point no-one reads the references anymore, people just refer to the standard references. The folklore as to "what is" the Monty Hall problem evolves. Science is a cultural, a social phenomenon, as much as anything else. I did not know about te Seymann commentary till I read about it here. He expresses my own gut feelings, I'm glad that that has been written down before. Long live Wikipedia, long live amateur science! You guys are doing the work which the so-called "professionals" (like me) don't have time to do anymore, since we need to spend all our time writing grant applications and grant reports and going to department meetings and doing politics just in order to survive. Gill110951 (talk) 18:16, 18 January 2010 (UTC)

My understanding is that Selvin first brought up the conditional formula a few months after his original letter to the journal. Other than contriving the 'host bias', but not a 'car placer bias', thereby creating an entirely new and different puzzle which is not about a game show, what did Morgan contribute? Glkanter (talk) 18:45, 18 January 2010 (UTC)

The MHP in economics and game theory

Here is what some say is the first solution of Monty Hall by Game Theory:

Barry Nalebuff (1987) Puzzles: Choose a Curtain, Duel-ity, Two Point Conversions, and More. Economic Perspectives vol. 1 nr. 1 pp. 157--163

http://www.jstor.org/pss/1942987

"Puzzle 1" is our very own Monty Hall. I'll try to collect more literature references (and find out what the contents are). But if doesn't belong on the talk page but somewhere else, please move it. Gill110951 (talk) 20:00, 18 January 2010 (UTC)

This is the same Nalebuff reference that's in the article (referred to in the History section). If you want more literature references you might look up the Barbeau references that are in the article. Many of the folks commenting here don't seem to realize this, but it really is quite a good article. Wikipedia's featured article standards are quite high - at least aspirationally equivalent to Brittanica. The sources are generally the original sources for the points that are made, and are the sources that other sources refer to. For example, if you read Rosenhouse's recent book you'll find its references look mighty similar to the references in the article. -- Rick Block (talk) 21:07, 18 January 2010 (UTC)

Rick, what about the issue described above on linking sources? Shouldn't all sources be linked to be addressing the same thing? Would such a presentation be able to show some prominence? Please react above, if you wish. Heptalogos (talk) 21:20, 18 January 2010 (UTC)

The History section of the article contains most of what I have been able to find about, well, the history of the problem. Barbeau's survey (the 1993 one in particular) is quite detailed, but contains nothing in the gap between Selvin's publication in 1975 and Nalebuff's paper in 1987 (which Barbeau does not mention). I haven't been able to find anything that was published in this interval although Nalebuff says "This puzzle is one of those famous probability problems, in which, even after hearing the answer, many people still do not believe it is true" - clearly implying it was famous (at least within academia) by that point. Nalebuff doesn't say where he got it from. There was a mention of it in Mathematical Notes from Washington State University newsletter shortly before vos Savant's first column (I don't have this source). I don't know where Whitaker heard of it. It would be interesting to compare Whitaker's version to the one in Mathematical Notes from WSU. Following the publication in Parade the problem was extremely widely known, both in popular sources and academia. Others on this page have claimed it was an example problem in probability classes at MIT - this is informally supported by a scene in 21, the movie about the MIT Blackjack Team - although I don't know how to pin down exact dates for this. -- Rick Block (talk) 02:38, 19 January 2010 (UTC)

"The basis to my solution is that Monty Hall knows which box contains the keys and when he can open either of two boxes without exposing the keys, he chooses between them at random." - Steve Selvin

The American Statistician, August 1975, Vol. 29, No. 3

http://montyhallproblem.com/as.html

Glkanter (talk) 22:47, 18 January 2010 (UTC)

And vos Savant completely overlooked the "when he can open either" part of this. Do you have a point you're trying to make? -- Rick Block (talk) 01:38, 19 January 2010 (UTC)

vos Savant published a reader's letter in a general interest magazine. As authors/critics in a peer-reviewed scientific journal, Morgan, et al (and/or the peers) have the responsibility of understanding the existing literature on the subject, which certainly includes Selvin's letters. Selvin's letters were in the same journal Morgan published their critique in. They shouldn't have been hard to find. Selvin completely puts the kibosh any any contrived 'host bias'. Without 'host bias', Morgan's statement that all simple solutions are 'false' has no legs to stand on. So, one more time, what is noteworthy about the Morgan paper? How did it advance the understanding of the MHP? Glkanter (talk) 12:59, 19 January 2010 (UTC)

Here's more from the same letter:

"Monty Hall wrote..."Oh and incedentally, after one [box] is seen to be empty, his chances are no longer 50/50 but remain what they were in the first place, one out of three. It just seems to the contestant that one box having been eliminated, he stands a better chance. Not so." I could not have said it better myself." - Steve Selvin

So Monty doesn't mention door numbers at all, talks about 50/50, (the probabilities) remain what they were, and 1/3. Steve Selvin says "I could not have said it better myself."

But Morgan and Rick know what the paradox 'really' is, and they claim this isn't it. 16 years and 35 years after Selvin himself already told us it is.

So, let's talk about how much weight to give to various sources. And I don't mean sources in the 'publication' sense. I mean 'sources' as in which author is (most) reliable.

Did the paradox change due to Morgan's paper? For Rick's interpretation to be right, it must have. It didn't. I covered this topic in more detail here. Glkanter (talk) 02:59, 19 January 2010 (UTC)

Are you suggesting some change to the article? If so, please say what it is. Thank you. -- Rick Block (talk) 03:20, 19 January 2010 (UTC)

Well, my contribution to the discussion would be along the lines of, "Selvin directly contradicts Morgan's contrivance of a 'host bias', and also directly contradicts Rick Block's interpretation of what the MHP paradox is. In this light, along with all the other errors and fallacies in the Morgan paper, and presumably those that rely on it, I recommend that Morgan's emphasis in the article be reduced to no more than a footnote in an appendix. Near the end of the article." But that's just my opinion, based on the reliable sources. What would you suggest, Rick? Glkanter (talk) 04:20, 19 January 2010 (UTC)

I would suggest that Morgan et al. is unarguably a WP:reliable source and that since its viewpoint (that the MHP is fundamentally a conditional probability problem) is consistent with a large number of other reliable sources and is a standard (if not the dominant) academic viewpoint, that this viewpoint should be prominently mentioned in the article. -- Rick Block (talk) 05:20, 19 January 2010 (UTC)

These last 2 paragraphs are instructive. I based my recommendation on how to edit the article on the words of the sources. You base your recommendation on your personal, unsupported opinion of the 'academic viewpoint' and a non-comparative use of the phrase 'large number of other reliable sources'. However many there may be, if they're based on Morgan, clearly they're ill-advised. This reliance on your personal interpretations, rather than the sources is consistent with the pro-Morgan POV of the article, and in the veto power you and Nijdam continue to yield over the editing of the article. Glkanter (talk) 13:10, 19 January 2010 (UTC)

You have this exactly backwards. I base my recommendation on what the predominant sources say. If you'd like me to list them again I'd be happy to. You base yours on your own OR and that of a few others commenting on this page who have concluded (on their own, not from ANY published sources) that Morgan et al. contains "errors and fallacies" and your argument that what this paper says contradicts what Selvin says (??!!). Even if this latter point were true (it's not), that would make the POV of this paper simply another POV that by the fundamental Wikipedia content policy of WP:NPOV SHOULD be included, "in rough proportion to [its] prevalence within the source material." Rather than try to assess this prevalence you say "However many there may be, if they're based on Morgan, clearly they're ill-advised." What you're saying is that you don't want the POV of these sources, no matter how many there are (!), fairly represented in the article. You are the one here relying on your personal interpretation, rather than the sources. -- Rick Block (talk) 14:27, 19 January 2010 (UTC)

Yes, I'm familiar with this aggregation technique you subscribe to. I learned about it here.

Of course Morgan's paper is full of errors and fallacies. They mis-quote vos Savant, they have a math error discovered by Martin and Nijdam, they contradict Selvin by creating a 'host bias'. There's plenty more. Glkanter (talk) 14:01, 20 January 2010 (UTC)

Morgan includes a very brief history of the problem, referring only to two sources of the "prisoner's dilemma" from the 1960s. No mention of Selvin. I just did a search of the Morgan article. No mention of Selvin. Not so good for a piece titled, "Let's Make A Deal: The Player's Dilemma" that criticizes the work of others, twice accusing vos Savant of 'false' deeds in the introductory paragraph. Glkanter (talk) 20:59, 19 January 2010 (UTC)

This is not relevant to their point. They're not criticizing Selvin. They're criticizing vos Savant (who never mentioned Selvin either). -- Rick Block (talk) 22:22, 19 January 2010 (UTC)

Yeah, just why did these 4 uninformed hooligans pick on that nice general interest magazine lady in their prestigious peer-reviewed professional journal? Glkanter (talk) 22:30, 19 January 2010 (UTC)

Well, what do the sources say? Ah yes, right at the end of the intro of Morgan et al. they say that although, under certain circumstances, her answer is correct, her methods of proof are not. This is their POV. You apparently don't agree. But you're not a reliable source. Gillman is. He says (presumably untainted by Morgan et al. since he published essentially simultaneously) "Marilyn's solution goes like this ... This is an elegant proof, but it does not address the problem posed.". Grinstead and Snell is. They say (in their textbook, 14 years later, plenty long enough for Morgan's and Gillman's egregious errors to have been discovered) "This very simple analysis [the outcome of a predetermined "stay" strategy compared a predetermined "switch" strategy], though correct, does not quite solve the question that Craig posed." (not quite as direct as Morgan et al. or Gillman, but it means the same thing).

How many sources would you like me to produce before you're willing to call this a mainstream POV? -- Rick Block (talk) 01:33, 20 January 2010 (UTC)

There are reliable sources that criticise Morgan, Seymann, of course, who was published in the same journal and Rosenhouse, in his book devoted entirely to the MHP. He refers to the paper as 'their bellicose and condescending essay', in which they 'presumed to lay down the law regarding vos Savant’s treatment of the problem'. Of Morgan's criticism of vos Savant, Rosenhouse says, 'Rather strongly worded, wouldn’t you say? And largely unfair,...'. Martin Hogbin (talk) 10:14, 20 January 2010 (UTC)

And, how many sources would you like me to produce before you're willing to call this a mainstream POV? Even if some sources criticize Morgan et al., given that there are other reliable sources supporting the POV that vos Savant doesn't quite address the question this is a perfectly valid POV. -- Rick Block (talk) 13:09, 20 January 2010 (UTC)

That depends, Rick. Is the Wikipedia article about the Monty Hall problem, or a critique of vos Savant's column? Glkanter (talk) 13:20, 20 January 2010 (UTC)

I'll be more specific. The POV is that unconditional solutions (such as those used by vos Savant and most popular sources) don't quite address the Monty Hall problem. -- Rick Block (talk) 13:37, 20 January 2010 (UTC)

No, Rick. Morgan criticizes vos Savant and doesn't mention Selvin. Gillman criticizes 'Marilyn', and Grinstead and Snell refer to 'Craig'. This is all from your diff earlier in this section.

Rick, are those sources criticizing vos Savant, or the simple solutions to the Monty Hall problem? In their words, please, not your interpretation of what they might have meant. Glkanter (talk) 14:08, 20 January 2010 (UTC)

Do they rely on a 'host bias' to make their point? How does the 'Combining Doors' solution fail to address the Monty Hall problem? Glkanter (talk) 13:44, 20 January 2010 (UTC)

No, Morgan et al. generically criticize all unconditional solutions. Their "false" solutions F1, F2 (one of vos Savant's), F3 (vos Savant's experiment), and F5 are specific unconditional solutions, but it's clear they are criticizing all unconditional solutions:

• "Thus is the player having been given additional information, faced with a conditional probability problem."
• "The distinction between the conditional and unconditional situations here seems to confound many"
• "The correct simulation for the conditional problem is of course to examine only those trials where door 3 is opened by the host. The modeling of conditional probabilities through repeated experimentation can be a difficult concept for the novice, for whom the careful thinking through of this situation can be of considerable benefit."
• "In general, we [meaning anyone] cannot answer the question 'What is the probability of winning if I switch, given that I have been shown a goat behind door 3?' unless we either know the host's strategy or are Bayesians with a specified prior."

Gillman criticizes Marilyn directly, but the reason he states for his criticism is that Marilyn's solution does not address the conditional probability: "Game I [what he considers to be the MHP] is more complicated [than Game II in which "you have to announce before a door has been opened whether you plan to switch" - italics in the original]: What is the probability P that you win if you switch, given that the host has opened door #3? This is a conditional probability, which takes account of this extra condition." (italics in the original). His POV is clearly that the MHP is a conditional probability problem, and that this is not addressed with an unconditional solution.

What Grinstead and Snell say (from above) is clear as well. They're not directly criticizing Marilyn, but saying that the problem Craig posed (that vos Savant addressed unconditionally) is a conditional probability problem.

Do they rely on host bias to make their point? What difference does it make?

How does "Combining Doors" fail to address the Monty Hall problem? It is an unconditional solution (are you arguing that it isn't?). -- Rick Block (talk) 16:09, 20 January 2010 (UTC)

Selvin rules out any 'host bias' in the Monty Hall problem. The 'Combining Doors' solution shows a goat revealed by the host. Then the contestant decides whether or not to switch. I don't even call it 'probability', let alone 'unconditional'. I call it 'simple'.

Do any of these sources mention Selvin? Glkanter (talk) 16:52, 20 January 2010 (UTC)

What sort of obligation did Morgan, et al have as writers in a peer-reviewed professional journal?

The title of their paper is "Let's Make A Deal: The Player's Dilemma".

Where they expected to know the history of the Monty Hall problem? In 1991, they mention the "prisoner's dilemma" from the 1960s, but not Selvin's letters of 1975 to the very same journal Morgan was published in. Glkanter (talk) 14:15, 20 January 2010 (UTC)

Do you have some point related to editing the article? It would be helpful if you make whatever point you're trying to make in your initial post without making someone ask what your point is. -- Rick Block (talk) 16:19, 20 January 2010 (UTC)

Rick, surely you're familiar with the 'Socratic Method'? I could call them 'lazy' or 'uninformed' or 'unprofessional', but that would be my personal opinion. This way, qualified people will hopefully answer my question, and that these guys were 'lazy' or 'uninformed' or 'unprofessional' and wrote a paper that is not a reliable source for the Monty Hall problem article (other than a footnote, perhaps) will become self-evident. Just maybe not to everyone. Glkanter (talk) 16:59, 20 January 2010 (UTC)

So, you're saying because they don't mention Selvin they should not be considered a reliable source? Please read Wikipedia:Reliable sources. What makes a source reliable has nothing to do with your (or anyone else's) judgment of the content. What you're effectively arguing is that we should ignore sources that don't match your POV. This is contrary to the fundamental content policies of Wikipedia (specifically WP:V and WP:NPOV). It is NOT our job to "fact check", or "ensure the logical consistency", or apply any value judgment at all pertaining to what reliable sources say. We need to understand what they say, and fairly represent it in the article. That's all. -- Rick Block (talk) 17:47, 20 January 2010 (UTC)

I'm saying by asking a question, the answer becomes self-evident. Each editor who takes part in the consensus editing will make his own judgment based on his understanding of the source materials and Wikipedia's policies and guidelines. It's The Monty Hall problem article, not the 'Marilyn vos Savant's Parade Magazine Column Is Not Fit For Use As A University Text Book On Probability' article. Glkanter (talk) 18:16, 20 January 2010 (UTC)

This Archiving Is So Stupid...

I would rather we just wrote in the archives, and then someone told us when to start a new one. It would make more sense than what's going on now. Glkanter (talk) 17:11, 20 January 2010 (UTC)

This page is very big — still over 350k. It seems to me that it's easier to let the bot prune the deadwood discussions than to do it manually, but you can change it if you wish.

—WWoods (talk) 07:59, 21 January 2010 (UTC)

Thanks for you prompt response. You indirectly pointed out a pet peeve of mine. The talk pages and archives are maintained for the benefit of the users. The way the archiving is taking place does not maintain the chronology of the sections. In fact, there may be older sections in archive 13 than in archive 12. So in order to use this bot, the users are not nearly as well served as they are with some other methods. Glkanter (talk) 10:02, 21 January 2010 (UTC)

Proposed Edits To The Probabilistic solution

"The basis to my solution is that Monty Hall knows which box contains the keys and when he can open either of two boxes without exposing the keys, he chooses between them at random." - Steve Selvin, The American Statistician, August 1975, Vol. 29, No. 3

Given that the originator of the Monty Hall problem clearly states that the host chooses doors (boxes) randomly when he has two goats (empty boxes) to choose from, any 'host bias' scenarios should be treated the same as the other 'variants' included in the 'Variants - Slightly Modified Problems' section.

Accordingly, I propose removing the following from the Probabilistic solution section:

From paragraph 1: These solutions correctly show that the probability of winning for all players who switch is 2/3, but without certain assumptions this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens.

From paragraph 1: The difference is whether the analysis is of the average probability over all possible combinations of initial player choice and door the host opens, or of only one specific case—for example the case where the player picks Door 1 and the host opens Door 3. Another way to express the difference is whether the player must decide to switch before the host opens a door or is allowed to decide after seeing which door the host opens (Gillman 1992). Although these two probabilities are both 2/3 for the unambiguous problem statement presented above, the conditional probability may differ from the overall probability and either or both may not be able to be determined depending on the exact formulation of the problem (Gill 2009b).

From paragraph 2: This analysis depends on the constraint in the explicit problem statement that the host chooses randomly which door to open after the player has initially selected the car.

Eliminate the second image. It's just the first image rotated 90 degrees.

I noticed that the only solutions offered in this section are the 2 detailed images of every possible outcome once a door has been selected. From the 'Probabilistic' images the reader can determine that the likelihood of his choice being the car is 1/3 by adding the 1/6 + 1/6 from the 'Total probability' column, or he can do the (probably unknown to him) calculation of the remaining door '(1/3)/(1/3 + 1/6), which is 2/3'.

Or he can use logic and derive the 1/3 for his selection hasn't changed, much like in Selvin's second letter to The American Statistician, "Monty Hall wrote..."Oh and incidentally, after one [box] is seen to be empty, his chances are no longer 50/50 but remain what they were in the first place, one out of three. It just seems to the contestant that one box having been eliminated, he stands a better chance. Not so." I could not have said it better myself." - Steve Selvin Glkanter (talk) 12:53, 21 January 2010 (UTC)

Rather than delete, there's another suggestion for revisions above, see #Revised version of probabilistic solution. Specific comments on these suggestions:

• My suggestion from above also deletes the "without certain assumptions" bit from the first paragraph. Is this what you're actually objecting to, and if so does the suggestion above address this concern? I think introducing this solution as "another way to analyze the problem" is about as flatly NPOV as possible.
• I think it's important to explain what the difference is between the two approaches (average probability vs. probability in the specific case). Does the version I suggest above address the concern here as well (which I'm thinking is actually the sentence starting "Although these two probabilities..."). In what I suggest above this is moved, and rephrased with the intent of making it clear that the average probability and probability in the specific case differ only in variations of the problem statement.
• The figure is rotated (showing all possible car locations against a constant initial player choice and door the host opens rather than all possible player choices against one arrangement of goats and car). I'd be fine with one image, but I strongly prefer this one rather than the other one. My reasons for this are
  1. This image matches the example case given in the problem description where the player has picked door 1 and the host has opened door 3.
  2. This image shows the symmetry between the host opening door 2 and the host opening door 3 (the image itself is symmetrical)
  3. This image does not have the cartoonish graphic.
  4. The column widths in this image match the probabilities. This creates a visual interpretation consistent with the text.
  5. With this image, it's easy to see the result where the host opens either door.

-- Rick Block (talk) 15:16, 21 January 2010 (UTC)

Thanks for the prompt response, Rick. I think just having a symmetrical condition solution section is instructive enough. And if both solutions rely on the same chart, pointing this out may be enough for the reader. The less ornate image is much easier to discern both the (1/3)/(1/3 + 1/6) and the 1/6 + 1/6 from.

Your description of how to do a conditional formula would be instructive. So, we could add:

Referring to either the figure or the tree, if the host then opens Door 3, switching wins with probability 1/3 if the car is behind Door 2 but loses only with probability 1/6 if the car is behind Door 1. The sum of these probabilities is 1/2, as the host will open Door 3 only 1/2 of the time. The conditional probability of winning by switching for players who pick Door 1 and see the host open Door 3 is computed by dividing the total probability (1/3) by the probability of the case of interest (host opens Door 3), therefore this probability is (1/3)/(1/2)=2/3.

I changed a couple of words. Glkanter (talk) 15:44, 21 January 2010 (UTC)

Again, not arguing, just trying to understand what you're saying. Is the "less ornate" image you're referring to the tree diagram as opposed to the table with the pictures at the bottom of the "Probabilistic solution" section? Which figure are you referring to in your conditional solution, and where are you suggesting adding this?

To clarify my comment to you, I'm saying I prefer the table-with-pictures image in the Probabilistic solution section to the similar one in the Popular solution section (for the reasons provided above). -- Rick Block (talk) 01:25, 22 January 2010 (UTC)

The tree/table that you added to this discussion is the one I want to keep. For both sections. For the reasons I gave above. There are currently 3 unique solutions in the Popular section. The first one would be replaced by my proposal, including the tree/table.

[moments later] I just got your point. You want to remove the image with the heads. The tree/table could be used for both the revised first solution and the 'you get the opposite' solution. Chun's chart shows both, as I mentioned to Martin. And I still want to delete the 2nd image from the probabilistic solution. Glkanter (talk) 01:49, 22 January 2010 (UTC)

... and keep the image with the heads in the Popular solution section? I'll note that there's no correspondence between Chun's diagram and the "heads image" (Chun's diagram is "rotated" in the same way as the "no heads image" in the Probabilistic solution section). -- Rick Block (talk) 04:56, 22 January 2010 (UTC)

No, the heads image gets removed. Chun works with Popular's revised '1/3 2/3' and 'opposite'. And Probabilistic. Glkanter (talk) 05:40, 22 January 2010 (UTC)

Proposed Edit To The Popular solution

I'd like to make this the first solution offered. Attributed to Selvin's 2nd letter; Chun 1991; Grinstead and Snell 2006:137-138),

Your initial chance of selecting the car is 1/3. This is unchanged by the host revealing a goat. Therefore, when offered, switching doubles your chance of winning the car from 1/3 to 2/3. Glkanter (talk) 13:14, 21 January 2010 (UTC)

I think the concept is good but the reason why the unchanged chance of having originally chosen the car mean that the chances of winning by switching are 2/3 is afar from obvious to most people. The solution needs further explanation, maybe along the lines of you always get the opposite of your original choice if you swap. Martin Hogbin (talk) 14:31, 21 January 2010 (UTC)

Actually, my proposal is just a re-write of what's already in there. But, let's flesh out your idea while sticking with reliable sources. There's already 3 sources for the current more-complicated statement in the article. Selvin agrees with Monty Hall that it remains at 1/3 in his 2nd letter to The American Statistician. In his 1st letter, he has a table of all 9 possible outcomes, 3 locations for the car * 3 contestant choices. He lumps each of the the two-goats choice as 1 line item. This chart, which shows all possible outcomes, could accurately be summarized as 'you get the opposite'. (Chun's and Grinstead & Snell's table also shows this 'opposite' result by switching.) Chun and Grinstead & Snell demonstrate the 1/3 in their chart in the Probabilistic section. I guess we could add the chart to the Popular solution. I think it would demonstrate the equivalency of the differing approaches if they both used the same sources. All they do in the chart is divide the original 1/3 contestant door by 2. Then all the contestant does is add the 1/6 & 1/6 back together. And, there are 2 other solutions already provided in the section. Glkanter (talk) 15:11, 21 January 2010 (UTC)

"This is unchanged by the host revealing a goat." is not what Grinstead and Snell say. I can't find my copy of the Chun reference, but I highly doubt this is what he says either (do you have a copy of this reference, or are you interpreting this from the tree diagram?). What Grinstead and Snell actually say is: "Using the “stay” strategy [deciding beforehand to stay with the initial choice whatever the host does], a contestant will win the car with probability 1/3, since 1/3 of the time the door he picks will have the car behind it. On the other hand, if a contestant plays the “switch” strategy, then he will win whenever the door he originally picked does not have the car behind it, which happens 2/3 of the time." Selvin doesn't say this either - he's quoting Monty Hall. If you want to include the full quote that's fine, but from a sourcing perspective it would be much better to have a secondary source for this.

And, you're completely missing the point of the table and the tree. The host only opens one door, not both. Before the host opens a door the player's chance is 1/3 (in the tree this is the 1/3 for Door 1 on the left). After the host opens either, but only one, of the doors the player's chance is 1/6 and the other door's chance is 1/3. Viewed as a conditional probability this means the player's chance, in this case, is 1/3 and the other door's chance is 2/3. You don't add the 1/6 and 1/6 to get "the player's chance". You divide 1/6 by the probability of being in this case (which is 1/2). -- Rick Block (talk) 15:45, 21 January 2010 (UTC)

Yes, I am using them as a source based on the diagram showing the Total Probability is 1/6 + 1/6 = 1/3.

Selvin says of Monty Hall's quote, 'I could not have said it better myself'. That means he agrees.

I can use that table any way that is mathematically sound. The column heading is 'Total Probability'. It's really nothing more than Selvin's original table from his first letter reduced to just 1 of 3 random contestant picks, plus the 2-goat options are split out. That I choose to add 1/6 + 1/6 rather than do the unfamiliar division is certainly a valid use of the figures. Just not what the authors expected. Glkanter (talk) 15:56, 21 January 2010 (UTC)

If you are interpreting the table the words should reflect that. As stated it sounds like you're saying these sources state this in words. To make it clear you're interpreting the table (which is a valid thing to do) I'd suggest you say what it is in the table that you're referring to.

My point about adding 1/6 plus 1/6 is that by doing this you're determining the player's chance of initially selecting the car (before the host has opened a door). This is the exact point we've been talking about for over year. It is NOT mathematically sound to add these together and call it the chance of the player's door hiding the car after the host has opened a door. This is not what the table shows. It's the chance of initially selecting the car. It is also the chance of winning the car if you ignore what the host does and stick with your original choice. But this 1/3 (=1/6+1/6) is not the chance in effect after the host has opened a door (say Door 3) where the opened door's chance is now 0. If you're saying the player's door's chance is 1/3 because it's 1/6+1/6, the chance of either of the other doors (by the same logic) is 0+1/3. The opened door's chance is 0 only conditionally, i.e. 0 divided by 1/2. Its total probability (considering all cases) is STILL 1/3! Look at the table. The other door's chance is 2/3 only conditionally, i.e. 1/3 divided by 1/2 - but its total probability is also still 1/3. Similarly, the original door's chance after the host has opened a door is also only the conditional chance, i.e. 1/6 divided by 1/2. The numeric answer is 1/3 which is the same number as its total probability considering all cases, but where this number comes from (according to the table) is not 1/6+1/6 "unchanged" by the host opening a door. It's the original 1/3, divided into two pieces (reflecting the cases where the host opens door 2 and door 3), and then divided by the probability of these cases (which is 1/2). The result is the numeric probability doesn't change, but the reason for this, as shown by the table or the tree, is that it splits it into 1/6 and 1/6 and then becomes 1/3 only as a conditional probability (i.e. by dividing by 1/2). -- Rick Block (talk) 18:03, 21 January 2010 (UTC)

The tables in the Probabilist solution section are derived from Selvin's first letter. He lists 9 cases and determines 3 lose when switching, for 1/3. The Probabilistic section's table shows only 1 of the 3 random contestant choices (door #1) and splits out the 1/3 for 2 goats by dividing by 2 for 1/6 and 1/6. This gives 2 cases at 1/3 each and 2 cases at 1/6 each in the Total Probability column of Chun's and Grinstead and Snell's table.

1. It looks like you can't solve the conditional problem without also solving the unconditional problem. That's what the 1/6 + 1/6 gives us. From the Total Probability column of the same table they use to solve the conditional problem. I have now solved the problem unconditionally.

2. To get to 1/6, they divided Selvin's 1/3 by 2. To solve this conditionally, you're telling me I need to divide by 1/2. OK. To determine my door's probability of having a car, I will divide by 1/2. Which is the same as multiplying by 2. So, (1/3)/2 = 1/6. 1/6 * 2 = 1/3 probability I have chosen a car. I have now solved the problem conditionally.

3. Boris suggested "The coexistence of the conditional and the unconditional can be more peaceful." What could be more peaceful than both solution sections referencing the same table of outcomes? Glkanter (talk) 18:48, 21 January 2010 (UTC)

I'm not sure I'm understanding what you're saying - I'm not arguing here but would like to echo back what I think you're saying.

Tree showing the probability of every possible outcome if the player initially picks Door 1

By the tables in the Probabilistic solution you mean this tree diagram (which is Chun's, not Grinstead and Snell's) and the large table with the images at the end of the section. The diagram and table show only one of the 3 possible initial player picks (i.e. using player picks door 1 as an example). [just a note - these don't derive from Selvin's table. Chun published the tree diagram, and the large table is simply another view of Chun's diagram. Grinstead and Snell's diagram is larger and shows all possibilities, not just the ones involving the player picking door 1]

1. The diagram and the table shows the probability across all cases, so you can solve either the conditional or unconditional problem. Unconditionally, if you don't switch you win the car with probability 1/6+1/6 (top two lines of the diagram - middle column in the table) and if you do switch with probability 1/3+1/3 (bottom two lines of the diagram, outer two columns in the table).

2. Where the 1/6 comes from is the original 1/3 (the chance of a player who has picked door 1 selecting the car before the host opens a door) divided by 2 (because the host can open either door). To make this the conditional probability you divide by the probability of the case of interest (e.g. host opens door 3) which is 1/2 (from the diagram this is one of the 1/6 lines plus one of the 1/3 lines, from the table it's either the right half or the left half). [just a note - perhaps a little more intuitively, what this is saying is the door you've picked has a 1/6 out of 1/2 chance of hiding the car in the case the host opens door 3] This is the conditional solution.

3. Boris suggested peaceful coexistence of the conditional and unconditional - what could be more peaceful than showing both in the same diagram and table?

You don't explicitly say it, but it sounds like you're suggesting you're OK with using these two figures (Chun's diagram and the table) in both the "Popular" and "Probabilistic" solution sections. Do I have this right? -- Rick Block (talk) 19:33, 21 January 2010 (UTC)

As a component of all the edits I suggested, yes, the 1 table (I already suggested removing the 2nd one) could be added to the Popular solution section. While the 2 1/6 lines are more detailed than necessary, I think we would be OK. Once the reader got to the symmetrical Probabilistic section and saw the same table, he might just grasp the coexistence we're trying to show.

I don't know what you mean by 'outer column'. All the values I referenced come from the 'Total probability' column.

I'm using the conditional method to derive the probability of my door being the car (1/6)/(1/2) which is the same as 1/6 *2. It's a much less complicated formula than door 3's. And it's a more consistent approach to always solve for my door.

I'd like to see some changes in the FAQs, too. Glkanter (talk)

I'm losing track of all the proposals. And already some time ago I made aa suggestion on this page: Talk:Monty Hall problem/Construction. There you find my proposal for the beginning of the article. Nijdam (talk) 22:16, 22 January 2010 (UTC)

Nijdam, do you intend to unconditionally revert any edits that I make? Do you agree with the reasoning I'm using to propose these changes? Glkanter (talk) 22:50, 21 January 2010 (UTC)

Let's carry on

As I read this discussion between Rick and the guy from the Mediation Committee, we shouldn't do anything different as we wait for the formal mediation.

Accordingly, I will continue proposing, and when appropriate, making changes to the article. Glkanter (talk) 17:07, 22 January 2010 (UTC)

What happens now?

I've shown that Morgan overlooked Selvins' 'the host acts randomly when faced with 2 goats (boxes)'.

Nijdam has been uncommunicative about his intentions and reasons.

I edited the article and Nijdam reverted me. I reverted him back. What's next? Glkanter (talk) 12:33, 23 January 2010 (UTC)

I've undone it. Your change to the popular solution is not good, because you make an unexplained jump, which is clearly explained in the original. Your changes to the probabilistic solution cannot be done by the argument that you proved Morgan wrong, or similar, which is POV. Morgan did not criticize Selvin anyway. It might me easier for you to make several logical subchanges, with less chance to be undone all. Heptalogos (talk) 13:18, 23 January 2010 (UTC)

Yes, had Morgan, et al even the vaguest knowledge of the puzzle's history, a simple letter to vos Savant referencing Selvin's 2nd letter would have allowed her, as Selvin also found it necessary, to clarify that the host chose randomly when faced with 2 goats. Glkanter (talk) 14:12, 23 January 2010 (UTC)

She did not even need to clarify, because it's perfectly reasonable to assume no host bias. But this discussion doesn't make any sense in discussing an encyclopedian article. Heptalogos (talk) 14:19, 23 January 2010 (UTC)

I think it is best to leave any editing until we get an input from the mediator. So long as this happens within a reasonable time. Martin Hogbin (talk) 13:28, 23 January 2010 (UTC)

You changed Sources of confusion into 3 parts: 50/50 paradox, assumptions, and (un)conditional. But the last one is hardly explaining the discussions concerning the number of the door. Also, the second one starts with "selvin", out of the sky. Heptalogos (talk) 13:55, 23 January 2010 (UTC)

'Increasing the number of doors' and 'N doors'

'Increasing the number of doors' is in the 'Aids to Understanding' section. 'N doors' is in the variants.

It seems confusing to me that the same idea would be appropriate for both sections. Glkanter (talk) 13:16, 23 January 2010 (UTC)

Sources of confusion

Most of the article's content is about confusion. I see three sources:

1. Incomplete information: we assume worlds outside and/or within the stated problem.

How is the setup? What is the behavior? What is the knowledge?

2. Surplus information: we decide which information to use and which not.

Game show, door numbers, host knowledge, talking host.

3. Paradox: the presentation of the information leads to wrong assumptions and decisions.

A door is not just randomly opened out of three or two, which is the missing key for many.

Source 3 is the basic reason for the problem.
Source 1 is the basic reason for the paradox.
Source 2 is the basic reason for the afterparty disagreements.

The article uses the same order: paradox first, then the assumptions explained, and then the discussion about the details included in or exluded from the conditions. I propose to setup the article using these headers, or similar:

• Problem statement: the usual and the correct answers
• Paradox: simply explained
• Assumptions: reasonableness and explicitness
• Conditions: possibilities and variations
• History: (conflicting) perspectives
• Bayesian analysis
• etc. Heptalogos (talk) 20:13, 23 January 2010 (UTC)

Your section 'Paradox:simply explained' needs to convince people of the right solution. I may not normally be the purpose of an encyclopedia to convince people but, unless readers believe the solution, the rest of the article is wasted. As most people get the answer wrong and many do not believe the answer even when explained to them, this point is of primary importance.

Split the article into 2 articles: The MHP Problem and Solution, and The MHP's Conflicting Perspectives. Because nobody, other than you guys, gives a shit. Glkanter (talk) 16:58, 24 January 2010 (UTC)

Where will you put the conditional solution? Martin Hogbin (talk) 16:08, 24 January 2010 (UTC)

I think the article should indeed not be basically explanatory, but that's already done now in the current article, e.g. in the popular solution section. This may be in the Paradox section. We need to end it with the explicit assumptions needed for this simple explanation(s) to be correct, which is then an inroduction to the next chapter. Is it an option to link to this, which is about the best primary source available? This will also provide an opportunity for people to understand it better. Marilyn offers some examples, like the sea shells. If people still don't understand, they should maybe visit an internet forum to discuss.

The assumptions section will be the introduction to the Conditions section. This seems to be the best section for those sources that go into detail, like Morgan. But Morgan may already be mentioned as a source in the assumptions section, or even in the last part of the Paradox section where the necessary assumptions are given. We'll need a source for that statement anyway. This setup might also be in line with the combined solutions idea of Rick. Heptalogos (talk) 18:48, 24 January 2010 (UTC)

I'm not seeing a significant difference between what you're suggesting and the current structure. You're missing the WP:LEAD (which needs to summarize the entire article - or is this what you mean by "Problem statement"?), but other than that it seems extremely similar. I think it would be helpful if you mapped out where the existing content (by section) would go in this structure and if you're suggesting deleting anything explicitly mentioning that. Without this, it's a little difficult to understand what you're really talking about. For example, where would the content currently in "Aids to understanding" go? -- Rick Block (talk) 20:32, 24 January 2010 (UTC)

Roger. I think I need to shuffle this into a new article on my personal page. Nothing needs to be deleted principly. Heptalogos (talk) 20:46, 24 January 2010 (UTC)

For those who don't give a s...

Glkanter: How about near the beginning of the article we state, For a simpler and less comprehensive summary see this version or some such? hydnjo (talk) 20:40, 24 January 2010 (UTC)

Interesting link from that page. A reliable online source that treats the problem simply. Monty Hall Problem Martin Hogbin (talk) 20:51, 24 January 2010 (UTC)

Is Selvin's Unconditional Solution 'False'

Morgan, without any acknowledgment of Selvin, including his 'equal goat door constraint', calls all unconditional solutions 'false'.

Selvin used a 9 row table (3 car locations * 3 contestant choices) to solve the puzzle he created. Glkanter (talk) 15:02, 25 January 2010 (UTC)

And, in his initial solution, he didn't make all his assumptions explicit. In his second letter, he said "The basis to my solution is that Monty Hall knows which box contains the keys and when he can open either of two boxes without exposing the keys, he chooses between them at random." [emphasis added] With these qualifications, his initial solution is fine. Without them, not so much. Let's review. Selvin poses the problem and solves it unconditionally. When questioned, he says what the critical assumptions are behind his solution. 16 years later, vos Savant publishes the problem. When questioned, she omits one of the critical assumptions. Morgan et al. (and others) follow through the consequences of omitting this assumption, concluding the probability of winning by switching (without this assumption) is not 2/3 but 1/(1+p) and, if we take p to be 1/2 (which is what the omitted assumption does) the answer is 2/3. If not, we don't exactly know the probability of winning by switching but since it's in the range of 1/2 to 1 it makes sense to switch anyway. The analysis that leads to the 2/3 solution does so regardless of what we assume p is. What Morgan et al. (and others) consider the "proper analysis" doesn't. Does that roughly match your understanding? -- Rick Block (talk) 02:54, 26 January 2010 (UTC)

You seem to be accepting that in the symmetrical case, a solution that does not distinguish between legal doors opened by the host is acceptable. I agree.

I also agree that vos Savant failed to make the assumption of symmetry explicit in her explanation, but as I have demonstrated, however you treat the problem the only logical and consistent assumption is that the host chooses a legal door randomly. Martin Hogbin (talk) 09:39, 26 January 2010 (UTC)

Let's review, Rick.

Selvin introduces his puzzle in The American Statistician

He solves it unconditionally, with the 9 line table of all possible outcomes

A few months later he clarifies that Monty Hall knows where the car is, and that Monty Hall chooses randomly when faced with 2 goats

15 years later vos Savant answers a reader letter and leaves some of Selvin's premises unstated

1 year after vos Savant, while never mentioning Selvin, Morgan claims all unconditional solutions are false. This is also published in The American Statistician, in a piece called 'Let's Make A Deal: The Player's Dilemma', not 'A Critique Of vos Savant's Parade Column'

Selvin's puzzle and solution leave no ambiguities as to the premises, or insistence that it 'must be conditional' or room for 'without certain assumption'. Glkanter (talk) 11:46, 26 January 2010 (UTC)

I never noticed this. I will add it to my Morgan criticism page. The 'certain assumptions' had already been answered, and in the same journal as the Morgan paper! It makes Morgan's attack on vos Savant even less justified. Unless you assume that Whitaker was asking about one of the many other game shows with one car and two goats ;-) Martin Hogbin (talk) 12:49, 26 January 2010 (UTC)

Selvin corresponded with Monty Hall himself. Morgan calls their paper, 'Let's Make A Deal: The Player's Dilemma'. There's only 1 game show that these two parties can be talking about. Although Morgan was apparently oblivious to Selvin's letters, and it shows. Glkanter (talk) 14:16, 26 January 2010 (UTC)

Yes, I was joking. Obviously Morgan are talking about the same show as Selvin. Odd that they did not spot his letters, but that would have spoiled their fun. Martin Hogbin (talk) 17:03, 26 January 2010 (UTC)

A few months later he clarifies that Monty Hall knows where the car is, and that Monty Hall chooses randomly when faced with 2 goats

He solves it as a conditional probability that does not depend in any way on specific door numbers and/or box letters. No mention is made of the possibility that the host might prefer one door/box over another, or that placement might be similarly biased. So it did not address "the conditional problem" as Morgan poses it, and would be lumped in with their "false solutions." JeffJor (talk) 17:39, 26 January 2010 (UTC)

15 years later vos Savant answers a reader letter and leaves some of Selvin's premises unstated

It states the same premises as Selvin's original letter did, and her answer was intentionally worded so as to imply the same premises Selvin clarified. Whether or not you agree she succeeded, she has stated that was her intention.

1 year after vos Savant, while never mentioning Selvin, Morgan claims all unconditional solutions are false. This is also published in The American Statistician, in a piece called 'Let's Make A Deal: The Player's Dilemma', not 'A Critique Of vos Savant's Parade Column'

And conspicuously omitted a dependency that is more important to their solution technique than q, the bias toward opening a specific door. Allowing for placement bias is just as necessary. So they did not address the full "conditional problem," and their solution is just as "false" as those they criticize. We can't use that, I know; but we can acknowledge it in how we use their work.

Later, the assertion that the MHP is "a conditional problem" that depends on door numbers is reiterated in three other works: Gillman, Grinstead & Snell, and Krauss & Wang. Interstingly, the first two do not justify this assertion, whcih clearly is interpreted by others differently. And the third says it is not true but, that people do use door numbers in the same way that Selvin did, which is not the "conditional problem" Morgan raised.

The problem continues to appear in popular literature, most publically in the NY Times, and most recently in Leonard Mlodinow's book A Drunkard's Walk: How Randomness Rules Our Lives. None of these treat it as "the conditional problem" as Morgan did; and in fact, their solutions are among what Morgan calls "false." And even when sources acknowledge Morgan's assertion (i.e., Rosenhaus), no discussion of why the problem is important or controversial mentions "the conditional problem" at all. Rosenhaus even says Morgan's issue "was hardly the point at issue between vos Savant and her angry letter-writers," and that "what vos Savant discussed was surely what was intended," thus allowing Morgan's assertion to be discounted, officially, in Wikipedia. JeffJor (talk) 17:39, 26 January 2010 (UTC)

It is not clear at all that Whitiker's letter - which uses doors, cars, goats, a proffered switch - derived from Selvin - which uses boxes with keys, empty boxes, and an offer to buy the box back for cash (the contestant raises the possibility of a switch). Selvin's game, despite his mispelling "Monte Hall," is true to the game show he mentions, while Whitaker's is not. Vos Savant has even disavowed any connection to it. (It's possible she edited that out.) Morgan treats it the same way, except in their title. Which may be as much a reference to the vernacular usage that led to naming the show "Let's Make A Deal", as to the game show itself. JeffJor (talk) 18:00, 26 January 2010 (UTC)

That seems a remarkable coincidence to me. Are you seriously suggesting that there is no connection between the two problems?Martin Hogbin (talk) 18:27, 26 January 2010 (UTC)

That's what I also raised in "relations between sources". Who thought it to be the same problem? The same question applies to the game theory source mentioned, which is about curtains. I think we only have Vos Savant and all sources explicitly linked to that. All other problems are related, similar, variants, whatever, including three prisoners. Heptalogos (talk) 20:35, 26 January 2010 (UTC)

Which means that it should be called the "Game Show Problem", which is indeed better. In fact, MHP is not correct at all. Heptalogos (talk) 20:44, 26 January 2010 (UTC)

Should I put a "citation needed" to "A well-known statement of the problem was published in Parade magazine"? Where's the Monty Hall connection? This seems to have been editor's consensus in the past. In the Dutch article it is even named after a Dutch show host! I don't even care so much about the name, but the quoted phrase above is not in line with "puzzle based on the American television game show". Wiki-rules-Rick, what's your reaction? Heptalogos (talk) 21:54, 28 January 2010 (UTC)

My statements:

• It's unknown whether the puzzle is based on a specific show.
• There are many puzzles which may be similar to many persons, but for the article we should let the sources decide which are similar and should therefore be mentioned.
• The main source that most sources use is Parade, after which the problem should be named. (Game Show Problem)

Heptalogos (talk) 22:32, 28 January 2010 (UTC)

Very interesting this all (See also Keller.), especially for the historic part. What nowadays is called the MHP, is mainly the K&W version, a well defined probability puzzle. And of course one studies weaker versions, like Morgan did. Nijdam (talk) 01:20, 29 January 2010 (UTC)

That's from a maths perspective, which is quite narrow. Heptalogos (talk) 08:44, 29 January 2010 (UTC)

There is a long tradition in mathematical and logical puzzles that they should be framed in a way that makes the solution simple and as intended by the originator. Even if there are minor loopholes on the puzzle statement it is usual to overlook these. Martin Hogbin (talk) 10:14, 29 January 2010 (UTC)

The history section of the article mentions Nalebuff's version, which appeared three years before the initial Parade column. Here's his version (reference is in the article):

The TV game show "Let's Make a Deal" provides Bayesian viewers with a chance to test their ability to form posteriors. Host Monty Hall asks contestants to choose the prize behind one of three curtains. Behind one curtain lies the grand prize; the other two curtains conceal only small gifts. Once the costumed contestant has made a choice, Monty Hall reveals what is behind one of the two curtains that was not chosen. Now, Monty must know what lies behind all three curtains, because never in the history of the show has he ever opened up an unchosen curtain to reveal the grand prize. Having been shown one of the lesser prizes, the contestant is offered a chance to switch curtains. If you were on stage, would you accept that offer and change your original choice?

As far as I know, no one has claimed that this is where Whitaker's version came from or that this version is a restatement of Selvin's, although the similarities in both directions are striking. Dozens of sources refer to the Parade version as the "Monty Hall problem". Since these earlier sources also mention Monty Hall and the problems are obviously isomorphic, saying these are different problems seems like a fairly absurd stance. -- Rick Block (talk) 15:17, 29 January 2010 (UTC)

What seems absurd to me is that you kept saying that it doesn't matter a bit what we all think is very obvious, because only reliable sources matter. And now you're talking about 'striking similarities' and 'obvious isomorphic problems'. These dozens of sources are very welcome to provide the missing link here, that's all I'm saying. Heptalogos (talk) 21:45, 29 January 2010 (UTC)

Wait a minute. What I've said all along is that what we all think about whether the problem must be addressed conditionally doesn't matter a bit because only reliable sources matter. What I'm saying here is that reliable sources call "vos Savant's problem" the Monty Hall problem which is the same thing Selvin calls his problem, and that Nalebuff uses the same host (and same problem set up). If this is not enough, I'm perfectly willing to provide more sources. How about Barbeau's literature survey (referenced in the article): "The problem, known variously as Marilyn's Problem, The Monty Hall Problem, and The Car-and-Goats Problem reads as follows ..." BTW, if you're interested in Barbeau's statement of the problem it's this:

A contestant in a game show is given a choice of three doors. Behind one is a car; ehind each of the other two, a goat. She selects Door A. However, before the door is opened, the host opens Door C and reveals a goat. He then asks the contestant: "Do you want to switch your choice to Door B?" Is it to the advantage of the contestant (who wants the car) to switch?

And, Barbeau lists as references for this problem Selvin's two letters, vos Savant's Parade columns, Morgan et al., Gillman, Falk, ... (about 40 altogether). -- Rick Block (talk) 02:34, 30 January 2010 (UTC)

I don't need more sources, I need any source! You did not yet mention one that connects all, but now you mentioned Barbeau. A few others would be welcome indeed, thank you. I am willing to draw the picture, to see how everything is connected. I think this should be part of the article in any way, because otherwise it's not clear what is the scope of the article anyway, and what the scope is based on. Heptalogos (talk) 15:28, 30 January 2010 (UTC)

Question: Will Rick answer the question directly asked by the section header: Is Selvin's Unconditional Solution 'False'? Glkanter (talk) 03:37, 30 January 2010 (UTC)

Would Morgan et al. consider this solution a "false" solution? I believe the answer is yes.

Does Rick believe this solution is "false"? We could talk about this on the/Arguments page if you'd like, however just like I believe for the purposes of this page it does not matter in the least what you think about this, I believe for the purposes of this page it doesn't matter in the least what I think about this. What editors believe (as opposed to what reliable sources say) is completely irrelevant as far as editing is concerned. Will you ever understand this? -- Rick Block (talk) 05:32, 30 January 2010 (UTC)

What editors believe is important, Rick. It says so right here

"Likewise, exceptional claims in Wikipedia require high-quality reliable sources, and, with clear editorial consensus, unreliable sources for exceptional claims may be rejected due to a lack of quality (see WP:REDFLAG)." Glkanter (talk) 13:31, 30 January 2010 (UTC)

OK, fine. I'll amend my statement. If something is said by multiple high-quality reliable sources, whether editors agree with it or not is irrelevant. Also see below. -- Rick Block (talk) 02:41, 31 January 2010 (UTC)

FYI

Hate to spoil the conversation but thought y'all might be interested in this. hydnjo (talk) 04:16, 30 January 2010 (UTC)

Yet another source supporting Morgan's POV

See Lucas, Rosenhouse, and Schepler Lucas, Rosenhouse, and Schepler (it's the same Rosenhouse as the recent book). Published in Math. Magazine, 82(5) 332-342 (Dec 2009) - another refereed journal (this is targeted at the undergraduate level, see http://www.maa.org/pubs/mm-guide.html). Some selected quotes:

• "The general principle here is that anything affecting Monty’s decision-making process is relevant to updating our probabilities after Monty opens his door."

• "This [the "high numbered Monty" variant - Monty picks opens the highest numbered door without revealing the car] shows that any proposed solution to the MHP failing to pay close attention to Monty’s selection procedure is incomplete."

They reference vos Savant and Selvin, but not Morgan et al. -- Rick Block (talk) 06:04, 30 January 2010 (UTC)

I saw nothing in the Lucas and Rosenhouse paper that remotely supports Morgan. In fact, their problem description uses no door #s. I couldn't find those quotes, or references to Selvin or vos Savant. Is that the right link? Glkanter (talk) 10:44, 30 January 2010 (UTC)

The paper mainly addresses a more general form of the problem including variants where the host is known to pick non-randomly. Obviously in that case 'any proposed solution to the MHP failing to pay close attention to Monty’s selection procedure is incomplete'. The paper has little to do with this article. Martin Hogbin (talk) 12:20, 30 January 2010 (UTC)

Sorry, it was the wrong link (I've corrected it above). Like Morgan et al., this paper addresses 'Classic Monty" conditionally, even though their problem statement does not use door numbers (!), and addresses variants (including both "High-Numbered Monty" and "Random Monty") to draw attention to the fact that "any proposed solution to the MHP failing to pay close attention to Monty’s selection procedure is incomplete". And, to be clear, the way I read it this quote pertains to ANY version of the MHP - not just variants where the host is known to pick non-randomly. Again, like Morgan et al., they use these variants to show the truth of this statement. -- Rick Block (talk) 14:48, 30 January 2010 (UTC)

Fine, but you cannot use a variant to prove that a solution relating to the standard problem only is wrong. That would be exactly like using a non-right-angled triangle to prove Pythagoras wrong. Martin Hogbin (talk) 16:52, 30 January 2010 (UTC)

Yes, but if Pythagoras's logic never used the fact that the triangle is a right triangle, his proof would presumably apply to all triangles not just right triangles. We've been here before and it's really not a very productive argument. Rather than argue about this, I'll pose a question to you (and Glkanter). What is your suggestion for how, in an NPOV manner, to include the POV expressed by multiple reliable sources that unconditional solutions are missing something? In particular, I'm talking about the following (bold added):

• Morgan et al.: Ms. vos Savant went on to defend her original claim with a false proof and also suggested a false simulation ...

• Morgan et al.: Solution F1: If, regardless of the host's action, the player's strategy is to never switch, she will obviously will the car 1/3 of the time. Hence, the probability that she wins if she does switch is 2/3. ... F1's beauty as a false solution is that it is a true statement! It just does not solve the problem at hand.

• Morgan et al.: Solution F2: The sample space is {AGG, GAG, GGA}, each point having probability 1/3, where the triple AGG, for instance, means the auto behind door 1, goat behind door 2, and goat behind door 3. The player choosing door 1 will win in two of these cases if she switches, hence the probability that she wins by switching is 2/3. ... That it [F2] is not a solution to the stated conditional problem is apparent in that the outcome GGA is not in the conditional sample space, since door 3 has been revealed as hiding a goat.

• Gillman: Marilyn's solution goes like this. The chance is 1/3 that the car is actually at #1, and in that case you lose when you switch. The chance is 2/3 that the car is either at #2 (in which case the host perforce opens #3) or at #3 (in which case he perforce opens #2)-and in these cases, the host's revelation of a goat shows you how to switch and win. This is an elegant proof, but it does not address the problem posed, in which the host has shown you a goat at #3.

• Grinstead and Snell: This very simple analysis [as a preselected strategy, staying wins with probability 1/3 while switching wins with probability 2/3], though correct, does not quite solve the problem that Craig posed. Craig asked for the conditional probability that you win if you switch, given that you have chosen door 1 and that Monty has chosen door 3. To solve this problem, we set up the problem before getting this information and then compute the conditional probability given this information.

• Lucas, Rosenhouse, and Schepler: This [the "high numbered Monty" variant - Monty opens the highest numbered door without revealing the car] shows that any proposed solution to the MHP failing to pay close attention to Monty’s selection procedure is incomplete.

• Rosenthal Monty Hall, Monty Fall, Monty Crawl: This solution [what he calls "Shaky Solution" which basically says your original chance of selecting the car is 1/3 and this doesn't change since you knew the host would open a door revealing a goat] is actually correct, but I consider it "shaky" because it fails for slight variants of the problem.

Let's see. So far we have 2 peer reviewed academic papers (Morgan et al., and Lucas, Rosenhouse and Schepler), a textbook (Grinstead and Snell), an article appearing in American Mathematical Monthly by a past president of the Mathematical Association of America (Gillman), and an article in Math Horizons (another publication of the Mathematical Association of America). All these sources are basically saying the same thing, which is that solutions that don't address the conditional probability of winning by switching are shaky, or incomplete, or don't quite address the problem, or are (most bluntly) false solutions. If these sources aren't sufficient for you, I could find more (but, really, 5 impeccably reliable sources ought to be enough for anyone).

To repeat, my question is how would you like, in an NPOV manner, to include the POV expressed by these sources that unconditional solutions are missing something? -- Rick Block (talk) 02:31, 31 January 2010 (UTC)

Rick, in the 15 months we have been discussing this, I find that my conclusions upon reading the identical material, Wikipedia policies or MHP sources, are almost never the same as the conclusions you reach. Nothing you cited above causes me to think those sources are capable of telling either Selvin or vos Savant (as proxy for Whitaker) what they 'really' meant, when they both made it so clear in so many ways. In Selvin's MHP, the contestant's original 1/3 can never change, and no outcomes get removed. Only with a different problem, with different premises, can Monty reveal the car, causing the contestant's 1/3 to go to 0, which he cannot do in Selvin's or vos Savant's problem. And the combining doors solution does show an open door #3 with a goat. I could go on extensively, but for what purpose? Glkanter (talk) 05:36, 31 January 2010 (UTC)

Let's take this one step at a time.

Do you agree the 5 sources I've cited above are reliable sources by Wikipedia's standards? If not, why not? -- Rick Block (talk) 05:56, 31 January 2010 (UTC)

Rather than repeat myself here my comprehensive criticism of the Morgan paper can be found here. The paper also has its own criticism published in the same journal in the form of a commentary by Seymann, and many editors here think that it is unreliable. It was, however, published in a peer-reviewed journal so we are obliged to have some reference to it in the article. It is essentially an academic diversion.

Gillman and G&S just seem to be repeating Morgan with adding anything. They should be included as references to the academic diversion that is described by Morgan.

Lucas, Rosenhouse, and Schepler show that a different problem requires a different solution. This should be in the variant section.

Rosenhouse says the simple solution is actually correct, but he considers it shaky because it fails for slight variants of the problem. This is a useful reference as it goes some way to settling some of the debate here. For the symmetrical problem an unconditional solution is correct. Once we add possible variations, such as the host is known to choose a legal door non-randomly, the unconditional solution fails. This might form the basis of a link section between two sections of the article. Martin Hogbin (talk) 12:48, 31 January 2010 (UTC)

Lucas, Rosenhouse, and Schepler start off by formulating what they consider to be the Monty Hall problem. They are teachers of elmentary university mathematics, and they are anxious to show the power of probability theory. The question they pose is not "would you switch?" but "what would you do to maximize your chances of winning the car?". They make the explicit assumption that when Monty has a choice "he chooses his door randomly". Later it becomes clear that by "randomly" they mean "completely randomly" or "at random, with equal probabilities". They later give the doors equal probabilities 1/3 to be hiding the car, with no reasoning why, except at some point for a side remark "since the doors are identical..." In my opinion these are defects to the paper. They have a tool, probability theory, and they (re)formulate the MHP in order to show off their tool. True, the paper contains a lot of useful references, history, and variants. It takes a long time to discover that they are writing about the conditional version of Monty Hall, even though initially they don't mention any door numbers. It's clear to me why they choose the conditional version: otherwise there wouldn't be much to write! They don't discuss why they have formulated the problem this way; they just make lazy and conventional choices, many of them only implicitly; so they add a whole lot more restrictions to the formulation of the problem as it became famous (in Parade magazine). I don't think the existence of their paper changes the fact that there are several points of view as to "what" the MHP is. Gill110951 (talk) 16:13, 31 January 2010 (UTC)

Agreed. Many here think the Monty Hall problem is a mathematical puzzle and as such should be formulated in a way that keeps it simple, just like the Three Prisoners Problem is. Martin Hogbin (talk) 16:37, 31 January 2010 (UTC)

I'm now reading Rosenhouse's book. It's nice, in fact much deeper than the Lucas, Rosenhouse, and Schepler (LRS) paper, but still has some strange features. On page 35, at the beginning of Chapter 2: "Classical Monty" he writes down what he calls the canonical version of the problem. As in LRS he doesn't name any doors. His first sentence is "you are shown three identical doors". A lot of pages later, after having discussed many informal solutions and talking a bit about elementary probability, he turns to presenting "the solution". At this point he numbers the doors. We are now 12 pages further on and he wants us to write down probabilities for all the possible initial configurations of (door chosen by, you, door opened by Monty, door hiding the car). For simplicity he focusses on those situations in which you have chosen door 1, so there are only four possibilities (1,2,1),(1,2,3),(1,3,2),(1,3,1). In two of those four, the car is behind door 1. At about this point point he says "it is built into the statement of the problem that the car is equally likely to be behind any of the three doors". Together with the assumption that when the quizmaster has a choice, he chooses with equal probabilities, this forces the four probabilities to be 1/6,1/3,1/3,1/6. Rosenhouse finally computes the probability that switching will give the car, given that you initially chose door 1. Thus he neither solves the unconditional problem, nor the conditional problem (he has not conditioned on the door opened by the quizmaster). Only in the penultimate chapter does he consider explicitly the unconditional problem. BTW he also acknowledges wikipedia for some of his alternative solutions. It seems to me that the authoritative literature shows that there are a lot of different opinions out there, and that it is the charm of the MHP that it allows so many formulations. BTW he has some rude things to say about the Morgan et al paper, in particular, their dogmatic style. Gill110951 (talk) 17:04, 31 January 2010 (UTC)

@Martin, where does Rosenhouse say the unconditional solution is correct, but shaky? In the situation of total symmetry (all permissable choices by all parties uniform random) it is clear that the conditional and unconditional probabilities must be the same. Prob(switching will give you the car)=Prob(switching will give you the car|Your initial choice)=Prob(switching will give you the car|Your initial choice and door opened by quizmaster; moreover (in the case of total symmetry) Prob(switching gives you car)=1-Prob(not switching gives you car)=1-1/3=2/3. (This symmetry argument was Boris Tsirelson's genial observation). There is absolutely nothing shaky about all this. Without symmetry, conditional probabilities can vary, and we can't know much about them; it is only in pedagogical articles (for students and teachers of elementary probability) by pedantic mathematicians who choose dogmatically to fill in the missing details in such a way that the pretty answer remains true, that we are told dogmatically that everything is "random" or "identical". My POV is that a sensible player chooses her initial door uniformly at random; on being asked whether or not to switch, she'll certainly switch. It doesn't matter which door she happened to choose first, and which door Monty happened to open. Her behaviour is the same. In her situation, it helps not one d*** s*** to answer the question "what is the probability the car is behind the other door, given my initial choice and the door opened by Monty?". The value of this probability depends on a load of things which she doesn't know. And her (or our) "ignorance" does not mean that suddenly various probabilities get fixed at 1/3, 1/2 etc etc. No: her ignorance means that she cannot use her conditional probability (since she has no way of knowing it) as a guide to her choice. Gill110951 (talk) 18:43, 31 January 2010 (UTC)

Martin meant Rosenthal, not Rosenhouse. A reference above he presumably accidentally deleted that I just restored. -- Rick Block (talk) 19:00, 31 January 2010 (UTC)

You are quite right Rick. Sorry for the accidental deletion. Martin Hogbin (talk) 20:02, 31 January 2010 (UTC)

Rick, I see none of those sources state that an 'unconditional solution' cannot solve the problem. That is your interpretation. They state that they don't solve the exact problem, e.g. because they don't use the right condition, or don't make the right assumptions. That's what they all say. Heptalogos (talk) 21:36, 31 January 2010 (UTC)

Right, they state unconditional solutions don't solve the exact problem. Glkanter has brought up the version as of the last FARC (claiming it is full of POV and has been improved since then). Re-reading this I'm actually not seeing the problem. In this version there are two paragraphs between an initial unconditional solution and a conditional solution that say:

The reasoning above applies to all players at the start of the game without regard to which door the host opens, specifically before the host opens a particular door and gives the player the option to switch doors (Morgan et al. 1991). This means if a large number of players randomly choose whether to stay or switch, then approximately 1/3 of those choosing to stay with the initial selection and 2/3 of those choosing to switch would win the car. This result has been verified experimentally using computer and other simulation techniques (see Simulation below).

A subtly different question is which strategy is best for an individual player after being shown a particular open door. Answering this question requires determining the conditional probability of winning by switching, given which door the host opens. This probability may differ from the overall probability of winning depending on the exact formulation of the problem (see Sources of confusion, below).

This structure is very close to how Grinstead and Snell handle this issue (they present an unconditional solution first and then say how this addresses a slightly different question). I assume these are the paragraphs that Glkanter most objects to. Is there any way we can work on these two paragraphs to satisfy everyone's goal of making the article NPOV? -- Rick Block (talk) 22:09, 31 January 2010 (UTC)

Well, I'm not sure what you mean by 'unconditional solution', but it's good to see your pragmatism by bringing in some article text. I think the first paragraph is POV. The Morgan link probably refers to solution F3, which they call wrong because it does not use the number of the door. However, if the equal goat assumption would have been made, I think Morgan would not have called this wrong. And in the article, this assumption is actually made. That's why it is IMO not correct, but a specific interpretation of Morgan. Heptalogos (talk) 22:34, 31 January 2010 (UTC)

Proposal to complete the popular solution

I only just realized that the "popular solution" just needs the one word "symmetry" added in order to turn it into a mathematically complete and rigorous solution of what we call here the conditional problem, under the further assumptions that many people (but not me) consider the canonical extra conditions. I wrote:

In order to convert this popular story into a mathematically rigorous solution, one has to argue why the probability that the car is behind door 1 does not change on opening door 2 or 3. This can be answered by an appeal to symmetry: under the complete assumptions made above, nothing is changed in the problem if we renumber the doors arbitrarily, and in particular, if we switch numbers 2 and 3. Therefore, the conditional probability that the car is behind door 1, given the player chose 1 and Monty opened 2, is the same as the conditional probability that the car is behind door 1, given the player chose 1 and Monty opened 3. The average of these two (equal) probabilities is 1/3, hence each of them separately is 1/3, too.

Is the missing word "symmetry" the reason that Rosenthal found the unconditional argument "shaky"? If you just say "opening door 3 doesn't change the chance the car is behind door 1" you are certainly being shaky. *Why* doesn't it change the probability? Intuition can so easily be wrong in probability puzzles! So let's make this step in the argument rigorous, rock-solid. Symmetry does that for you.

Mathematicians love using symmetry, since they love to make results obvious, they hate calculations; they are looking for beauty (two important sources of beauty are chance, and symmetry). I notice that non-mathematicians are often not entirely convinced by the symmetry argument. They feel tricked, suspicious. Of course, they are not used to it. Using symmetry is using a meta-mathematical argument, ie, a mathematical argument about mathematical arguments. And such arguing can be tricky, think of Gödel! Gill110951 (talk) 10:31, 1 February 2010 (UTC)

Looks good to me. I have been arguing here for years that symmetry is a valid reason to accept the unconditional solution, using much the same argument as you. It is also worth noting why Rosenthal found the unconditional solution shaky, this was only because it did not apply to slight variants (such as host chooses non-randomly). Nobody finds Pythatgoras' theorem shaky because it applies only to right angled triangles.

There is also the argument that random information is no information. That is a standard point of information theory. Again, this argument needs to be carefully made. Martin Hogbin (talk) 13:17, 1 February 2010 (UTC)

The issue is what source would you use for this? -- Rick Block (talk) 13:34, 1 February 2010 (UTC)

I won't object the symmetry argument on the ground of lacking source. And to Martin: Gill solves the CONDITIONAL problem! Nijdam (talk) 13:58, 1 February 2010 (UTC)

I am not the one who is hung up on the conditional/unconditional issue. If you just want to use the word 'conditional' that is fine with me, so long as you do not try to complicate the problem and solution by insisting that it matters which door the host opens, or say that the solution does not address the exact problem 'as asked'. Martin Hogbin (talk) 23:17, 1 February 2010 (UTC)

I am not the one who is hung up on the use of the word conditional. Never said so. But I do insist that the simple solution, and equivalently the combined doors solution, is not addressing the exact problem 'as asked'. And I want the article to make this clear to the reader. Nijdam (talk) 23:43, 1 February 2010 (UTC)

Is that not what this thread is all about? If the host chooses randomly then the simple solution does exactly answer the question as asked (after the host has opened a door). The reason - symmetry. Martin Hogbin (talk) 23:49, 1 February 2010 (UTC)

If i understand that correctly defining symmetry in such a way basically forces the host to pick between 2 goat doors at random with p=1/2 (it seems to be an equivalent assumption). I don't mind to modify the article in such way, but i agree with Rick to the regard that we would need a source for that.--Kmhkmh (talk) 18:00, 1 February 2010 (UTC)

@Kmhkmh. It's the other way round. Suppose we want to solve the conditional problem. IF the host chooses *uniformly* at random when he has a choice, and IF the car is initially equally likely behind every door, then all the conditional probabilities Prob(car is behind door x| You chose x, Monty Hall opened y), where x,y are any two different door numbers, are the same. In particular,

Prob(car is behind 1| you chose 1, MH opened 3)

=Prob(car is behind 1| you chose 1, MH opened 2).

We have been told already that

Prob(car is behind 1|you chose 1)=1/3.

But by the law of total probability,

1/3 = Prob(car is behind 1|you chose 1) =

Prob(MH opened 2|you chose 1) x Prob(car is behind 1| you chose 1, MH opened 2)

+ Prob(MH opened 3|you chose 1) x Prob(car is behind 1| you chose 1, MH opened 3)

Denoting the unknown probability by p, we have

1/3 = Prob(MH opened 2|you chose 1) x p + Prob(MH opened 3|you chose 1) x p = p

Gill110951 (talk) 09:35, 2 February 2010 (UTC)

Would Selvin's letters satisfy the requirement? Where he says this:

"The basis to my solution is that Monty Hall knows which box contains the keys and when he can open either of two boxes without exposing the keys, he chooses between them at random." Glkanter (talk) 19:08, 1 February 2010 (UTC)

Strictly speaking "choosing at random" is not enough, since that does not define the distribution (i.e. you need p=1/2 for each goat door). However "choosing at random" is often meant to implicitly assume a uniform distribution, which is how i would read Selvin's description. Conclusion : If you are picky, there some ambiguity in Selvin's description as well.--Kmhkmh (talk) 20:39, 1 February 2010 (UTC)

That is getting overly picky. I am sure Selvin meant that the box choice was 'uniform at random'. Martin Hogbin (talk) 23:09, 1 February 2010 (UTC)

How about the following. I'm not exactly happy with the wording and somebody will have to find appropriate references, but I think this more or less captures what folks are saying.

<unconditional solution here>

The reasoning above directly addresses the average probability across all possible combinations of initial player choice and door the host opens (some reference). This means if a large number of players randomly choose whether to stay or switch, then approximately 1/3 of those choosing to stay with the initial selection and 2/3 of those choosing to switch would win the car. This result has been verified experimentally using computer and other simulation techniques (see Simulation below).

A subtly different question is which strategy is best in a specific case such as that of a player who has picked Door 1 and has then seen the host open Door 3. 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). Because of the symmetry of the problem, the average probability as determined above applies to any specific case as well (this must have a reference). The probability of winning by switching in a specific case can also be determined as a conditional probability, given which doors the player picks and the host opens. Although this is the same as the average probability of winning by switching for the unambiguous problem statement as presented above, in some variations of the problem the conditional probability and average probability may be different, see Variants below.

<symmetric conditional solution here>

The idea would be to have ONE solution section, sort of like the version as of the last FARC [1] but with these two paragraphs between the unconditional and conditional solutions. -- Rick Block (talk) 04:32, 2 February 2010 (UTC)

The simple symmetrical solution does address the question 'which strategy is best in a specific case such as that of a player who has picked Door 1 and has then seen the host open Door 3?'. If the host action is random, it does not matter that the host has opened a specific door, because we know that the door opened makes no difference, by reason of symmetry. Can any one else explain this better? Martin Hogbin (talk) 09:15, 2 February 2010 (UTC)

I have tried Martin, above Gill110951 (talk) 09:35, 2 February 2010 (UTC)

On the other hand if, between the two solutions we had something like, If it is considered that the host might not choose between the two available doors randomly, the door actually opened by the host may give information which changes the probability that the player has originally chosen the car and thus it becomes important 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) I would be happy. Martin Hogbin (talk) 09:37, 2 February 2010 (UTC)

Sources present three different kinds of solutions

Does anyone disagree that most solutions presented in sources are one of the following three types:

1) Completely unconditional, i.e chance of initially picking the car is 1/3 and a goat 2/3, and if you switch these flip.

2) Assuming the player has picked (for example) door 1, i.e. vos Savant's table:

Door 1	Door 2	Door 3	result if switching
Car	Goat	Goat	Goat
Goat	Car	Goat	Car
Goat	Goat	Car	Car

3) Assuming the player has picked (for example) door 1, conditional given the host has opened (for example) door 3, e.g. any of the "conditionalists". These end up as (1/3) / (1/3 + 1/6) = 2/3.

There is clearly conflict among sources about these solutions and clearly conflict among editors about these solutions, so how about a single solution section somewhat like this:

---

Solution

Different sources present solutions to the problem that directly address slightly different mathematical questions.

The average probability of winning by switching

This is the simplest kind of solution. The player initially has a 1/3 chance of picking the car. The host always opens a door revealing a goat, so if the player ignores what the host does and doesn't switch the player has a 1/3 chance of winning the car. Similarly, the player has a 2/3 chance of initially picking a goat and if the player switches after the host has revealed the other goat the player has a 2/3 chance of winning the car. (some appropriate reference, perhaps Grinstead and Snell)

What this solution is saying is that if 900 contestants all switch, regardless of which door they initially pick and which door the host opens about 600 would win the car.

The probability of winning by switching given the player picks Door 1

If the player has picked, say, Door 1, there are three equally likely cases.

Door 1	Door 2	Door 3	result if switching
Car	Goat	Goat	Goat
Goat	Car	Goat	Car
Goat	Goat	Car	Car

A player who switches ends up with a goat in only one of these cases but ends up with the car in two, so the probability of winning the car by switching is 2/3. (some appropriate reference, perhaps vos Savant)

What this solution is saying is that if 900 contestants are on the show and roughly 1/3 pick Door 1 and they all switch, of these 300 players about 200 would win the car.

The probability of winning by switching given the player picks Door 1 and the host opens Door 3

Tree showing the probability of every possible outcome if the player initially picks Door 1

This is a more complicated type of solution involving conditional probability. The difference between this approach and the previous one can 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 probabilities in all cases where the player has initially picked Door 1 can be determined by referring to the figure below or to an equivalent decision tree as shown to the right (Chun 1991; Grinstead and Snell 2006:137-138 presents an expanded tree showing all initial player picks). Given the player has picked Door 1, the player has a 1/3 chance of having selected the car. Referring to either the figure or the tree, if the host then opens Door 3, switching wins with probability 1/3 if the car is behind Door 2 but loses only with probability 1/6 if the car is behind Door 1. The sum of these probabilities is 1/2, meaning the host opens Door 3 only 1/2 of the time. The conditional probability of winning by switching for players who pick Door 1 and see the host open Door 3 is computed by dividing the total probability (1/3) by the probability of the case of interest (host opens Door 3), therefore this probability is (1/3)/(1/2)=2/3.

Although this is the same answer as the simpler solutions for the unambiguous problem statement as presented above, in some variations of the problem the conditional probability may differ from the average probability and the probability given only that the player initially picks Door 1, see Variants below. Some proponents of solutions using conditional probability consider the simpler solutions to be incomplete, since the simpler solutions do not explicitly use the constraint in the problem statement that the host must choose which door to open randomly if both hide goats (multiple references, e.g. Morgan et al., Gillman, ...).

What this type of solution is saying is that if 900 contestants are on the show and roughly 1/3 pick Door 1, of these 300 players about 150 will see the host open Door 3. If they all switch, about 100 would win the car.

A formal proof that the conditional probability of winning by switching is 2/3 is presented below, see Bayesian analysis.

Car hidden behind Door 3	Car hidden behind Door 1		Car hidden behind Door 2
Player initially picks Door 1
Host must open Door 2	Host randomly opens either goat door		Host must open Door 3
Probability 1/3	Probability 1/6	Probability 1/6	Probability 1/3
Switching wins	Switching loses	Switching loses	Switching wins
If the host has opened Door 3, these cases have not happened		If the host has opened Door 3, switching wins twice as often as staying

---

I'm not overly attached to any of the specific wording used, but I think presenting these as three different types of solutions and including with the last one the essence of the controversy is an NPOV approach. -- Rick Block (talk) 15:41, 2 February 2010 (UTC)