Talk:Monty Hall problem

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many people say switching is better, but are still wrong

many people will say switching is better because your odds are 1/2. This is wrong of course, because your odds are 2/3. I think this particular confusion needs to be made explicit in the article —Preceding unsigned comment added by 76.126.238.69 (talk) 23:41, 9 February 2011 (UTC)

How about the information angle ?

How about first establishing what actual information is present after the host opens the door instead of playing with probebilities that become irrelevant once the door gets opened? Call it a methodological problem if you like. Probabilities of guesses can be mapped to presence and absence of information. The difference is that information based reasoning points clearly what's relevant and what's irellevant. Relevant is that player had new info which gives him 1/2 chance if he tosses a coin and chooses again at random. His first choice is irellevant since he didn't get any info about that choice, he only got infor that new choice with better ods is possible. If we believe that a random choice under uncertainty gives the best odds, then he will improve his chances only if he makes a new random choice, not a forced one.

Information-wise the 2 cases from initial state are merged when new information is provided - they don't exist anymore in the new state, they are indistinguishable. Same applies to the case with 1 mil doors. If player makes forced choice he remains in the prior state with much lower odds - he's not using new information. If he wants to use new information then he has to toss the coin again in order to realize the new state.

Another way to look at it is to realize that in an assembly only a random choice can select new configuration and all forced, non-random transitions are equivalent and confined to the same configuration.

His actual odds still improve from 1/3 to 1/2. ZeeXy (talk) 13:21, 3 November 2010 (UTC)

(Attempt to) Pick wrong and increase the chances of switching right, or (Attempt to) pick right and not know until it's too late. Metaphysically speaking, I'd rather wing it. 70.15.11.44 (talk) 05:28, 4 November 2010 (UTC)

When wondering where the car is, you shouldn't just use the hard information which you have in front of you, but also the likelihood that that information came to you under the different scenarios which concern you. You chose Door 1. The host is twice as likely to open Door 3 if the car is behind Door 2 than if the car is behind Door. 1. When the game is repeated many times, the car will be behind Door 2 twice as often as it is behind Door 1, within those occasions that you chose Door 1 and the host opened Door 3.

Forget about probability, forget about information. This is about very simple arithmetic. Richard Gill (talk) 07:20, 5 November 2010 (UTC)

Wrong. If a host opens #3 you don't know if that's because you missed or guessed #1 and no imagination can help you. If you automatically switch to #2 that's equivalent to picking #2 in the first place -- he'd still open #3. Now what? :-) He could even let you keep switching till you turn blue. That's why he can be "generous" - coz you are at 50% ignorance and there's nothing that can help you. You could have started with 100 doors and he could have let you switch every time he closed a door and you'd still be in exactly the same state. That's why I said that information-wise prior states are merged as a result of the new info - they are indistinguishable, there is no observable difference. Your first 98 choices are irellevant for the new state -- you are still in a state of 50% ignorance. The only thing you can do to improve your chances in any 50% ignorance state is to toss a fair coin.

Wrong, wrong. We are told as part of the problem statement that the quizmaster knows where the car is hidden, that he will always open a door revealing a goat, and that he will always offer us the opportunity to switch to the other still closed door. Richard Gill (talk) 13:22, 6 November 2010 (UTC)

Here's example with a real statistical ansamble

Say you were choosing among 3 presidential candidates and one got killed. Does that automatically make the one you haven't picked a 2/3 winner? :-) Then it has to apply to all other people who picked one of these 2. All these people are now real statistical ansamble - genuine massive sample of random choices with equal probabilities and you can clearly see that your particular initial choice is irellevant. Assume totally split election, every candidate having 1/3 before one got killed and assume no one really cares - everyone just wants to vote for the winner since then they get a coin if they voted for a winner. Suppose they all follow your automatic switch tactics and the 3rd voter set gets split equally. No one wins. What's the best chance of voting for the winner you have? 50% What the sole way to insure that someone does win and thereby 50% of you achieve the goal -- that no one does anything authometically but everyone tosses a coin. Why? Because fair coin toss is never 100% fair with finite number of tosses. Only the act of everyone tossing insures the winner. This is probably the best illustration how irellevant and blocking automatic switching is and how you do need to toss in order to actually realize the chance presented by new information.

This is all very different from a situation in which there would be some underlying cause and your sampling is really just measuring it since then you would expect sampling to converge with a very high probability. Without underlying cause it takes infinite number of samples to realize your statistics and that's strictly historical and completely irellevant for a particular singular trial. That's the part people easily forget when they start deluding thelselves with abstract statistics -- it's irellevant for a particualr sample unless there's underlying cause which will ensure rapid convergence. That's the sole thing that makes sampling worth anything -- if there is underlying cause there will be fast convergence. Have to remind you that theoretical limit for a big number rule to apply is infinite number of samples.

Go tossing coins and see how long runs of equal values you are going to get and how huge deviations from imagined 1/2 you are going to get. ZeeXy (talk) 12:44, 6 November 2010 (UTC)

Maybe it would be best if the participants in this discussion only made reference to reliably published sources, as a method of discussing changes to the articles. I'm sure there are various more appropriate forums elsewhere on the internet that welcome spirited debate on the mathematics and logics of the MHP. Glkanter (talk) 12:52, 6 November 2010 (UTC)

This another typical response to the MHP article. Everyone wants to know why/how/if the answer is 2/3 rather than 1/2. Martin Hogbin (talk) 10:36, 6 November 2010 (UTC)

@ZeeXy - if you'd like to discuss the mathematics behind the problem, I'd suggest we move this thread to the /Arguments subpage. If you're suggesting a change to the article, please say what change and on what source (or sources) you'd base that change. -- Rick Block (talk) 15:16, 6 November 2010 (UTC)

Rick, there shouldn't *be* an 'arguments' page. Arguing the math/logic of the MHP is no more appropriate for a Wikipedia article talk page than discussing the greatness of your favorite musical performer with like-minded fans. The 'arguments' page should be deleted, rather than encouraged. Talk pages are for discussing editing Wikipedia articles. Glkanter (talk) 15:31, 6 November 2010 (UTC)

The point of the /Arguments page is to have a place for these sorts of discussions, which are not directly related to editing the article, to be held. It is more or less like the Wikipedia:Reference desk - but with a specific focus on the MHP. There's one for various other articles on controversial topics, like 0.999.... You are absolutely correct that THIS page is for discussing editing the article. If you strongly feel the need to see a community consensus about whether the Arguments page should be deleted, please open a discussion at Wikipedia:Miscellany for deletion. -- Rick Block (talk) 17:23, 6 November 2010 (UTC)

Thanks for the suggestion. I may do that. If its a Reference desk item, well, then it belongs at the Reference desk. Or, I could just take the page off my Watchlist... Glkanter (talk) 21:55, 6 November 2010 (UTC)

Recent overhaul and state of the mediation

First of all thanks to all who put effort in the recent overhaul, which from my perspective works well overall.

I minor nitpicking I'd have though is the (incomplete) quotation of Behrends in the sources of confusion section. It should be mentioned while Behrends considers both answers as correct he does consider them as 2 slightly different problems or questions at least.

Another I'd like to know is whether the conflicting parties in the mediation are happy with the current version (or at least can live with it) or whether we still have (major) disagreements and an potenial editing conflict down the line. --Kmhkmh (talk) 15:46, 14 November 2010 (UTC)

I think I am reasonably happy with the article as it is now. I did not realise that the article was being actively edited during the mediation so I am assuming any edits made during mediation to be non-contentious ones.

As you will see on the mediation page, I have suggested that we start discussion based on the article as it is now, rather than rewriting large chunks of it from scratch. I guess you support this proposal. Martin Hogbin (talk) 10:41, 9 December 2010 (UTC)

Just out of curiosity, what was the mediation over? The MHP is a stats problem, which does not strike me as something prone to violent arguments. --Ludwigs2 17:39, 8 February 2011 (UTC)

Best look through the last couple of years' talk pages. I do not think anyone wants a re-run. Martin Hogbin (talk) 22:02, 8 February 2011 (UTC)

Without looking through the archives, why is there a long list of references, but no in-line citations? Cla68 (talk) 00:07, 10 February 2011 (UTC)

The article uses Harvard style referencing. There are plenty of inline references. -- Rick Block (talk) 00:37, 10 February 2011 (UTC)

I'm pretty much happy with the present article. But I did some more OR in the direction of creating a synthesis between simple and conditional solutions, see [1]. Richard Gill (talk) 22:50, 10 February 2011 (UTC)

Proposal to add some alternative conditional solutions

I would like to see some more mathematical solutions to the conditional problem, just as there are various informal solutions to the unconditional problem. I think they all give additional insight into MHP. There exist at least three solutions which follow a simple chain of logical reasoning, and which can be converted step by step into equivalent mathematical formalism (this is a useful exercise for the beginning student of the formal probability calculus who has to learn how to connect the formalism with ordinary logical reasoning and insight into the structure of the problem), but which avoid calculations or formula manipulation. These are: simple plus symmetry, Bayes' rule, and symmetry plus simple.

Simple plus symmetry: by symmetry the probability that the car is behind door 1 cannot depend on whether the host opened door 2 or door 3. The unconditional probability was 1/3. Therefore the two conditional probabilities are equal to 1/3 too. Reference: Bell (1982).

Bayes rule: the odds that the car is behind door 1 (the door chosen by the player) are initially 2 to 1 against. Whether or not the car is behind door 1, the chance that the host opens door 3 is the same, 50%. (In the one case because if the car is behind door 1, the host is equally likely to open either other door, in the other case, because if the car is not behind door 1, it is equally likely behind either other door, and the host's choice is forced.)

Symmetry plus simple. Pretend for a moment that the player's choice is also completely random. After the host's action, we can refer to the doors as: door chosen by player X, door opened by host H, door remaining closed (to which the player may switch) Y. From the simple solution we know that the door hiding the car, C, is either door X or door Y, with probabilities 1/3 and 2/3 respectively. By symmetry, the triple of door numbers (X,H,Y) is a completely random permuation of the numbers (1,2,3), and C either equals X or Y, with probabilities 1/3 and 2/3, independently of which of the six permutations is the permutation (X,H,Y). This tells us that the specific door numbers are irrelevant to the player who wants to maximize his chance of getting the car. The actual numbers are completely independent of the relationship of C to X,H,Y.

Nothing is changed by fixing the value of X, X=1 say. Now there are just two permutations possible, (1,2,3) and (1,3,2). By symmetry they are equally likely, and this is independent of whether or not C=X.

Each of these alternative proofs has pedagogical value for students of probability and statistics since they use extremely valuable tools. I think they each give further insight into "why you should switch". Each of the proofs is intuitive, you don't need a formal mathematical training to appreciate the ideas used in them. All the proofs explain why the ordinary lay person is perplexed by the argument that the simple solutions are "wrong" and that you have to learn Bayes's theorem and formal probability calculus to solve MHP properly - because each of the proofs make clear in a different way why the ordinary lay person is completely right not to be too bothered about the specific door numbers. Each of the proofs ties in with Vos Savant's wording "say, Door 1", and "say, Door 3", since we see that the specific door numbers are indeed irrelevant to the chance that switching will win and to the decision process of the player.

Reliable sources: the various uses of symmetry go back to discussants of the Morgan et al. (1981) paper, especially Bell (1982). The use of Bayes' rule is promoted by Jeff Rosenthal (2006?) (who by the way is a prominent mathematician and probabilist, as well as a prominent popularizer of probability and statistics whose "popular" writings are appreciated both by lay persons and by experts). See also a prepublication by me, [2], which is based entirely on what I learnt from fellow editors on the MHP page. Richard Gill (talk) 09:16, 11 February 2011 (UTC)

I think it is generally useful to understanding have several ways of looking at a problem covering all levels of understanding and interest. The only thing I would want to keep in mind is that the MHP is essentially a simple problem that most people get wrong so we should start with simple and convincing solutions> After that, something for the experts. Martin Hogbin (talk) 11:06, 11 February 2011 (UTC)

Those are all great ideas, Richard. Too bad the Conditional solution section is so grossly polluted with variants, hypotheticals, OR, NPOV and UNDUE WEIGHT violations, and other assorted crap intended to diminish the simple solutions, but which serves only to confuse the reader.

By the way, is that great authority, Jeff Rosenthal, the same one who says of a simple solution?:

"This solution is actually correct, but I consider it "shaky" because it fails for slight variants of the problem. For example, consider the following:"

"Monty Fall Problem: In this variant, once you have selected one of the three doors, the host slips on a banana peel and accidentally pushes open another door, which just happens not to contain the car. Now what are the probabilities that you will win the car if you stick with your original selection, versus if you switch to the remaining door?"

Maybe you could clarify for me what the English term, 'actually correct' means and what bearing 'it fails for slight variants of the problem' has as to the 'actually correct'-ness of that simple solution as a technique for solving the Wikipedia article's subject, which is the symmetrical MHP? Glkanter (talk) 12:00, 11 February 2011 (UTC)

@Martin: yes indeed. My proposal would be to place these solutions in place of the present formal mathematical proof via Bayes' theorem. That proof can be replaced by a reference to the article Bayes theorem where it already figures as an example.

@Glkanter. Yes, Jef's words are incomprehensible. What is wrong with something which is right? What is the relevance that "it" fails for a different problem? I think Jef was using the word "solution" in two different sences without realizing it. The "answer" (2/3) is correct but the "argument" is not. Because the same argument gives the same answer, 2/3, for a different problem Monty Crawl, where 2/3 is the wrong answer. And the argument is not even applicable to Monty Fall. But why don't you ask him yourself? So I guess this was a momentary lapse. But I don't have e.s.p., so I can't tell what was going on in his mind. I can only guess. Richard Gill (talk) 19:05, 11 February 2011 (UTC)

*I* don't have any problem comprehending his words. Nor do I find any 'lapse'. The only reason I bring it up, again, is because you have repeatedly agreed with Rick Block and others that those words constitute a 'criticism' of all simple solutions. That's an unsupportable conclusion by you people. His only point is to show why his 'discovery', or whatever, has utility in some other applications, which the simple solutions, of course, would fail in. This (these?) other method(s) is the whole point of his paper. It is in no way a 'criticism' of simple solutions as a solution to the symmetrical MHP at all. Hopefully nonsense like this will be exposed in arbitration, don't you think? And wouldn't it be nice if the Conditional solution section, which also talks about variants for 90% of the time was cleaned up to make room for your suggestion? Glkanter (talk) 19:18, 11 February 2011 (UTC)

If you are expecting arbitration to settle the longstanding content disputes on this article, you will probably be disappointed. Content disputes are outside ArbCom's remit, and while sometimes they try to help resolve a content dispute by giving disruptive editors a "time out," as a rule they won't address the content dispute directly. Woonpton (talk) 05:40, 12 February 2011 (UTC)

So the whole arbitration is about Rick Block's complaint about my conduct? Glkanter (talk) 06:58, 12 February 2011 (UTC)

@Glkanter: I think @Woonpton is right, and also @David Tombe knows about these things through personal experience and probably has good advice on how to survive. @Rick Block has complained about your behaviour, you can see his complaint somewhere. Richard Gill (talk) 12:29, 12 February 2011 (UTC)

@ Richard. I note that implicitly you now take the "conditional formulation" (decide after the door has been opened) as the MHP to be presented. Your "simple plus symmetry" solution, better is called "conditional using symmetry" solution, as the characteristic thing of a simple solution just is not considering any conditional probabilities. This solution and the one using Bayes' have always been considered correct solutions. Let us not discuss your third option, as IMO it is not suited for Wikikpedia. As for the term "simple", let's keep it for the simple solution, the one not being a correct solution to the conditional formulation of the MHP. The article should mention it with the criticism. Nijdam (talk) 11:15, 12 February 2011 (UTC)

Again incorrect? Not paying regard to "which door" has been opened? Please avoid that misleading, fallacious and nebulizing strategem. Based on the question asked, the identity of the door opened cannot give any "secret additional info" on the actual location of the car, unless you "assume" something totally unknown. So "before" and "after" are identical. Anyone is free to "assume" what he likes, but in that case he preferably should clearly disclose his assumption. Gerhardvalentin (talk) 12:50, 12 February 2011 (UTC)

And, just to acknowledge but also to solve that world famous conflict on Wikipedia: Have a look there:

"From the mathematical point of view, one can completely solve MHP by first using symmetry to show that the relationship between the doors when identified only by their role - door chosen by player, door hiding car, door opened by host - is statistically independent of the numbering of the doors (at least, this is true when the player's choice is also random). This implies that the player's decision whether to stay or switch might just as well be made in advance,
ignoring the door numbers in question. They give him no further information."

". . . They give him NO further information."  But every now and then here unchangibly is repeated, and repeated, and repeated, and repeated, and repeated: "Most people will be confused by the unconditional formulation of the simple solution",
or: "the final answer about the required probability depends on the way the host acts",
or: "the simple solution is not a solution to the MHP, and that's the crux of the problem",
or:"There is a difference between the probability before the host opens a door and the probability thereafter",
and: "Anyone with basic understanding of probability theory should know the difference",
or: "It's some reasoning, which fails in the simple solution.",
or: "By watching the shows, and keeping track of how many times ... - an outsider sees the result of the host's bias. It is this history..."
or: "these tallies may show different odds of winning by switching (depending on which door the host opens).",
or: "the individual chance (by door) may be different (depending on the host bias).",
or: "the simple solution, the one not being a correct solution . . . The article should mention it with the criticism",
and so on, and so on, and so on, and so on, and so on, and so on, and so on, for more that two years now. – Most gratifying for all observers. :::::Gerhardvalentin (talk) 21:22, 12 February 2011 (UTC)

@Gerhard, I am trying to solve Vos Savant's question and I am using Laplace's definition of probability (subjectivist-Bayesian) based on the primitive concept of "equally likely". I do this because this is how the man in the street approaches probability and because MHP is a popular brain-teaser. "Equally likely" means equally likely from your (subjective) personal point of view. All you know is what Vos Savant's tells you. You have not watched the show a hundred times in the past! "Equally likely" is not something which is verified by statistics from past observations. It refers solely to your personal expectations of one single play of the game. The definition of "the probability of X is 2/3" is no more and no less than "there are three equally likely outcomes and for two of them X is true, for the other one it isn't". No more and no less. The car is behind the initially chosen door with probability 1/3 given your initial choice and the choice of the door opened by the host means no more and no less than the situation that your initial choice was door 1, and the door opened by the host was door 3, can be decomposed into three equally likely situations, in two of which the car is behind door 1 and in one of which it isn't. Because we know nothing, to begin with the car is equally likely behind each door. For us it is equally likely, when we have chosen door 1 and the car is behind door 1 too, that the host opens door 2 or door 3. It is nothing about historically observed repetitions of knowledge about how the host's brain works. It is about our lack of any information to believe any more strongly that door 2 or 3 would be opened.

With this approach, we might as well imagine the player choosing his initial door at random, and afterwards consider the special case that it was Door 1. Since from our initial information it makes no difference at all how we initially choose our door. Now we notice that because of our total lack of any information, all six possible values of (door chosen, door opened, door left closed) are equally likely: (1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2), or (3,2,1). Whether or not door chosen=door hiding car cannot, by symmetry, depend on what is the numbering of our triple f (door chosen, door opened, door left closed) . Obviously the probability is 1/3 that the initially chosen door hits the car. Therefore the probability is also 1/3 that the initially chosen door hits the car, given that (door chosen, door opened, door left closed)=(1,3,2). The total symmetry tells us that the decision to switch or not can be taken independently of the actual numbers of the doors chosen, opened, left closed in the case at hand.

I think that the concept of independence is more fundamental than the concept of conditional probability. Conditional probability is a derived concept. So a solution based on independence and symmetry is better for a broad public, than a proof based on conditional probability. Richard Gill (talk) 15:48, 13 February 2011 (UTC)

Are you aware of this arbitration? Glkanter (talk) 21:30, 12 February 2011 (UTC)

@Glkanter. I'm sorry, I disagree. Jeff is a mathematician. He is interested in correct arguments just as much as correct answers. I find it annoying that he doesn't explain *why*, from his point of view, you *have* to compute a conditional probability, but I do believe that that is his point of view. I also know good reasons for having that point of you. Of course, mathematics can not ever tell you what you have to do. It has no legal or moral authority. But it can tell you what it would be wise to do. Richard Gill (talk) 12:29, 12 February 2011 (UTC)

Yes, but you ignore his English language answer in order to disagree. I am not required to, nor should I, make the same error. Glkanter (talk) 21:30, 12 February 2011 (UTC)

Since his written English language taken at face value is self-contradictory, it could be wise to try to figure out what he might have meant. It could be wise to try to appreciate the context in which he is writing, to appreciate the likely understanding of his intended audience. My claim to have some understanding in this direction (I'm a member of the same academic and professional community as Jeff Rosenthal) lead you to accuse me of claims of extra sensory perception! Richard Gill (talk) 16:08, 13 February 2011 (UTC)

@Nijdam. I am tired about bickering about what is THE Monty Hall Problem. You and I disagree. I think there are many mathematizations and any decent mathematization of course allows many different decent solutions. You may call the approaches which I listed by any name you like. I think the names I gave will be understandable to everyone interested in MHP and active on these pages, not just to the people who share your point of view, which I find dogmatic and inflexible. PS my third solution is the most beautiful of all especially since it is a mathematization of the (correct) intuition of all intelligent laypersons that specific door numbers are irrelevant, the only thing that counts is the probabilistic relation between the roles - both manifest and (for the player) hidden - of the doors. Indeed, the relation between the roles is independent of the numbering. Vos Savant's "say, Door 1" and "say, Door 3" was spot on. These words can be placed in parentheses, they can even be deleted altogether - at least for a rational Bayesian/subjectivist, like Laplace and like all ordinary people. Laplace built his theory of probability on symmetry. He would like solution number 3. Richard Gill (talk) 12:34, 12 February 2011 (UTC)

Now I'm even more confused about what the bone of contention here is. It seems to me that the problem in the MH problem is not statistical (the statistics are actually quite clear on the matter), it's that people (even experts) make the error of treating a non-probablistic act (the opening of a door with a goat) as though it were in fact a probabilistic act. I mean, this would be obvious for two doors (you know there's a goat behind one door and a car behind the other, you open the door to reveal the goat, what's the probability that the car is behind the other door?); This is more like a word game than a problem in statistics. are you all just arguing over your personal favorite ways to talk about the stats? --Ludwigs2 22:39, 12 February 2011 (UTC)

The fights about solutions are "much ado about nothing". Fights about whether the solution which you might want to give to students in a mathematics class on introductory probability, should be seen as superior to the solutions which ordinary folk can understand. So it's also a fight about demarcation, about ownership, about arrogance of power. What's the use of a wikipedia page which gives a list of solutions which everyone can understand, tells them these solutions are wrong, and then presents solutions using concepts which ordinary people don't know about? And moreover, does this from a position of authority and dogmatism. Never explaining *why* they think that one shoud solve the problem in a particular way. Mathematics can never tell you what you *must* do. It can only tell you what it is *wise* to do. And the good mathematician should be able to explain why it is wise.

In a number of publications I took the trouble to explain *why* it could be wise to solve the problem in a certain way. I also pointed out that the simple solutions make less assumptions hence are of wider applicability. I also pointed out that how you want to solve the problem, and what assumptions you are able to make, could well depend on what you understand by "probability". Which is an ongoing unresolved debate lasting for at least three hundred years, and no sign that it is about to stop. My own opinion is that it is a matter of taste. Richard Gill (talk) 16:00, 13 February 2011 (UTC)

Richard, You sound a little annoyed, and I in your place also would be annoyed with myself. But don't blame the messenger. You d... well know one big issue here is about which solution suits which version. So, behave scientific, and do not mention just solutions without referring to the version of the problem. It's not in the interest of the article, nor in the interest of the readers, and in the end not in your interest if you leave this point unclear. Nijdam (talk) 09:12, 13 February 2011 (UTC)

That's only one of many issues. MHP is not owned by any particular person or community. It is certainly not owned by the community of teachers of Bayes' theorem in introductory discrete probability. I am in favour of diversity. Against dogmatism. Richard Gill (talk) 16:00, 13 February 2011 (UTC)

Richard, strange that you can be sharp as a knife and precisely to the point, but often also seem to have trouble to stick to the subject of discussion. And that is: speaking of solution, without reference to what is is supposed to solve, Got it? For the rest: which are the other other issues? List them in short as a help in the arbitration. Did I say MHP is owned by some particular person?? I also like diversity, as long as it is not confusing. Nijdam (talk) 22:07, 13 February 2011 (UTC)

There are countless reliably sourced solutions that are 'conditioned' on the 100% likelihood that the host will reveal a goat behind another door & offer the switch. And unless the contestant knows these facts before he selects a door, the odds aren't necessarily 2/3 & 1/3. So he can start solving the puzzle without waiting for the host to open a door.

2/3 of the time the contestant will select a goat.

Therefore he should switch.

...Is reliable sourced, mathematically correct, and meets the narrative of the puzzle, "Suppose you're on a game show...". Conditional door-based solutions also solve the same problem statement, but are not required. No matter what these guys tell you. We're in arbitration about this right now. Glkanter (talk) 23:56, 12 February 2011 (UTC)

You are right, Glkanter, because the symmetry of the problem, when we are using probability in the man-in-the-street subjectivist sense (Laplace, 1814), tells us that the specific door numbers of door chosen and door opened are irrelevant. The pedantic maths teacher will say that your solution is not complete if you don't mention this fact. He will say that your answer is correct but your reasoning is not complete. Possibly you were indeed smart enough to see that you don't need the door numbers, but if you don't write this down explicitly, he can't decide whether you are smart but fast, or careless and not aware that you are missing a possible issue. The correctness and completeness of the argument to get to the answer is as important as the answer itself, for the pedantic teacher, in the maths or logic classroom. Richard Gill (talk) 16:14, 13 February 2011 (UTC)

I don't have to do anything. The reliable sources are what matter. Your opinion about them, not so much. I don't see any ambiguity in what Rosenthal writes. Glkanter (talk) 17:03, 13 February 2011 (UTC)

Further, my original point was that the Conditional solution section is horrible, and that your proposed stuff would be a lot better than the horseshit in that section today. Glkanter (talk) 17:07, 13 February 2011 (UTC)

Based on your responses, Richard, it would seem Rosenthal could be handled in one of 3 ways:

1. Based on the English words he wrote - he is *not* a critic of simple conditional solutions - (Glkanter)
2. Declare his writings incoherent - [at best, then, his paper should be disregarded] - (Richard says the paper is contradictory)
3. Decide for him what he really meant to say - despite what he wrote, "This solution is actually correct, but I consider it "shaky" because it fails for slight variants of the problem.", he is a critic of the simple conditional solutions - (Richard, The Most Noblest Of The High Priests)

The first one seems right to me. You have argued for the 2nd option, (but not the conclusion in the brackets), in order to support #3. I don't see how. And you have also told me the third option is the best, because you have 'lived the life', and that you alone, can know what he 'really' means. Preposterous for Wikipedia purposes. Or for any discussion, anywhere else on the planet, really. Glkanter (talk) 08:48, 14 February 2011 (UTC)

I think you are jumping to conclusions, @Glkanter. If one sentence appears incoherent but the rest of the text appears very professional and correct, and if there is a simple rewrite of that one sentence which makes it coherent with the rest of the text, then I would go for the rewrite. But, if *you* are unable to understand the rest of the text then *you* are stuck between Scylla and Charybdis: either you trash the whole text, or you trust an expert. (Fortunately Bayes' rule is intuitive and easily "internalizable" so you should not find it hard to understand the rest of Rosenthal's paper). Richard Gill (talk) 11:21, 14 February 2011 (UTC)

Jeff Rosenthal wrote me the following in an email today and I quote it here with his permission Richard Gill (talk) 17:47, 14 February 2011 (UTC)

"I apologise that my article's one sentence about the "Shaky Solution" wasn't sufficiently clear. What I meant was that this solution does give the correct answer, but only because it so happens that in this particular case, conditioning on the fact that the host has opened a non-car door does not change the probability that the original guess was correct. This last is a very subtle point which requires justification (e.g. using Bayes' Rule). I believe that many people who quote the Shaky Solution do not realise the importance of this point nor how subtle it is. As illustration, I believe that many people who quote the Shaky Solution would believe it also applies to the Monty Fall Problem even though it does not (and gives the incorrect answer in that case). In summary, I would say that the Shaky Solution can be "made" to be correct, but only by providing a clear justification for why this conditioning does not change the probabilities; without such justification, the solution is incomplete and insufficient."

Thank you for pursuing that, and sharing it, Richard.

My opinion is unchanged. His paper is *not* a criticism of simple conditional solutions as interpreted and prostelicized by Rick Block and Nijdam.

What you have shared is simply his opinion, in a private correspondence, that the version of the simple solution and/or the problem statement, he, himself restated in his own words, was 'incomplete and insufficient'.

Which was his intention all along:

4 Monty Hall Revisited

The Proportionality Principle makes the various Monty Hall variants easy. However, first a clarification is required. The original Monty Hall problem implicitly makes an additional assumption: if the host has a choice of which door to open (i.e., if your original selection was correct), then he is equally likely to open either non-selected door. This assumption, callously ignored by the Shaky Solution, is in fact crucial to the conclusion (as the Monty Crawl problem illustrates). Monty Hall, Monty Fall, Monty Crawl Jeffrey S. Rosenthal (June, 2005; appeared in Math Horizons, September 2008, pages 5{7.)

The generally accepted (Selvin, K & W) MHP makes the 50/50 host bias premise very clear, along with revealing the goat behind another door and the certainty that the switch will he offered. Glkanter (talk) 18:24, 14 February 2011 (UTC)

For some puzzle solvers, "Suppose you're on a game show..." in the problem statement says all they need to know about symmetry. Other will only feel 'complete' when the solution says '...and therefore, due to symmetry..." The entire world will fall somewhere in-between these points, inclusively. I reject any assertion as to the 'only right' way to phrase a solution. Who says 'certain ' premises (Suppose..., uniformly at random car & goats, 50/50 host bias) must be repeated in the solution? As per whose doctrine?

Rosenthal's telling of the problem leaves the phrase 'Suppose you're on a game show...' out. It's not really the MHP as per Selvin, vos Savant, or K & W, is it? Nor was that really his intent, obviously: "...callously ignored by the Shaky Solution..." Otherwise, there's no raison d'être for his paper, is there? Glkanter (talk) 12:55, 15 February 2011 (UTC)

I recommend you read Rosenthal's very nice popular book "Struck by Lightening". It has a good chapter on MHP. (Also I recommend you read Jason Rosenhouse's popular book on MHP, especially the chapter on probability interpretations). Jeff Rosenthal, a professional / academic mathematician / probabilist, and a prominent popularizer of probability and statistics, knows that our ordinary probability intuition often lets us down. As Persi Diaconis said (interviewed in the NY Times article on Vos Savant's article), "our brains just aren't wired for probability". If Rosenthal is preaching anything, he's preaching Bayes' rule. Which I would like to preach, too. If you don't have the right insight to solve the problem in a flash all on your own, you had better learn a reliable route to get to the right answer.

Once you've found the right answer, you can start wondering if there was a short cuts to find it.

Bayes' rule (as opposed to the horrible formulas usually known as Bayes' theorem) is simple to state in words, insightful, easy to internalise.

How do mathematicians find beautiful theorems.? First by stumbling across them by accident after a lot of sweat and hard work, seeing a pattern and recognising that it isn't just chance, there is something wonderful going on there, thereby discovering the deep connections which make the result obvious, once you do the necessary "parallel thinking". Bayes' rule should be called Bayes' theorem. What is usually called Bayes' theorem is just the result of substituting, twice, the definition of conditional probability. The formula gets longer on each substitution... big deal! Bayes' rule follows by dividing two instances of "Bayes theorem" by one another (conditioning two different "target" events both on the same "given" event). A load of crap cancels out; what is left is beautiful and memorable and powerful and useful. Richard Gill (talk) 14:47, 19 February 2011 (UTC)

Mathematical formulation section

The section with the mathematical version using Bayes theorem was wrong, the symbols C, S, H standing both for random variables and for possible values taken by them, and consequentlly a number of the wordings were totally garbage. For instance, "the probability P(C)" is nonsense, and writing Bayes theorem with P(H|S,C) etc is nonsensical - Bayes theorem is about events, not about random variables. @Glopk changed it back again but I have undone his undoings. Maybe some mathematicians would like to take a look. If there's to be a mathematical section showing students of probability theory how to do it by routine (automatic, brainless) application of Bayes' theorem then it should be done properly. Richard Gill (talk) 08:12, 14 February 2011 (UTC)

At last. Nijdam (talk) 09:07, 14 February 2011 (UTC)

Fie. If P(C) is nonsense, then so are the thousands of referred papers and textbooks using this notation (including the ones referred to in the article). Yes, it is a shorthand, and yes, it is common in the literature (and on Wikipedia). On the other hand, please explain how you can keep in your head a the same time "I want to write P(C=c | H=h)" and "Writing P(... | I ) is rendundant because the background information is always assumed". glopk (talk) 21:19, 14 February 2011 (UTC)

@Glopk, I have a question and an answer.

Question: Please give me some references (papers, textbooks, or wikipedia articles) where $P(X)$ is used as notation for the probability that the random variable $X$ takes on some value $x$. Such wikipedia articles certainly need correction. Especially if the same text goes on to talk also about $P(Y)$ and $P(3)$. Students who need to see an explicit computation of the text-book (but rarely used) form of Bayes' theorem also need to learn correct notation. What people actually use in practice is Bayes' rule.

Answer (to your question about the consistency of my thought processes): Of course all good mathematical notations "suppress" information which is understood to be present but need not be mentioned specifically, since it never changes throughout a whole text, or because it is crystal-clear obvious from the context. Tell me, what is the point of writing $P(...|I)$ *throughout* a whole text, where I stands for background information which is never specified and never changes? I think it would be wise to agree in advance that $P(...)$ will be used as shorthand for $P(...|I)$. That would be a service for reader and writer. Richard Gill (talk) 16:29, 15 February 2011 (UTC)

I have added mathematical solutions using Bayes rule, and symmetry in two different forms. Richard Gill (talk) 11:16, 14 February 2011 (UTC)

I would now propose to delete the formal computation with Bayes' theorem, replacing it with a link to Bayes theorem where it already is an example. Richard Gill (talk) 11:23, 14 February 2011 (UTC)

Oppose The solution using the so-called "odd form" of the Bayes Theorem is much less referenced in the MHP literature than the full Bayes expansion, therefore placing it prominently in the article would give it undue weight. Further, all three "math formulation" paragraphs added by Richard Gill are unsourced, and the last one appears to be OR. I propose to delete (and will boldly do so soon unless I hear a convincing argument to keep them).glopk (talk) 21:26, 14 February 2011 (UTC)

"Odds" not odd, @Glopk. Richard Gill (talk) 08:26, 15 February 2011 (UTC)

All three proofs are sourced and all three can be sourced even more if that is desired. Richard Gill (talk) 08:46, 15 February 2011 (UTC)

Oppose. I personally like the 3 step modelling and I see no problem with that in the article. Moreover it is bugging me, that this is opening up another completely needless edit conflict. In particular if people said above they are essentially happy with article, why do they have to start another round of edit conflicts rather than leaving it alone.--Kmhkmh (talk) 22:57, 14 February 2011 (UTC)

I personally also like the three step modelling. I also like alternative approaches. The article will give a very biased picture of the literature if it is written as if there is only one way a mathematician can do a formal proof of the important result. It would be "undue weight" and a great disservice to all students of Statistics 101 out there, only to show the least attractive and least useful mathematical treatment. Richard Gill (talk) 08:41, 15 February 2011 (UTC)

Personally I also am quite happy with the present article and I think it's time we move on from the conflicts, and get back to constructive editing. The discussions of two years have actually opened up a lot more literature and knowledge to all of us, and generated several peer reviewed publications. MHP does not stand still. Richard Gill (talk) 08:46, 15 February 2011 (UTC)

Oppose. This is Wikipedia, source of info for all kind of readers. This application of the (now) correct form of Bayes' formula is widely accepted as a way of calculating the (necessary) conditional probability. Nijdam (talk) 23:25, 14 February 2011 (UTC)

Yes, and it is already there on the Bayes theorem page. So a reference to that article would be good enough. MHP does not stand alone. It is connected to the rest of the world, to the rest of wikipedia. Let's work on cleaning up the Bayes theorem page, it is quite a mess. But in the mean time, a decent version can of course be kept here on the MHP page, if that's what everyone wants. Richard Gill (talk) 08:46, 15 February 2011 (UTC)

You say calculating the (necessary) conditional probability. Of course, a smart mathematician realizes that it is not necessary to compute at all, because of independence. The task of mathematics is to replace calculations by ideas (Riemann). Let's not forget that. The task of mathematics teachers is to teach students ideas and how to use them, not teach them to be calculators. Richard Gill (talk) 08:50, 15 February 2011 (UTC)

Not much fun, is it, Richard? Glkanter (talk) 23:51, 14 February 2011 (UTC)

On the contrary, great fun! This is what collaborative editing is about. Discussions and controversies, learning and teaching. The sum is more than its parts. Wikipedia MHP has to be accessible to a huge and varied readership. The present editors are representative of the future readers.

To business. Of course a conditional probability can be calculated using Bayes' theorem. It can also be calculated using Bayes' rule. The article on Bayes theorem contains this very calculation as an example. So in my opinion it is unnecessary to have it here, but of course I bow to any concensus (how could I do otherwise?).

Saying that doing it by Bayes rule is "undue weight" is in my opinion rather silly. Jeff Rosenthal in his article and his book does it this way, others do it this way too, and it is a method which gives insight into "why". This is the "modern" way to do it; via Bayes' theorem is a rather dull old-fashioned way to do it. It merely translates the numerical calculation already done back into formulas! What's the point of that. The formal derivation using Bayes' theorem just shows that it can be done by an automatic proof computer, but it does not give insight. It is an exercise for a class on Bayes' theorem, not a contribution to the understanding of MHP (in my opinion, that is).

My third solution is sourced. It is Vos Savant's solution! The door numbers are irrelevant by symmetry and can be ignored. You do not need to compute a conditional probability because there is no need to condition on anything, by independence. I will add some more references. I know that Persi Diaconis has done MHP in this way and others too. I think it adds insight and moreover builds a bridge between simplists and conditionalists. It therefore serves to unify the wikipedia page. Obviously editors who are still embattled in old positions and not ready for reconciliation and synthesis, will object. Then this proof will only be added to wikipedia in ten years time after it has been around in more easily accessible reliable sources for a longer time. Too bad for the wikipedia readers of the intervening decade.

Now about notation in the present formal proof. $C$, $H$, $S$ were introduced as random variables. It would be quite standard notation to use small letters for possible values of those variables, and moreover, it would be quite standard notation to use a small $p$ to stand for probability mass function, though many authors also use $f$, a probability mass function can be seen as a density function, and we then get the same "theorems" for mass functions as for densities. Properly one should attach to the mass function some indication of which variable we are talking about. So you could write for instance $p(c)$ as shorthand for $P(C=c)$ but $p(3)$ would be ambiguous and one should write something like $p_{C}(3)$. Anyway: if the section with the formal proof via Bayes' theorem is to stay, as some kind of help to students of probability and statistics classes, it had better well use correct notation, otherwise it will be no use for them at all.

$P(\cdot )$ - note the capital letter - is only ever used, in standard texts, as far as I know, for "Probability of". You can have "probability of an event" but you can't have "probability of a random variable". Richard Gill (talk) 08:23, 15 February 2011 (UTC)

Mathematical formulation

As the lemma Bayes' Theorem - The Monty Hall problem already shows the full Monty Hall problem in mathematical formulation, as talked to students of conditional probability in textbooks, and in order to avoid unnecessary redundancy, I hope that there will be no objection to delete the redundant formal computation with Bayes' theorem here, and to replace it with the proper link to Bayes' theorem - The Monty Hall problem. It's a matter of avoiding unnecessary redundancy. Gerhardvalentin (talk) 18:14, 11 March 2011 (UTC)

This article stands on its own. IMO, the full Bayes expansion is useful here. It's not overly long. Why delete it? -- Rick Block (talk) 18:39, 11 March 2011 (UTC)

Rick, in Bayes' Theorem,  the "MHP" is a welcome example to show how to use various assumptions in conditional probability theory and -calculus. And there it does not need to question the sense of licentious assumptions and licentious presuppositions. No need. It's shown there for the sake of understanding conditional probability. And there it belongs, and it is needed there. I repeat: It belongs there.  Period.

But, for the reader of the MHP, to understand the paradox, never needed flawed assumptions nor "conditioning" on any completely irrelevant "(im)possible might be-s". The numbers of the doors are irrelevant. No need to condition on their irrelevant "numbers".  Please pay respect to what all reliable sources say. Serious "conditional probability", even intentionally pretending to "know exactly" what no one can ever know, and with intentional pretentiousness considering absurd additional information received on a silver platter, never was able to show that staying could ever be better than to switch. Even intentionally "using" the absurd illusion of additional silver platter information. So all of that illusion is of no avail for the MHP.  But, in contrast, it's a matter of "conditional probability". Please notice that this is "two quite different pairs of shoes".

All of that illusion, that conditional probability – even in gathering never to be given "silver platter information" – could ever advise another decision than to always switch, is of no avail whatsoever for the MHP. Please understand what the lemma is about. It is about a paradox that more than 90 % of people get wrong. So the lemma should help to clearly see the obvious paradox and how you can match for it. So: No more confusing mind games, please.

Once more, Rick: The value of the "conditional probability" is totally irrelevant for the paradox called "MHP". So you should show all of that absurd mockery later in the article, for people interested in. And there should be the link to the Baye's Theorem, also. You are free to present all of that academic odds and ends there. But not in the first place where the secret of the paradox should be accessible for the interested reader. I strictly oppose to shoo interested readers with uninteresting mockery, but that gimcrack mockery should be broadly shown later, for the delectation of readers interested in. So I will be going to do as I proposed and set a link to Bayes' Theorem. Please help the article to be accessible and intelligible for the readers. Thank you. Gerhardvalentin (talk) 21:37, 12 March 2011 (UTC)

There are 4 editors here who have opposed deleting this from the article, which at least IMO would make this a change for which there is not consensus - i.e. would make this simply disruptive. -- Rick Block (talk) 22:15, 12 March 2011 (UTC)

Thank you. Date of your accredited warrant of attorney? Rick: It is for honesty, it's for the readers, it's for what the sources say. Please help the lemma be what it should be: The unbelievable paradox, and its dazzling history. I strongly hope you will help to "solve the problem". It's for the sake of Wikipedia and for the sake of the readers, to get all of that in a clear manner, not disintegrated, nor misty convoluted, anymore. Hope we can focus on the clearness of the lemma. Thank you. Gerhardvalentin (talk) 22:38, 12 March 2011 (UTC)

False dates, missing references

The new recently added alternatives solutions seem to cite publications from The American Statistician with false dates (1980, 1981 - probably meant to be 1990,1991?). Also both papers are not listed under references either.--Kmhkmh (talk) 04:33, 15 February 2011 (UTC)

Yep, exactly what I meant by saying that they are "unsourced" above. glopk (talk) 07:27, 15 February 2011 (UTC)

Sorry, those were typos. I will fix the dates and add the references, if we don't have them yet in the big list. They are both part of the reactions to Morgan et al. and well worth reading. Richard Gill (talk) 08:01, 15 February 2011 (UTC)

@Kmhkmh and @glopk, are you telling me you don't actually know the literature on MHP? And these are two very readable papers at the very centre of the Morgan et al. controversy. Fie on you! Do your homework! I have a collection of pdf's of all this literature which I can email to you privately if you like. Alternatively, I could set up a dropbox.com shared folder for the use of us wikipedia MHP editors - anyone interested?. Richard Gill (talk) 08:29, 15 February 2011 (UTC)

I'm telling you that the correct/complete references were missing. Whether I've read them or not, whether they are particularly readable or not or whether you have personal pdf copies is rather irrelevant. What's relevant here, is that if you use them please reference them properly. And yes in case I haven't read all the possible interesting publication on MHP nor do I intend (= I got better things to do). However if you don't mind I'd indeed appreciate a copy of the pdf files (I#ll send you an email).--Kmhkmh (talk) 15:51, 15 February 2011 (UTC)

Great! Sorry for annoying you, I should at least have added  ;-) to my reaction above. Richard Gill (talk) 16:13, 15 February 2011 (UTC)

"In its usual form the problem statement does not specify this detail of the host's behavior"

So what?

The last paragraph of the MHP page summarizes the conditional/unconditional controversy. It seems to me that this whole section is written with a strongly frequentist slant. Randomness is seen in the host's behaviour, and probability is seen as measuring randomness. Yet according to a subjectivist view of probability, probability is a measure of our knowledge or lack thereof. Probability does not measure physical randomness, but it measures our personal uncertainty. For a subjectivist, the host's behaviour is irrelevant. The subjectivist who hears Vos Savant's words has those words, and those words only, to go on, plus some general/common knowledge about quiz shows etc. The problem description gives us no reason to assign any special meaning to any particular door numbers. "Say, Door 1", and "say, Door 3" could just as well have been any other pair of doors numbers. For a subjectivist, all three doors are equally likely to hide the car, not because of any particular randomization procedure used by the quiz-team, but just because of lack of information to the contrary. Similarly, if coincidentally the player initially would pick the door hiding the car, the host would be equally likely to open either door, not because Monty uses a fair coin toss to determine his choice, but because the player would bet at equal odds for or against either choice, because for him the two doors are exchangeable.

Thus for a subjectivist, the fact that the problem statement does not specify the host's behaviour is totally irrelevant.

I think this needs careful thinking about. Rosenhouse has a whole chapter on this topic in his book. Either the text in the article should be neutral as to probability interpretation, or there needs to be a small discussion about the issue. Laplace (1814) builds his whole theory of probability on top of the *subjective* ("primitive") notion of "equally likely" and shows how symmetry, whether of knowledge or of physical laws, determines subjective probabilities in a whole range of problems. Richard Gill (talk) 13:46, 15 February 2011 (UTC)

But as I have said many times before, even from a frequentist POV Morgan's original conclusion is not justified. From a strict frequentist POV the problem is simply insoluble as we are not told how the car is initially placed.

Host bias is only important if we choose to make it so by selecting it for special attention from all other factors that might affect the outcome. Martin Hogbin (talk) 11:24, 19 February 2011 (UTC)

Yes. And with a subjectivist's (ordinary man-in-the street's) interpretation of probability, and with only Vos Savant's info to go on, what we think about host bias doesn't matter a bit - even if we believe that he might well have a bias, we are still totally ignorant of its direction, whatever its size, because our opinions are neutral to relabelling of the doors. This should have been the big point of Morgan et al's computation, the one they got wrong and which you, Martin, corrected: with a symmetric prior on host bias, your personal expectation that switching gives the car remains 2/3. As must also be the case, by symmetry. Richard Gill (talk) 13:58, 19 February 2011 (UTC)

Conditioning on the day of the week and other acuteness

The famous PARADOX on the one hand, and quite other dissentient issues on the other hand, that do not relate to the famous paradox question, actually are "mixed together" in the article. In order not to befog, such strange sidelines and stratagems should not appear every now and then in the lemma, they imho should be presented and treated in a different section, because that issues really belong to a different section, e.g.:

Some just feel free to predetermine high-handed and licentious haphazard, shaky and unproven assumptions. Quite outside the famous PARADOX. They may be free to do so.

In the absence of permanence lists they decide for example (just for their own pleasure) that the host on Sundays never does pay respect to the actual placement of the car, and this way on Sundays therefore in 1/3 of cases just will show the car instead of a goat. And thus on Sundays he limits Pws to 1/2.

And on every Wednesday, they let him only open his nearby adjacent door, whenever possible. And they decide that they know exactly about that fact. On Wednesdays he only will exceptionally open the distant door if his nearby door actually is hiding the car, and then Pws clearly is "1". And, to get the appropriate cognition, are applying conditional probability terms just for practice. So they like to need the appropriate day of the week as a condition, and they draw conclusions of "coincidence and evident inference" depending on the day of the week. And they are conditioning on the appropriate day of the week and other acuteness to handle such coincidental.

Such forgery and adulteration, even if it has been said so, never is element of the "famous paradox" itself. Not to befog, all of that should not be interspersed in the article, but be shown in a separate section.  Gerhardvalentin (talk) 18:51, 15 February 2011 (UTC)

Nice paradox: a frequentist cannot solve MHP because he doesn't know anything (unless he stops being passive, and randomizes himself). A subjectivist can solve it, for the very same reason, namely, that he knows nothing! Ignorance is bliss. It seems to me that Vos Savant's question has to be solved with subjectivist (aka Bayesian) probability because she does not give us any information. If anyone wants to quote this, it is the concluding sentence of my recent paper in Statistica Neerlandica. But I can't push "own research" on wikipedia. Hopefully this wisdom will permeate into the standard literature on MHP in a few decennia. ;-) Richard Gill (talk) 20:32, 15 February 2011 (UTC)

If the above paragraph is High Priest talk for "when you're on a game show, of course it's reasonable to assume uniformly at random distributions unless told otherwise', I've heard it (and said it) 1,000 times before.

Does that change if the contestant makes his selection by 'lucky number' rather than by using a random number generator? I didn't think so. Glkanter (talk) 20:43, 15 February 2011 (UTC)

Subjective probability in, subjective probability out. Objective probability in, objective probability out. But yes @Glkanter you may take this as High Priestly authorization of the usual assumptions for the purposes of solving a well known brain teaser in pubs, at parties, etc. The biased host is irrelevant when we solve the problem using ordinary man in the street probability. He doesn't know anything about it, either way, so for him, for this specific game, it's 50-50. And that's all we are talking about. For the rest, you are welcome to your opinion about the difference between choosing by lucky number or by random number generator. (But I thought you didn't care for the opinion of high priests). Richard Gill (talk) 21:08, 15 February 2011 (UTC)

Richard, I would not say so. No, the biased host is irrelevant not only for the use of the ordinary man in the street probability. Any pretense of bias is useless and completely in vain to "solve" the paradox, because you have no permanence list for that "one and only game" the question is about, that incidentally never was on stage in reality, in exactly this manner. It's not about the "door numbers #1 and #3", it's not about the "necessity of conditional probability theorems", it's just about worthless and cheeky assumptions that never can be given. It's just about cheeky, unproven assumptions. Because, in effect, you really do know nothing at all about all of that. Marilyn explicitly excluded such "additional hints". Such senseless assumptions therefore never affect the famous question in any way. You have no permanence list! Such assumptions are an alien issue, completely outside the famous paradox. Such assumptions are really a waste and never can "help to solve the paradox".

To say "If you knew that the host uses to give closer information on the actual location of the car in each and every game, then you would know better" is a brain teaser quite outside the famous paradoxon. And for that you do not need indispensable conditional probability theorems, all what you need is your assumptions. Full stop. That never are to be given. Only those "assumptions" are the "condition" you base on, then. Not "before and after", not door numbers, not "on Sunday or on Wednesday". Your only condition is your actual "shake off the cuff never to be given assumption". That's the only thing you *need*. But it will not help you for your decision, as an answer that is based on unproven assumptions will never be "a better answer" than the plain simple solution is giving: You should switch.  

And "selling unproven assumptions" belongs to a separate section, not to intersperse them over the whole article.  Conditional probability is fine, as long as we aren't using it to sell unproven assumptions, slyly without naming them as what they are.  Gerhardvalentin (talk) 23:28, 15 February 2011 (UTC)

The article is already structured so that the first part (up to "Variants") discusses solutions using the usual (symmetric) assumptions. What change are you suggesting? -- Rick Block (talk) 01:01, 16 February 2011 (UTC)

Except for the 90% of the Conditional solution section that talks about host bias variants, and all the stuff in Sources of Confusion that calls simple and/or unconditional solutions 'the sources of confusion', Rick. Glkanter (talk) 01:11, 16 February 2011 (UTC)

 

@Rick, those famous lousy "variants" that only can be based on senseless "unproven assumptions" (because everyone is in a complete lack of knowledge thereon), and that not ever can be able to contribute to any serious "solution", and that all belong to a separate "variants-section", begin with the "Sunday's version" (host does not pay regard to what's behind his doors) in line 61:

This is different from a scenario where the host simply always chooses between the two other doors completely at random and hence there is a possibility (with a 1 in 3 chance) that he will reveal the car.

And again in line 67:

This example can also be used to illustrate the opposite situation in which the host does not know where the prize is and opens doors randomly.

And in line 84 it reads:

Some sources, however, state that although the simple solutions give a correct numerical answer, they are incomplete or solve the wrong problem.    -   and:

but without additional reasoning 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.

And more weasel words, without expressly naming the underlying irresponsible "assumptions" in line 86:

the conditional probability may differ from the overall probability and the latter is not determined without a complete specification of the problem

Btw it's not just "the conditional probability" that differs, it's just the underlying assumptions! - And in line 92:

This analysis depends on the constraint in the explicit problem statement that the host chooses uniformly at random which door to open after the player has initially selected the car (1/6 = 1/2 * 1/3). If the host's choice to open Door 3 was made with probability q instead of probability 1/2, then the conditional probability of winning by switching becomes (1/3)/(1/3 + q * 1/3)). The extreme cases q=0, q=1 give conditional probabilities of 1 and 1/2 respectively; q=1/2 gives 2/3. If q is unknown then the conditional probability is unknown too, but still it is always at least 1/2 and on average, over the possible conditions, equal to the unconditional probability 2/3.

Line 195:

. . . but if the host can choose non-randomly between such doors then the specific door that the host opens reveals additional information. The host can always open a door revealing a goat and (in the standard interpretation of the problem) the probability that the car is behind the initially-chosen door does not change, but it is not because of the former that the latter is true. Solutions based on the assertion that the host's actions cannot affect the probability that the car is behind the initially-chosen door are very persuasive, but lead to the correct answer only if the problem is completely symmetrical with respect to both the initial car placement and how the host chooses between two goats

Line 196:

depending on how the host chooses which door to open when the player's initial choice is the car (Morgan et al., 1991; Gillman 1992). For example, if the host opens Door 3 whenever possible then the probability of winning by switching for players initially choosing Door 1 is 2/3 overall, but only 1/2 if the host opens Door 3. In its usual form the problem statement does not specify this detail of the host's behavior, nor make clear whether a conditional or an unconditional answer is required, making the answer that switching wins the car with probability 2/3 equally vague. Many commonly presented solutions address the unconditional probability, ignoring which door was chosen by the player and which door opened by the host; Morgan et al. call these "false solutions"

And only then, one line later, follows the relevant section:

Variants – slightly modified problems

 

All of that should be presented in one separate section, as a dazzling display of fireworks in literature, not to be tangent to the famous question. All of that is equal senseless as words such as: "By watching the shows, and keeping track of how many times ... - an outsider sees the result of the host's bias. It is this history..."  

We have no knowledge on the correctness of any of those "possible assumptions", and - as they never may be given - we don't need them for getting guaranteed "better results". It's just as similarly helpful as a sure-fire roulette system. – Because it's about ONE game, and everything else is completely unknown.  

Once more: It's not on "conditioning", it's about irresponsible assumptions. It is very important to show all of that as "interesting meanders", in a later and separated section, those allegations, pretending to "know" what no one can ever nor will ever know. To show that clearly. The lemma should not obfuscate. It clearly should underline them as being unproven assumptions, and show what they really are. Never being able to contribute to any better decision than the only one "correct" solution that is honoring what really matters: Our complete unawareness of any further "conditions".  Regards,  Gerhardvalentin (talk) 03:06, 16 February 2011 (UTC)

Nicely done, Gerhardvalentin! Glkanter (talk) 03:21, 16 February 2011 (UTC)

Unfortunately this polemic piece regarding conditioning and assumption misses an important point entirely. It is not up to us to decide whether assumption are meaningful/useful/helpful or whatever, but it is up to reputable sources. What you or I personally think of various assumptions or solutions doesn't really matter much as far as WP (and this article) is concerned. As long as people are pushing for "their way" to treat the problem rather than the "sources" way, there is no way to resolve the edit conflicts of this articles. An evangelist approach doesn't work, if different faiths have to cooperate. All we do this way is recreating and extending the conflicts and bickering of the academic community in here, while our real job should be to neutrally report on their (endless) bickering rather than creating one of our own.--Kmhkmh (talk) 07:34, 16 February 2011 (UTC)

Thank you, Kmhkmh, for commenting on my really polemic piece. And yes, I fully agree with you. But that's the problem here: I ask you to just take into consideration that, as all of the important sources and aspects have to be shown, those aspects should be presented successional and in a coherent manner. Clearly arranged for the reader. To make the lemma meaningful and readable. And we should stop to present a motley conglomerate of secret and hidden shaky assumptions, clad in mathematical theorems, neither naming them as what they are, not showing what is meant and what is addressed. Just boldly annunciating the importance of "conditional probability" to solve the problem, without naming, but hiding the real underlaying absurd shaky "assumptions", naming them conditions.

The MHP is perceived with little, but *necessary* assumptions. First, that it is a question about just ONE game show. Not on a dozen, and not on millions. Secondly, that the host not shows one goat and offers to switch just because the contestant has luckily chosen correctly. That's the first two steps, just to provide any sense. But as you *never* will dispose of "permanence log lists", and the behavior of the host is completely unknown - as a third step - that, besides of showing a goat, he will not be giving away any additional hint whatsoever on the actual location of the car, behind the two still closed doors. Not by words, not by gestures and not by some bizarre hint concerning the door he opens, if in that game he should have disposed of two goats: He will "equally likely" have opened one door. These assumptions make sense, as you indeed have no knowledge whatsoever about anything else.

All of that has to be supposed to make the question meaningful, otherwise you could grasp it just as a bad joke. If he's supposed to give additional hints, or just offers to switch because the guest has first chosen the car.

But then, being without further info, some reliable sources liked to turn things back again, from a meaningful question about an obvious paradox to a bad-joke-question. Adding additional shaky "assumptions", contradicting and going far beyond and grossly exceeding your just defined primary knowledge about ONE hypothetical game show.

The circumstances should be presented in a clear way. saying what the sources mentioned really do address. Not mixing contradicting aspects motley in a discretionary way. No, we should show that in a very clear way, naming and frankly saying what the sources expound. We should present that in a succession of all that additional, contradicting "assumptions" they present.

Please help to finally stop presenting motley mixed "contradicting assumptions" without naming them as what they are. That's a matter of courtesy to the reader.

And once more: All important sources should be shown, in the "variants of contradicting assumptions"-section. And YES, mathematics is helpful and should be shown as necessary, preferably in odds-form. But, when dealing with additional contradicting assumptions you should distinguish mathematics and assumptions and underline that it is not about mathematical "truth" then, but about "additional, boldly contradicting assumptions". Kind regards,  Gerhardvalentin (talk) 13:56, 16 February 2011 (UTC)

Well personally I don't care much for the particular ordering and what goes into which chapter as long as everything is covered. Your argument however has the same problem as before, it is really not up to you (or me) to decide what the "real" MHP and what merely a "variant" is. The same holds again for your notion (or mine) of reasonable/unreasonable or required/unnecessary assumptions, they are irrelevant, It only matters what the reputable sources consider as reasonable or not (yes the reputable sources don't really agree either, but that in doubt the article needs to reflect that). There is no objective/universally agreed method to settle which approaches, aspects and assumptions are "bad", "good" or "best", since the decision ultimately depends heavily on personal taste, interest, as well as personal schools of thought one adheres to (for instance frequentist version bayesian) and last but least ego. In short we can argue the various aspects until the end of time, which is each side believing (with some justification) to be right. It's almost like arguing faith/religion and that's why I said before the evangelist approach won't work here as long as involved parties adhere to different faiths.--Kmhkmh (talk) 03:45, 17 February 2011 (UTC)

Thank you again, Kmhkmh, and you are right again. Wikipedia is a forum where you can decide whether edits are good or bad, and it's not up to the editor to decide whether the sources are good or bad. But we should not hide what the sources are actually talking about, their underlying initial assumptions concerning the rules of the game. Once more: their differing underlying assumptions. And my strong "belief" is that the lemma, last but not least, is there for the reader, from student to Grandma, this should not be left out. The aim is to help the reader to grasp what the sources actually are talking about, their differing underlying initial "rules of the game".
That it is possible and conceivable to view and to evaluate the paradox from different sides, based on differing initial conditions, from differing rules. And to clearly distinguish those two groups of sources and to clearly emphasize this difference by distinctly separating them in the lemma.

Are they starting from (1: "the standard problem"), or do they allow for (2: "an additional host's hint") on the actual location of the car behind the two still closed doors that the host can and will be giving, by his considered possible special "behavior" in opening one of his two doors. Will (1) the conceivable probability to win by switching of 2/3 on average be the only reasonable and admissible answer for the actual game show, or can it be recommended and useful to include the (2) host's hint in your determination of the actual probability to win by switching. To clearly keeping apart that two kinds of "sources" will be of real benefit for the reader. For the reader, the difference of those two quite differing "solutions" should distinctly be made obvious and clear. A Gordian knot to be offered, as in the past, should be avoided at all costs. Whether the sources are good or bad is up to the reader to decide. But it's up to the editors to decide whether it is "good" to hide everything behind the alleged "truth of mathematics". It would be fine if you could help to clearly show the difference of the two groups of sources. Your help is needed.  Regards,  Gerhardvalentin (talk) 09:02, 17 February 2011 (UTC)

The MHP paradox is why it's 2/3 & 1/3 rather than 1/2 & 1/2. Any premises that change that are variants. Glkanter (talk) 05:56, 17 February 2011 (UTC)

I agree completely, that the different approaches and assumptions should be treated separately and clearly for what they are and where they differ. However imho one of the main reasons, why this has not worked out that well in the article so far, is that various factions here insist on their approach or assumptions being the "real ones" and that they need to be treated as "the" essential MHP and that's exactly the evangelists again. They care less for an overall lucid explanation of all aspects, but rather that their favoured version is most prominently featured. The current partially less lucid state, is result of all those evangelists trying putting their version on the top creating a "obfuscating" mix of everything.--Kmhkmh (talk) 14:05, 17 February 2011 (UTC)

Kmhkmh, thank you so much for your neutral evaluation of the current status. And yes, your words are  *the*  long-awaited clearance and liberation. You see the field and are raising your voice from a higher vantage point, and I would like to hope this will be of appreciable effect. And I am hopefully that we are closer to an end of the long lasting solidification now. Thank you once more!  Regards, Gerhardvalentin (talk) 15:01, 17 February 2011 (UTC)

Yes!

Kmhkmh, your approach is the basis of my current suggestion in arbitration, that we have the article in two parts, for the benefit of our readers. The first part has only simple solutions, without disclaimers, not because they are 'The Truth' but because they are what most people will want to understand. After we have dealt with that, we we can have a scholarly expostion of all the other issues relating to the MHP.

This is not some dastardly plot to promote my truth above all others, it is the way that most good text books and encyclopedia article work. Give a simple explanation first, maybe glossing over some technical details, then discuss the more complicated stuff. This effectively brings this long argiument o a close and enables us to work cooperatively again.

In the past you expressed your approval of this approach, which is pretty much how the article is now. Perhaps you could confirm that you still support this idea. I believe that it is the only way to move on. Martin Hogbin (talk) 11:18, 19 February 2011 (UTC)

I don't think my "approach" extactly matches your suggestion. I have no issue with the "simple solution" being presented first or without caveats/pointers to morgan style crticism. I do however have an issue with it being represented in an obfuscating manner (there I agree somehwat wih nijdam). Meaning we should clearly state what the original simple solution computes the overall probability for winning. No caveats but clearly stating what it does. You may also argue that the overall probability is identical with the conditional one under certain assumptions, but then again that needs to described in lucid manner (rather than what you personally consider as "obvious"). However the detailed argument here might better be pushed to a later article, to keep the "simple solution" indeed short and simple.

Having said since I dont intend to play the role of a "lucid description everywhere" evangelist. I'd live with your (imho mathematically obfuscating) suggestion just for the compromise's sake. I.e. if the rest of the editors is fine with your approach I won't object as long as a more lucid description regarding the details is further down in the article in a separate chapter. But alas so far neither the other "conditionalists" are willing to budge nor are you.--Kmhkmh (talk) 16:13, 19 February 2011 (UTC)

Kmhkmh, could you please indicate which reliable sources that actually offer a simple solution say the things you describe above about 'overall probability of winning'? If it comes from critical reliable sources, please reference them. Thank you. Glkanter (talk) 16:29, 19 February 2011 (UTC)

In the mean time I hear that I am being crucified on the arbitration page for being an expert and for trying to give expert advice where relevant and for patiently trying to explain the expert's point of view on the talk pages. It seems that a lot of people don't have much of a sense of humour, and in particular, are not able to appreciate it when people do not take themselves seriously, but in fact make jokes at their own expense. I guess it is rather British. Richard Gill (talk) 16:06, 17 February 2011 (UTC)

No, that's not accurate. You are not being crucified on the arbitration page for being an expert, far from it, although a false accusation has been made to that effect... As a statistician and a forensic expert, I would think you would be able to tell the difference between real evidence and a false accusation. My answer to that false accusation here As for the sense of humor issue, I had ignored that as it doesn't adequately explain away the evidence IMO, but I will respond to that accusation on my talk page, where you first raised the issue. Woonpton (talk) 16:31, 17 February 2011 (UTC)

Yes, you are right @Woonpton! One must distinguish between evidence and accusations. And certainly, at face value, the evidence presented is strong. I need to make more use of a sandbox when composing some text for wikipedia. By the way, you are falling into my own humour-trap here, because my calling what is going on "being crucified" is poetic license, hyperbole, exageration for dramatic effect. (Though, those were the very words a worried friend of mine used, to me.). I'm a typical somewhat autistic mathematician, poor communication skills. As long as I can remember, people have been alternately angry with me for appearing to be arrogant, or angry with me for being over modest, self-effacing. Fortunately, now I'm nearly 60 I don't have to care so much what people think. In Dutch there is a phrase "whether or not the host trusts his guests tells us whether he is an honest man himself". I honestly think that I act on wikipedia in good faith and I assume good faith on the part of anyone else. And to be sure, I'm human with all human failings, like all of us. Richard Gill (talk) 14:17, 19 February 2011 (UTC)

It shows that Rick Block's claim about variants not polluting the article is his usual hogwash. The variants monopolize the Conditional solution section at the expense of clarity. That's fact, not dogma. The positioning of extensive discussions of variants in the article as a means of criticizing the simple solutions is also UNDUE WEIGHT and a POV that is not supported by a significant minority of reliable sources. That makes it OR. Glkanter (talk) 07:41, 16 February 2011 (UTC)

Too many assumptions

The implicit assumptions list in the lead was too long--- the assumption that the car is placed randomly is unnecessary, if your first guess is uniformly random, it doesn't matter how the car is placed. There is no need to assume that the host will chose the goat door to open at random either. The only assumption that is needed is that the host will open one goat-door no matter what your initial guess. That's it.69.86.66.128 (talk) 07:00, 16 February 2011 (UTC)

Splendid, Mr (or Mrs) 69.86.66.128 ! I have been pointing out this solution for several years and it is mentioned several times in the literature but no one here is interested in this point of view. Finally I wrote a couple of reliable sources giving this solution and linking it to game theory. Richard Gill (talk) 07:09, 16 February 2011 (UTC)

Again there is a connection with the interpretation of probability. If you are a frequentist you don't know enough to solve the problem so you take action yourself - you randomize and switch and win the car with probability 2/3. The 2/3 is a property of the apparatus you use to choose your door (dice, tossing coins...). If you are a subjectivist you know nothing and hence your subjective probability that switching will give you the car is 2/3, independent of which door numbers are seen chosen and opened. The 2/3 is a property of your (non)knowledge. Richard Gill (talk) 07:17, 16 February 2011 (UTC)

Request

@Richard: please do not save every time you change one or more letters. And wait with your next changes until the other parties had the opportunity to react. Nijdam (talk) 08:36, 16 February 2011 (UTC)

Those are "minor edits". Richard Gill (talk) 08:43, 16 February 2011 (UTC)

By the way, you recently reverted some text near the start of the article so as to promote your Point of View. This was "solutions are almost always based on the assumptions ...". Lots of solutions are indeed based on the usual assumptions of random location of car and random choice of host when he has one. However lots of solutions are also based on the *only* assumption that the player's initial choice is random. See the last section started by an anonymous editor. I think "almost always" is a gross exageration. Richard Gill (talk) 09:11, 16 February 2011 (UTC)

Richard, W.Nijdam just only did revert an IP edit - back to your version. Gerhardvalentin (talk) 11:05, 16 February 2011 (UTC)

There are assumptions, and there are assumptions. The assumption that the host will behave the same way whether you choose a car or a goat is essential, without this assumption the problem is ill posed. The assumption that the car is randomly placed, or that the host has to open a goat-door (as opposed to opening a car door sometimes, thereby giving away the answer), or that the host has to choose the two goats at random are irrelevant distractions, which do not change the analysis of the problem or the answer. One must never conflate irrelevant assumptions with relevant ones.69.86.66.128 (talk) 11:07, 16 February 2011 (UTC)

Exactly, @69.86.66.128. The assumption is essential that the host is always going to open a door and reveal a goat. Most of Vos Savant's readers understood her to mean this, she also later said that she meant that, and Selvin from whom the problem originated (and before him, Gardner) have the same assumption, explicitly. All sources thereafter, as far as I know, also make this assumption. Other assumptions are up to the reader - there is not a concensus though there might be said to be a fairly clear majority opinion. My personal *opinion* is that if you use probability in a subjectivist sense, as I think do most ordinary people - thus probability is a measure of *your* personal (un)certainty - then the assumptions that all doors are equally likely or that either of the host's choices, when he has one, are equally likely, are automatic (logical) consequences of Vos Savant's problem statement (cf. Laplace (1814), founding subjectivist probability as a rigorous mathematical science - everything is defined in terms of "equally likely", in terms of symmetry, in terms of knowledge and lack thereof). All we have to go on are Vos Savant's words. If however you use probability in the frequentist sense, as many but by no means all scientists do, and many but by no means all statisticians and probabilists do, then my personal opinion is that the problem is ill posed, unless of course you allow the player the option of introducing randomness himself by choosing his door initially at random. —Preceding unsigned comment added by Richard Gill (talk) 14:12, 16 February 2011 (UTC)

How do the words, 'Suppose you're on a game show..." affect the above response? I read that as equivalent to 'a fair die is thrown in a fair manner'. I would say every American who has watched 3 - 5 hours of game shows on TV every day since birth would have the same interpretation as me. Glkanter (talk) 14:18, 16 February 2011 (UTC)

The wording in the lead is a summary of referenced wording in the "Problem" section, i.e. we're not talking about what editors think but what sources say. For example, the sentence "The resulting set of assumptions gives what is called "the standard problem" by many sources" is referenced to Barbeau (2000), which says: "The standard analysis of problem M is based on the assumption that after the contestant makes the first choice, the host will always open an unselected door and reveal a goat (choosing the door randomly if both conceal goats) and then always offer the contestant the opportunity to switch." -- Rick Block (talk) 15:35, 16 February 2011 (UTC)

What a sad, valueless rebuke, Rick Block. My comments are wholly consistent with the sources you mention, as well as Selvin, vos Savant, and K& W. All of whom are Americans (U.S) referring to a puzzle made famous in American (U.S.) periodicals about an American (U.S.) game show. Unless you were making those comments to some other editor, which isn't clear at all. Glkanter (talk) 16:29, 16 February 2011 (UTC)

Yes Rick, I agree that *many* sources call these assumptions the standard assumptions. Now, here's a specific case: I objected recently to @Nijdam's choice of wording "almost all sources". Do you support my recent alteration of "almost all" into "many"? On the arbitration page I am being viciously attacked for COI and supposedly promoting my own research and for again becoming an active editor of the page: but I think I am not promoting own research at all: I'm doing my best to collaboratively edit the article according to wikipedia principles. (Editors who find my own publications in this area totally unimportant ought to remove them from the reference list as soon as possible, I wish somebody would do that. They used to be on the talk pages, in order to share information.)

Amusingly, Barbeau (2000) presents only a simple solution, one which makes no use of the "random choice of the host" assumption at all. Richard Gill (talk) 17:24, 16 February 2011 (UTC)

My point, Richard, is that a puzzle about a game show, and Barbeau's problem statement includes the words 'game show', the "random choice of the host" premise exists simply by the common understanding of the term 'game show'. You, Rick, glopk, kmhlmh or anybody else can continue to argue that point. For whatever reasons you so choose. I will no longer do so. No more than I would argue that '6 is a prime number'. Glkanter (talk) 17:38, 16 February 2011 (UTC)

I am not arguing, @Glkanter, I think we agree. Is the wikipedia article only for the layperson who learnt about the problem in a discussion at a bar, or is it also for the student of Statistics 101 who learnt about it in his class? I think it's for both. Richard Gill (talk) 16:28, 17 February 2011 (UTC)

Rick Block says above:
Barbeau (2000) [...] says: "The standard analysis of problem M is based on the assumption that after the contestant makes the first choice, the host will always open an unselected door and reveal a goat (choosing the door randomly if both conceal goats) and then always offer the contestant the opportunity to switch."  

And I repeat: (choosing the door randomly if both conceal goats).  –  Exactly as Marilyn vos Savant was underlining of having been her "starting point". Evidently Barbeau knew what he is commenting about. And then I read above:
Barbeau (2000) presents only a simple solution, one which makes no use of the "random choice of the host" assumption at all.
And we surely can take Barbeau to be knowing what he says, can't we? Being aware of the obvious requirement of the elementary supposition of "randomness", that he emphasized explicitly, as an elementary rule for any approach to give an answer to the famous question, he nevertheless presents a "simple solution".

Am I right in concluding that this obvious requirement of "randomness" that he emphasized explicitly, also for him is the elementary basis for the simple solution he presents?  What, imho, for Barbeau like for vos Savant evidently implies the premise of an unbiased host for the "simple solution" they present.

Rick Block, am I correct in reading it this way?  Regards, Gerhardvalentin (talk) 06:33, 18 February 2011 (UTC)

The solution Barbeau presents is this: "The contestant initially selects a door concealing a goat with probability 2/3. With a policy of always switching, she will win a car with this probability." This is the same sort of wording Grinstead and Snell use for what they call their simplified version (that analyzes the probability of always switching vs. always staying), and the same sort of wording Carlton uses for his "intuitive solution" ("imagine you plan to play ... and employ the switching strategy"). This solution makes no mention of and has no dependency on how the host chooses between two goat doors. It is not saying the (conditional) probability of winning by switching if you've picked door 1 and have seen the host open door 3 is 2/3. It is instead saying if you decide ahead of time to switch and switch regardless of which door the host opens ("with a policy of always switching"), then your probability of winning is 2/3. If you want to know the conditional probability given you've picked door 1 and have seen the host open door 3 you have to figure it out (somehow). This is the point Morgan et al., and Gillman, and Rosenthal, etc etc make. Hypothetically, adding "everything is symmetrical with respect to the doors, so the probability in any individual case must be the same as the 'always switching' probability" would be enough (although one might wonder how you know everything is symmetrical, even given that the host must pick randomly between two goats) - but the sources presenting simple solutions typically say nothing like this and make no mention of the critical assumption. As Rosenthal puts it "This assumption, callously ignored by the Shaky Solution, is in fact crucial to the conclusion" (where the conclusion is that the conditional probability of winning by switching is 2/3).

The distinction between the probability of winning "with a policy of always switching" and the (conditional) probability of winning given you've picked door 1 and have seen the host open door 3 is what Morgan et al. are referring to when they say "The distinction between the conditional and unconditional situations here seems to confound many." -- Rick Block (talk) 18:28, 19 February 2011 (UTC)

This is the same 'reasoning' you've been monopolizing this article with for years. In the 2/3 & 1/3 MHP paradox, the 50/50 host bias is a stated premise (it also comes from the definition of a game show). There is no requirement that any solution to a problem uses every premise (piece of information provided) to solve it. You're just making that BS up, sadly. The simple solutions *are* conditional. As I've shown with the 100% conditional decision tree derived from Carlton/Morgan/vos Savant/Selvin/Hall. You make grand leaps of assumptions, and cannot provide a reference that gives a conditional solution that says any of that crap above. Glkanter (talk) 18:46, 19 February 2011 (UTC)

Please @Glkanter try to concentrate on the real issue: how the wikipedia article should take account of the hard fact that the sources do not agree how MHP should be solved. You can't alter the fact (except by destroying internet and burning down all the university libraries in the world) that a load of sources state loud and clear that more work has to be done to deduce that the odds are 2:1 on winning by switching given you chose door 1 and the host opened door 3, than to deduce that the odds are 2:1 on winning by switching given only that you chose door 1.

Yes, Richard, I *will* 'try to concentrate on the real issue'. Can you please clarify what you mean about '...a load of sources...'? I understand there are many different solutions, from many disciplines. That's *not* anybody's problem. Except Nijdam's, perhaps. The problem is that the simple conditional solutions are being prominently and repeatedly bad mouthed with UNDUE WEIGHT and NPOV violations and OR. I get really tired of having to repeat this. Look at the Conditional solution section and all that simple-solution-bashing variants crap in paragraphs 1, 2 & 4, please. We've been over this before, countless times. Glkanter (talk) 20:48, 19 February 2011 (UTC)

A load of sources: Carlton, Morgan, Rosenthal; then also just about every introductory probability or statitics text I have ever looked into (and that's a lot). The book by Grinstead and Snell is quite typical. Read it. It's free, it's on internet, it's promoted by the American Mathematical Society. It's quite well written except in my opinion for the Monty Hall section, where they too engage in the rewriting of history which Morgan and his friends went in for. Try working your way through it, so you learn to understand the mind of your opponents. That's how to win a fight. Richard Gill (talk) 10:09, 20 February 2011 (UTC)

Yes, there are various solutions. We all know that. And agree they belong in the article. And 1 problem statement. And no critics of the simple solutions. Except in the minds of certain editors. Nowhere else, though. Glkanter (talk) 10:17, 20 February 2011 (UTC)

I recall you never ever answered my question whether *you* thought these two matters were the same or different. But I do remember that you were delighted when Boris Tsirelson told you that the single word "symmetry" (which is implied by indifference for a subjectivist like you and like most ordinary folks) was enough to bridge the gap. Later Rick pointed out to me that Boris' approach was already in the literature: William Bell (1992) criticised Morgan et al. for making a mountain out of a molehill. Altogether I found three pretty symmetry-based proofs in the literature and put them in the conditional section, as an alternative to the lengthy formula manipulation which is sometimes used as an exercise or an illustration for students learning formal probability calculus (you can find it where it belongs, as an illustration in the wikipedia article on Bayes theorem!). Richard Gill (talk) 20:19, 19 February 2011 (UTC)

Yes, that symmetry *should* satisfy Nijdam, as I understand it. To the best of my understanding, Nijdam still considers the simple conditional solutions wrong, however. Hence, it is of little value to me in these stalemated discussions. In any case, what you describe above should change the argument from 'wrong' to 'not written very well by this particular High Priest' (What?! How can that be!?) or ignorant commoner. But that doesn't seem to have happened, either.

Rather, I prefer to argue logic. Cutting out the High Priests. The only way the solution, from the contestant's SoK, is 2/3 & 1/3 is if he is told *before* he makes his door choice all those things that the host is certain to do (that is, the contestant is *not* playing a variant problem with different results). So the contestant, being a thinking, sentient being, just like us Wikipedia editors, can start analyzing and deciding right then, indifferent to the door #s. Or he can wait until later. Lastly, the 'Carlton' decision tree shows that simple solutions *are* conditional, and that the contestant *will* be facing 2 closed doors and 1 open door. And he's indifferent to the door #s. I've said all this thousands of times. All that stuff about before/after, etc is make up and/or grossly over-interpreted and/or grossly over-exaggerated BS, not derived from or representative of a significant minority of reliable sources. Glkanter (talk) 20:48, 19 February 2011 (UTC)

I too prefer to argue logic. Remember, Laplace builds probability calculus on logic. It is nothing more than logic, the logic of the concept "equally likely". And "equally likely" is usually invoked because of symmetry - physical symmetry of dice, coins, or symmetry of our knowledge, e.g. indifference. That's why I like my third alernative conditional solution best. It replaces the technical concept of "conditional probability" with the logical concept of "independence". Independence coming again from symmetry, coming from indifference. The door numbers are irrelevant. All that matters for the player is their role (both visible and hidden to the player). He's two times more likely to hit a goat than a car first time, so he should switch. End of story. Richard Gill (talk) 08:38, 20 February 2011 (UTC)

It really is all the fight between frequentists and subjectivists. The present intro is pure frequentist. The host's choice is random (the host uses a fair coin toss). No! The player is indifferent to the host's choice. For the player, the host might as well have used a fair coin toss. The intro to the article already is enforcing a particular POV. Richard Gill (talk) 08:38, 20 February 2011 (UTC)

Is there any chance you'll respond directly to the issue I raised above, rather than taking us on another long and useless trip through Richardworld?

• My first point is that there is only 1 MHP problem statement, the one that has an outcome of 2/3 & 1/3 [please, show me in the literature any others that are *not* variants], and that the contestant has been told the rules before he watches them played out. So from the single 'standard' problem statement, any of the solutions is acceptable, and none is unacceptable.

Until this issue is resolved, everything else is wasted energy. And it sounds like, despite it being OR from each and every one of you other editors, this is what you guys want to argue about. Except it's made up. It doesn't exist in the literature. All that F0 and S0 and stuff from Nijdam is bogus. It's phony. A contrivance.

• I entirely agree with your first point. (Of course, there are examples of solutions whose argumentation is faulty. Eg Devlin, who wanted to do the conditional version of the problem, but overlooked a crucial step. We shouldn't refer to solutions where the logic is patently wrong. Devlin himself retracted his wrong argument.) Richard Gill (talk) 10:12, 20 February 2011 (UTC)

Well, then help me put the kibosh on all that wasteful rhetoric that Nijdam, Rick and Martin go on endlessly about. Any source that has renounced a paper in public I would want to know about. I'm not aware of Devlin's retraction, I have read Morgan's, and I disagree with your interpretations of Rosenthal. Glkanter (talk) 11:00, 20 February 2011 (UTC)

• Tell me why Nijdam ignores what you and Boris have come up with, that the literature had all along?

• How can I tell you how Nijdam's mind works? It's incomprehensible to me. (Signed HP = High Priest Richard the Omniscient).

• Tell me why the Conditional solution section in the article is 90% about variants.

• I don't know why, they don't belong there. (HP)

• Tell me why you and the others ignore the 100% conditional decision tree that shows the simple solutions *are* conditional? Meeting Nijdam's 'requirement' that the contestant is looking at 2 doors?

• Because everyone else uses the word "conditional" in a different (and more technical sense) than you. (HP)

It's not more technical, they do that to make their argument seem stronger than it is. It's a confusion tactic, using vague words as meaningless 'jargon'. My terminology, with greater precision, shows the canard. I'm disappointed you don't agree with me, and seem to defend their tactics. Glkanter (talk) 11:00, 20 February 2011 (UTC)

• Tell me why the unconditional interpretation of the problem, and simple solutions are included in the Causes of confusion section.

• Tell me why. It's a mystery. Could it be part of Rick Block - Nijdam - Glopk attempt to hijack MHP from the people, to whom it belongs, and hide it in a Statistics 101 class? (HP)

• Tell me why Nijdam and the others continue to insist that *Probability* is the only tool that may be used to solve the puzzle. Despite the facts, and despite the literature.

• Tell me why. I imagine they didn't know that subjective probability is just logic, anyway. They are slaves of the very calculus which was invented to serve us, they don't see any other way to do these problems except through the "official calculus". But it's not always the best tool for the job. (HP)

• Tell me how Selvin and vos Savant are both equally too stupid to realize they can't solve the puzzle they each created with a simple table of all possible outcomes. But Selvin's a High Priest. How can he be a High Priest and be wrong? Oh nos!

• Selvin is not a high priest. He was a guy having fun telling his mates about a stimulating brain teaser. Which can be looked at in all kinds of different ways. I think Vos Savant is so smart that she didn't realise that some people would need to have the symmetry spelled out for them. Also, she is using probability in a subjectivist sense while the Morgan's and others are clearly thinking in a frequentist frame. (HP)

The guy is a math PhD writing in a PEER REVIEWED PROFESSIONAL JOURNAL, right?. Saying that he is not a 'High Priest', just like you, or Nijdam, or Rosenthal or Morgan, et al is silly. Any difference that *may* exist is meaningless to us commoner dumbasses.

Which misses the point, anyways. They wrote the problem. They know what they meant, fer crissakes! Glkanter (talk) 11:00, 20 February 2011 (UTC)

But please, stop talking in High Priest code that doesn't address these issues. Hopefully, they'll topic ban you along with those hustlers. Then, maybe some old editors will return. Then all the babbling can come to an end, and the article can become what it should have been all along. Glkanter (talk) 09:24, 20 February 2011 (UTC)

@Glkanter, please try to learn some of what you call High Priest Code. Did you read Laplace (1814) yet? He was writing for ordinary intelligent people, not for mathematicians! Did you study the wikipedia page on the interpretation of probability? Have you studied the first chapters of a decent introduction to probability theory, so that you finally know what people mean when they write conditional probability, and so that you come to learn and love Bayes' theorem? I suggest this not to make you change your opinion, but so that you can fight your opponents better. I'll answer your questions "in situ" in a moment. Richard Gill (talk) 09:44, 20 February 2011 (UTC)

Intelligence: the ability to make finer distinctions. How to increase intelligence: by reading and thinking. [3]

I've read plenty. I read your papers, even the one that contained the folklore about the dumb Americans that couldn't figure out the game in real life. And I've read all your contradictions, and vagueness, and all sorts of things. I've read the so-called critics. So, stop trying to belittle me. Now, are you done editing, so I can post my responses without getting another edit conflict? The 'preview' button works wonders. As do the browsers with the in-line spell check. They're all the rage, you know! Glkanter (talk) 11:00, 20 February 2011 (UTC)

Too bad you can't focus on the above items, as I have begged you to do forever. We could make the article so much better if we we worked together. But, I insist on not learning, and you insist on teaching. And never the twain shall meet. Of course, you also insist on talking about optimal solutions, game theory, minmax, dense jargon, your papers, and all kinds of stuff that will never appear in the front of the article, where we need to focus our attention. Glkanter (talk) 11:00, 20 February 2011 (UTC)

You see, Richard, right off the top of my head I came up with half a dozen issues that are negatively affecting the article. And improving the article, for many of us, is why we're editing on Wikipedia. And it looks like you agree with some of my ideas as stated above. So, no, I don't need some greater level of knowledge in order to meet my goals. I need less interference and obstructions from editors with bogus goals and agendas. Maybe after we take care of the low hanging fruit I would be interested in pursuing other aspects of the MHP. But not yet. That's not what I've invested 2+ years for. Any interest in being part of the solution, Richard?

@Glkanter: That's exactly the perfect throw, showing how the lemma should be :  –  Without bloodcurdling inconsistence and without absurdities. Please help to get it that way. Regards, Gerhardvalentin (talk) 11:50, 20 February 2011 (UTC)

There big difference there, StatProb a size restriction, that we do not have, so we could (and imho should) treat the problem in greater detail (properly structured of course, separating a fast route with the most important aspects from the more detailed treatments and distinctions).--Kmhkmh (talk) 13:55, 20 February 2011 (UTC)

Sure, Gerhardvalentin, I'll "help to get it that way". I'll tell you what I'll do. I make those exact edits I describe above. Then those other guys will all revert my edits, no matter how many times I put them back in the article. Then I'll get reported for edit warring, and I'll get blocked. Meanwhile, you, and Richard, and Martin will pretty much just stand around doing nothing while I get my ass kicked. Sounds good, no?

Oh, wait, we all already did that last summer. Glkanter (talk) 15:11, 20 February 2011 (UTC)

@Glkanter: Please make a new section down here on the talk page, titled "Glkanters proposal", showing your intended changes, and paying regard to others comment (say you what, I'll be very critical), and please try first to convince.  –  And as to me, I will do so alike. Because I'm not as experienced in editing like others. Regards, Gerhardvalentin (talk) 15:29, 20 February 2011 (UTC)

The minute I know that the other editors are on board, I will do that in a heartbeat. In the meantime, I want to delete paragraphs 1,2 & 4 from the Conditional solutions section, and that 2nd image in that section. Replace Carlton's solution in the Simple solution section with his actual quote, and add the decision tree inspired by his solution. Remove all mentions of simple solutions form the Sources of confusion section. Just have 1 solution section with no subheadings or biasing narratives - simple, formal decision tree, Bayes. That's a start, anyways. Glkanter (talk) 15:50, 20 February 2011 (UTC)

@Kmhkmh, I like your words: "properly structured of course, separating a fast route with the most important aspects from the more detailed treatments and distinctions",  and also I would prefer to enforce visibility and to avoid messy confusing jumping around within the convolute, repeating the same several times, and leaving out what all is about. But structured in separated sections, from important to less important (details and variants). And noting that not all sources are equally important to fully understand and appreciate the paradox. Gerhardvalentin (talk) 16:35, 20 February 2011 (UTC)

 

Getting the Answer to the Question – Solving the Problem

Imo the problem (not to make it a bad-joke-question) is almost always solved using general assumptions, at least using the most general assumption (said by vos Savant to be implicit in her question and her own answers) that the host is certain to open a door showing a goat (which he can always do, because he knows the location of the car). The contestant has no knowledge on the location of the car, so usually it is either assumed that the car is equally likely to be behind each of the three doors, or that the player's own choice is completely random. Some solutions add to this the assumption that, if the host has two goats to show, having a choice which goat door to open, he is equally likely to open either. It's a question about just "one game show". Having no better knowledge at all, and just to stay serious you have no other choice, as you never can expect that show will or "must" be repeated, just to suit your special requirements.

Note that "must randomly choose" is putting randomness into the host's actions. This is obviously a frequentist idea of probability. But most people look at MHP subjectivistically, so chance comes from their lack of knowledge about givens, their lack of knowledge about the world, it is not in the physical world itself.

So, for a subjectivist, the two "equally likely" things (location of car, door opened by host) are not "extra assumptions". No, they are deductions or consequences of the problem statement.

And that's the basis of a reasonable answer to the question whether to switch or to stay. Gerhardvalentin (talk) 09:02, 20 February 2011 (UTC)

Very good, Gerhard! I frequently tried to change the present intro "Although not explicitly stated in this version, solutions are often based on the additional assumptions that the car is initially equally likely to be behind each door and that the host must open a door showing a goat, must randomly choose which door to open if both hide goats, and must make the offer to switch" into something like the following:

Though not explicitly stated, Vos Savant intended, as indeed most readers interpret her question, that we are to suppose that the host must open a door showing a goat -- something he always can do, since he does know the location of the car. Almost all solutions are furthermore based either on the assumption that the car was hidden at random, or that the player chose his door at random. Alternatively, since we have no information beyond what Vos Savant gives us, for us the car is initially equally likely to be behind any of the three doors (see Probability Interpretations). Some solutions furthermore assume that if the host has a choice of door to open, he determines his choice at random. Alternatively, in this situation, for us either choice is equally likely, since again we have no information beyond Vos Savant's words.

The present text is already biased to a frequentist view of probability and towards the conditional solutions. Of course, in introductory statistics texts, in their chapters on Bayes' theorem, such a bias is quite natural. But it is not a natural bias to the ordinary person hearing about MHP in a discussion at a bar or at a party. The problem has to be solved by logical analysis, not by the calculus of probability. Laplace (1814) built the calculus of probability explicitly on the logic of the concept "equally likely", the logic of symmetry, of indifference. And Laplace's probability is the sort used by ordinary people in their day to day lives. Back to basics! Richard Gill (talk) 09:37, 20 February 2011 (UTC)

Recall Martin Hogbin's words: no one can think deeply about MHP without pondering on the meaning of probability. Well, wikipedia has an article on that. MHP is not isolated from the rest of the world, it is not isolated from the rest of wikipedia. Good articles have plenty of cross-links. Nijdam should be finding wikipedia articles which explain his point of view, and if he can't find them, he should source the ideas and write the article.

Now, how about deleting that passage in the conditional solution with all the formulas? It is totally superfluous since already on wikipedia where it belongs (and where of course it came from!) as an illustration of Bayes theorem. Seems no one ever reads it either, last time I looked it was packed full of gobbledygook (even Nijdam approved of my corrections that time). Richard Gill (talk) 10:02, 20 February 2011 (UTC)

+1. Yes, because frequentist is just "one" aspect, dishonestly overstressed in the article. Unnecessarily filling and dominating the whole lemma. Just based on forever unknown (but firmly said to be given) repetitions. Farcical. Yes, the frequentist's aspect has to be arranged and integrated as it belongs. To straighten the hitherto absurdness of the lemma.

And to link accordingly to where it is legitimate.  Gerhardvalentin (talk) 10:17, 20 February 2011 (UTC)

@Richard: I'd wish you'd give that issue a rest, the objections from above haven't changed and as i said before is just (mostly needlessly) created another edit conflict. You should simply accept that some people prefer this treatment - period.

We immediately saw "objections" from three objectors, but no comments by anybody else, that's why I repeat this point here. Moreover, I find the objections very poor indeed. I suspect that they are more to do with conservatism than with caring for the interests of the readers of the article. So naturally, those who object will go on objecting till kingdom come, but I think we're all agreed the article is bloated with seondary detail. That derivation is a detail. And: it is elsewhere on wikipedia, so it can be replaced by a link! It is located where it belongs and where it came from: illustrating the workings of Bayes' theorem. It does not illustrate MHP. On the other hand, various proofs with symmetry or independence or the odds form of Bayes' rule give insight into Monty Hall Problem which you could share with your grandma. And insight into the difference between simple and conditional. And all of them used in the past by authoritative, clever, insightful writers to give complete and elegant analyses of the standard problem. The article as a whole doesn't have enough links to the outside world inside wikipedia. It is a little microcosm for editors who only care about MHP and don't know about anything else. That's not good for an important encyclopedia article. And it needs to be linked to, from elsewhere, for the same reason. Richard Gill (talk) 20:06, 20 February 2011 (UTC)

I know that you think that. The issue here is that you learn to live with others differing here and let it go.--Kmhkmh (talk) 21:26, 20 February 2011 (UTC)

I can live with others differing in their opinion from me, and I do let that go. I am angry that a minority is holding on to their minority point of view which has become engrained into the present article, blocking a majority from allowing the article to progress back to a balanced reflection of the literature. Richard Gill (talk) 06:42, 25 February 2011 (UTC)

As far as problems frequentist arguments are concerned, one should keep in mind although apparently favoured by some here, it is not without problems eithe. The assigning of priors but not be as obvious/justified as one might think. Another thing is that the concept of various implicit rules (don't show a car, don't open the candidates door, follow some rule at all) are somewhat associated with the notion of repeatability (the very nature of law or rule is repeated application in a way). Now if one insists one one time event nature, one might argue as well that you lose the rules and even symmetry (some publication(s) actually make that argument) and as consequence you loose 2/3 and simple approach altogether. --Kmhkmh (talk) 13:44, 20 February 2011 (UTC)

I disagree, Kmhkmh. I'm talking about the standard MHP, not some academic variation. Given only the words of Vos Savant (including her suggestive and parenthetical "say, Door 1", and "say, Door 3") together with the uniformly agreed addition that the host will certainly open a door becaues he always can, the subjectivist's prior is completely determined. I'm talking about the basic standard MHP, the one for ordinary people talking in a pub or at a party. Who are told vos Savant's question and the universally agreed clarification. And nothing else. Its a one time question with no other information, no previous experience. (Credit to Glkanter for hammering away at this point). Of course one can change the problem and things will become problematic. That's fine for some section for academic readers about variants.

BTW I agree with Martin Hogbin that for the frequentist, the same basic MHP is insoluble, precisely because we have no information to go on. The smart frequentist will therefore randomize his choice in advance of the show and switch whatever. His unconditional probability is 2/3, he doesn't know and doesn't care what his conditional probability is. He knows he's doing the best he can do, anyway (game theory) so why bother. Richard Gill (talk) 20:15, 20 February 2011 (UTC)

Well the problem everybody is talking about is first and foremost Whitaker's (and not vos Savant's) question. And the "uniformly agreed addition" is not uniformly agreed in literature either. It is just the most common assumption (or simplification) generally considered as reasonable to get a handle on the problem. And the real Monty Hall for instance did not always open a door nor behave strictly regular in anyway.--Kmhkmh (talk) 21:23, 20 February 2011 (UTC)

Whitaker's question was I’ve worked out two different situations based on whether or not Monty knows what’s behind the doors. In one situation it is to your advantage to switch, in the other there is no advantage to switch. What do you think?

Marilyn explained regarding her own question that the words "say, Door 1", and "say, Door 3" were added to help you visualise the situation but are not part of the question. They are parenthetical comments. They can be deleted.

Morgan et al. and later authors changed the problem and then told Marilyn off for giving the wrong solution. Richard Gill (talk) 06:36, 25 February 2011 (UTC)

What you say above is consistent (derived from?) my insistence that there is only 1 MHP, *not* as you have said in the past, [paraphrasing] "an ever evolving MHP that no longer belongs to Selvin or vos Savant". I'm trying to make this very point in the arbitration to support my contention that Rick's & Nijdam's POV about the 'critics' is *not* supported by the reliable sources (they [except Morgan, who just flat out lies] change the problem in order to teach conditional probability, Rick incorrectly reads this as criticisms), and I put together that table in my evidence that some arbitrator decided to collapse. I think you'd be doing the arbitration, the article and me a lot more good if you would write something in your evidence section in support of what I have been saying all along, and which you seem to now agree with. Glkanter (talk) 11:23, 25 February 2011 (UTC)

My friend, please learn to make distinctions. This is the mark of intelligence.

There are Vos Savant's words, they are fixed, and will never change. How they are interpreted is not fixed. Morgan et al. interpret Vos Savant differently from how you do. Personally, I think that there are many legitimate formalizations of her question. Some more popular than others. My personal opinion is unimportant. Which formalization is most popular, can change and does change, in time. The meta-MHP is the problem to formalize Vos Savant's words responsibly into a problem which allows logical analysis. The answer is the answer. The path between formalization and answer has to be a logically correct deduction from starting point to end point.

So we have to distinguish three things: The exact problem which we are going to solve. A logical deduction from that problem. An answer which is the endpoint of the logical deduction. Richard Gill (talk) 14:49, 13 March 2011 (UTC)

Please provide examples of sources that give different problem statements (and hence, different premises than K & W or Selvin) that have a result of 2/3 & 1/3. The only ones are the so-called critics, not the mainstream. I have a table that shows this in the evidence section. You'll need to un-collapse it. Glkanter (talk) 15:13, 13 March 2011 (UTC)

Morgan, et al, but not the other authors, say they are quoting vos Savant, use quotation marks, then change vos Savant's problem by improperly and deceptively eliminating the 'say' part of 'say door #1' and 'say door #3'. They're bums. And their journal's peer-reviewers aren't much better. Glkanter (talk) 06:50, 25 February 2011 (UTC)

FAQ

When you fire up wikipedia MHP Talk page from a smart-phone you get to see the FAQ in all its glory, the rest is initially hidden. So I finally noticed we had an FAQ and I read it and added some stuff on symmetry. There is also a lot of symmetry at my latest peer-reviewed earth-shattering brilliant masterpiece contribution to the Annals of MHP Studies ;-) [4]. Extended version at [5]. Those texts are based on a lot of interaction with Boris Tsirelson at [6]. Richard Gill (talk) 10:27, 20 February 2011 (UTC)

Citing accurately

It is just a minor thing, but still in the symmetry solution we have:

In Tierney (1991), the mathemagician and Stanford professor Persi Diaconis stands up for vos Savant,

However in Tierney's cited article Diaconis does no such thing. What he in fact does is being sympathetic towards his colleague Sachs (not vos Savant) for getting it wrong and later in the article he even explicitly argues that strictly speaking the problem cannot be solved without knowledge about the host's behaviour rather than siding with vos Savant. Now Diaconis might have explicitly supported vos Savants approach elsewhere, but he certainly does no such thing in Tierney's article. In short either the sentence regarding Diaconis needs to be changed or another source would be required.--Kmhkmh (talk) 21:56, 20 February 2011 (UTC)

Sorry. That was careless of me. I was reading an article from New York Times, 1991 or so, and thought it was the same as the one referenced in the article (Google search on "Diaconis Monty Hall"). I took the wording "Diaconis stands up for Vos Savant" right from the article. I know Diaconis was also interviewed in another article. By the way, the point about the host's behaviour, Diaconis' point, is that we need to be told that the host will always open a different door and reveal a goat. And if we are frequentists we need explicit symmetry (uniform-random) assumptionsm (but Diaconis is usually Laplacian, and then symmetry follows from lack of information to the contrary and/or Vos Savant's indication that the specific door numbers are not meaningful. Which she also wrote somewhere, later). Diaconis says he had written on the MHP problem before Vos Savant had made it famous, he knew the Selvin version from years earlier. More literature search needed. Richard Gill (talk) 14:28, 21 February 2011 (UTC)

PS, one could also refer to Yours Truly (2011), but I am not going to promote that. COI, OR. Question is, is it useful? Since we're both mathematicians, Kmhkmh, we might privately agree on the Mathematical Truth. You work in geometry. Symmetry in geometry is as old as geometry, just as symmetry in probability is as old as probability. But I'm about to take a hopefully restful WP:WIKIBREAK. Richard Gill (talk) 14:41, 21 February 2011 (UTC)

I corresponded with Persi about all this. He agrees with me about the content. Of course a private communication is not a reliable source. But I just mention it, anyway. Richard Gill (talk) 06:32, 25 February 2011 (UTC)

See below for why non-uniform choice (by the host) should not affect the players' game expectation overall, although it does still affect their expectation after the door is opened. Rich Farmbrough, 11:35, 27 February 2011 (UTC).

Useful resource

Jason Rosenhouse's book's first chapter is on internet: [7]. (I don't agree with him on many issues, but still, this must be considered a highly "reliable source"). Richard Gill (talk) 11:30, 23 February 2011 (UTC)

N doors and biased choice

• I changed the N door section to reflect that N/p is the important number, not N or (number of doors -1). This is a factual correction (quite a risk saying that on Monty Hall talk.. if I'm wrong - revert obviously).
• The biased choice - if the host has a 100% bias to door 2 (i.e. he will open it whenever he gets the opportunity) then the "value" of his disclosure is split unevenly depending on the door he opens. Door 2 - no information probabilities are .5 .5 , door 3 full information probabilities 0, 1 (always switch, always win). The net expectation is unchanged at 2/3, and indeed (sensibly enough) the exception contribution of each door opening event stays at 1/3. We should include in the table "Other host behaviors" enough information to show that the biased choice does not a priori give the player an advantage, in some cases it works out as a disadvantage, in others as an advantage, on average makes no difference.

(Detail: Bayes theorem confirms 1/(1+p) as the expectation (from switching) given door 2 is chosen. The probability of the host choosing door 2 is 1/3+p/3+0/3=(1+p)/3, the expectation contribution form door 2 is therefore E₂= (1+p)/3 . 1/(1+p) = 1/3, by symmetry E₃ =1/3, the total expectation E= E₂+ E₃ = 2/3. RF.)

• "proves that they form the minimax solution." should this read "proves that they form a minimax solution." ? Given the above I would say yes - there are other strategies that work equally well for the TV company.

Rich Farmbrough, 11:27, 27 February 2011 (UTC).

What the cited source (Granberg's appendix to vos Savant's book) says about the N-door variant is that if the host opens one door there is always an advantage, but this one door advantage approaches 0 as the number of doors grows, and also that if the host opens all incorrect doors except 1 (p=N-2), the advantage increases as the number of doors increases and approaches 1. It's clearly true that if you hold p constant, as N/p grows the advantage drops to 0, and if p=N-2 then as N increases N/p approaches 1 (not 0) and the advantage approaches 1 - but this isn't what the source says. I think essentially everybody agrees (see Wikipedia:Arbitration/Requests/Case/Monty Hall problem) one of the main issues we've had with this article is editors injecting their own conclusions. How about if we either change this back to what it said, or attempt to clarify the wording while sticking to what the source says (and not adding any essentially WP:OR conclusions)?

Regarding the Nash equilibrium, I suspect when the current ruckus has died down the bit about this will be expanded into an entire section on "game theoretic" approaches. It's probably not worth worrying too much about "the" vs. "a" until then (although the specific variant currently described in the table is Richard Gill's rather than either of the variants discussed in the Mueser and Granberg paper, and for this variant "the" might be more appropriate than "a"). -- Rick Block (talk) 00:34, 28 February 2011 (UTC)

Clarifying is good. What was there before, though doubtless it was supposed to mean what you said, seemed to me wrong as it stood. Given the nature of the article linguistic precision reflecting the mathematical precision seems a good idea. I am sure you can clarify the wording without introducing OR. I was reluctant to actually go and read the sources, although I may look at Gill's if I find time, I'm sure you know what you are talking about.

Again as far as the biased door opener is concerned, I'm sure that my little bit of OR is not very O - and all I am suggesting is that we don't give the wrong impression - if RS say or imply something demonstrably false (which I am not assuming they do) we do not have to reflect it, if they don't say or imply it we certainly shouldn't.

None of these are big deals, and can wait happily for the page regulars to resolve, unless I find time weighing on my hands, which is unlikely. Rich Farmbrough, 01:32, 1 March 2011 (UTC).

I read most of Gill's 1 March 2010 paper, he does say "the" in "proves that these are the respective minimax strategies" but does not support uniqueness. He also mentions the extremum where the expectation from switching is a half, so he has considered at least the edge cases, and that also can be cited into the article. maybe he also states the expectation calculation I gave above elsewhere, which would be cool. Of course he cites these (talk) pages as a source so prepare for self-ref arguments! Rich Farmbrough, 23:53, 4 March 2011 (UTC).

Are you OK with the clarification I made [8]? This stretches the source a tiny bit (the source says "... if all of the incorrect doors except one are shown, the advantage of switching increases as the number of doors increases and approaches 1.0 when the number of doors is very large" - clearly what is meant is that the probability of winning by switching approaches 1.0 rather than the "advantage" of switching).

Regarding the effect of a biased host, Morgan et al. (in their conclusion) show that the unconditional probability of winning by switching is 2/3 regardless of the host's bias. This same paper shows the probability of winning by switching regardless of the host's bias is never less than 1/2 - so there's no particular reason to reference Richard's paper for the extreme cases. BTW - a fully general Bayesian solution, with a discussion of numerous variants involving host bias and other variables, is presented in a paper by Puza, Pitt, and O'Neill (Teaching Statistics, v27(1), Spring 2005). -- Rick Block (talk) 06:37, 5 March 2011 (UTC)

"The minimax solution" indeed should, of course, read "a minimax solution" unless we have also proved uniqueness. We agree that there are strategies of host which guarantee him maximally risk of 2/3 to lose the car, and strategies of player which guarantee him minimally chance of 2/3 to win the car. This proves that the 2/3 is "the" value of the game, by von Neumann's minimax theorem. Now, to any *other* strategy of the player it is clear that the host can figure out a strategy which decreases the host's risk from 2/3, but to do that, he must use a different strategy from "random, random". Conversely, for any other strategy of the host, the host can figure out a strategy which does better than 2/3, but it has to be different from "random, switch". This proves that the minimax solutions of both host and player are unique. So: yes, one may talk about THE minimax strategies, but one has to do a bit more work to check that indeed this is legitimate. Richard Gill (talk) 14:42, 13 March 2011 (UTC)

Morgan, "...the answer is 2/3, period.", and "False-ness"

In their 1995 paper, Morgan concludes:

"2. CONCLUSIONS"

"In general, we cannot answer the question "What is the probability of winning if I switch, given that I have been shown a goat behind door 3?" unless we either know the host's strategy or are Bayesians with a specified prior. Nevertheless, in the vos Savant scenario we can state that it is always better to switch. The fact that Pr(W | D3) ≥ 1/2, regardless of the host's strategy, is the key to the solution."

In their 2010 response to Martin and Nijdam, Morgan writes:

"To wit, had we adopted conditions implicit in the problem, the answer is 2/3, period."

Which can only mean that they recognize that from the contestant's SoK, doors 2 and 3 are equally likely to be opened. Which leads to the question, how many of the various claims of 'false' that Morgan makes in their paper do they no longer consider false?

• vos Savant's proof of her original solution
• vos Savant's simulation
• All 6 solutions that they reject

Posted by Glkanter (talk) 02:50, 12 March 2011 (UTC)

Subjectivist view missing

Perhaps Morgan (unlike Rosenthal) actually had a point in 1991. But his back and forth on the issue, failure to notice a highly questionable result due to an error in calculation (corrected in 2010!) and frankly muddled writing, makes them a questionable source. I think that this (WP:RS, possibly wp:secondary too, but this debatable) explains best the issue with the interpretation of probability as applied to this problem (please read all three pages in the source)--and, yes, under the subjectivist interpretation, the answer can even be 1/2 (because, it's argued, the player may not know they're playing MHP, so they may think they are playing Monty Fall). This should be covered in the article in the variations section, preferably under a "subjectivist interpretation" sub-section. However, the horribly complex proofs in this "FA" (which are still entirely frequentist) are of questionable value. I'm not at all surprised that most of the vituperation on this talk page has been on the completely wrong issue(s). Tijfo098 (talk) 10:09, 13 March 2011 (UTC)

Game-theory solutions WP:RS (viewpoint mostly missing)

Chun 1999 also in American Statistician, covers this correctly (by solving a much more general case than even Morgan, and particularizing it to that):

“ Thus, the answer is clear in the vos Savant scenario; switching doubles the player's chances of winning from 1/3 to 2/3 regardless of p and q. ”

(p is the preference of the host for one of the doors when he has a choice; p+q=1) And I bothered to prove that myself, duh. In other words, the host's strategy does not matter at all; no need to assume he's equally likely to open either goat as said in the lead for instance. So much for this being a "comprehensive FA". Tijfo098 (talk) 09:36, 14 March 2011 (UTC)

You should quote a bit more of this source:

“ In the vos Savant scenario, if follows from (11) that the player's unconditional probability of winning is z=1/3 + (qx + py)/3 and the player's optimal strategy is (x* = 1, y* = 1). Thus, the answer is clear in the vos Savant scenario; switching doubles the player's chances of winning from 1/3 to 2/3 regardless of p and q. When the host specifically opens door No. 3, which has no car behind it, the conditional probability of winning the car is shown to be ... ”

The fact that the unconditional probability is 2/3 regardless of p and q (BTW, Morgan et al. show this result) is what the following sentence and subsequent text in the Conditional solution section means:

The popular solutions correctly show that the probability of winning for a player who always switches is 2/3, but without additional reasoning 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.

Making this somewhat more clear has been one of the issues in mediation. -- Rick Block (talk) 14:26, 14 March 2011 (UTC)

[if I'm not clear enough, please ask and I'll elaborate, although I'd rather discuss some actual changes rather than the problem & solution, which we both clearly understand well enough already]: the conditional probability (z_c in Chun) can indeed be 1/2 in some cases (2/3 of them overall) if the host prefers a certain goat door (aka Monty Crawl in Rosenthal). But "[...] in this situation, the player cannot gain (or lose) by switching." (Chun, 1999, p. 47) So it obviously doesn't change the player's strategy, because in the remaining cases of Crawl (which happen 1/3 of the time overall), switching wins with probability z_c=1. In more mechanistic terms: if the host has a preference for a goat door, there's a redistribution of the unconditoinal probability of winning z between two groups of cases (that will have different z_c), but this redistribution is subject to the constraint that the sum is always 2/3 and the conditional probability in either group cannot be less than 1/2. This is indeed fairly messy to explain in plain English, and it's also not something that is assumed to be "standard" MHP by plenty of authors: Rosenthal, [9], [10]; I can surely find more, but surely others think this is what Savant meant for some reason. There's also the question of whether the player knows what the host's strategy is. I've not seen Savant state that. Since none of this actually affects the player's strategy, for didactic reasons (assuming the Wikipedia audience is a general audience), I'd prefer we leave the more complex discussion (Monty Crawl/Monty Small) as a variant rather than making it the standard problem. Perhaps we should mention in the lead that not assuming that Monty "must randomly choose which door to open if both hide goats" assumption changes the conditional probabilities for cases (some go up some down), but ultimately does not alter the player's optimal strategy, because the conditional probabilities never drop below 1/2 for switching. I deleted part of the "aid to explanation" section that actually solved the Monty Small problem (to use Rosenthal's terms). Using that as aid to understanding what the lead sets up as a simpler problem is a case of: "let me explain to you Galois theory so you better understand how to solve the quadratic equation" (Yes, I'm exaggerating a bit, but I hope you get the point.) Tijfo098 (talk) 15:42, 14 March 2011 (UTC)

Succinctly: Morgan's "vos Savant scenario" (as cited by Chun 1999) is actually Monty Small in Rosenthal 2005/2008 terminology. Tijfo098 (talk) 16:29, 14 March 2011 (UTC)

The plain English way to describe the difference between the unconditional and conditional probabilities (per Gillman) is whether the player's decision point is before the host opens a door, or after the host opens a door (with knowledge of which door the player picked and which door the host opened). The point that Morgan et al. make (and Gillman, and Grinstead and Snell, and many others) is that since the decision point is clearly after the host opens a door, the probability of interest is the conditional probability - for example, the probability of winning by switching for a player who has picked door 1 and has seen the host open door 3. This probability is 2/3 only if the host picks evenly between two goat doors, or if you change the problem to be restricted to the player's state of knowledge (and assume the player is ignorant of any host's preference - which is in effect moving the decision point to before the host opens a door), or if you (less realistically) assume the doors are indistinguishable (which, again, effectively moves the decision point to before the host opens a door).

I have suggested unifying the two different solution sections several times, e.g. [11] (in the show/hide box). Would this change help address the point you're bringing up? If not, can you propose some other change(s) that would? -- Rick Block (talk) 17:01, 14 March 2011 (UTC)

Well, I'm going to make some baby changes in that direction. At least all the back and forth on the symmetry argument (gap) in informal proofs should be in one place; it's in at least 3 different places now! Also, how do you feel about renaming "simple proofs" -> "informal proofs". The other are not formal proofs, but are more formal in the sense of mathematical proof#Nature and purpose. Tijfo098 (talk) 17:49, 14 March 2011 (UTC)

Picture edit request

Can someone change "total probability" to "case probability" (or perhaps "branch probability") in File:Monty_tree_door1.svg? Tijfo098 (talk) 17:49, 14 March 2011 (UTC)