Talk:Monty Hall problem/Archive 29

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Distinguishing between different scenarios

You are right, Scharzwald. Different scenarios must clearly be kept apart. No way out, just to show respect for the readers.

That's why the introduction ("extended problem description") of this world famous paradox explicitly says that the car and the goats were placed randomly behind the doors before the show and in any case the host is to open a door showing a goat, offering a switch to his second still closed door, and nothing is known about any possible host's preference, if in 1/3 he should dispose of two goats, as he then chooses one uniformly at random.

This lineup ensures that no additional information (whatsoever) can arise by the host's opening of a losing door, because this scenario ensures that we’ve learned nothing to allow us to revise the odds on the door originally selected by the guest (see Falk). Absolutely nothing. In this scenario the odds on the door originally selected by the guest remain unchanged 1/3, as we did not get additional information on the actual location of the car. Nothing to allow us to revise the odds on the door originally selected by the guest (see the academic literature).

In this lineup / scenario the only correct answer forever will be and forever only can be based on the average probability of 1/3 by staying vs. 2/3 by switching.

Irrelevant whether the guest selected door 1 and the host opened door 2 or door 3, and
irrelevant whether the guest selected door 2 and the host opened door 1 or door 3, and
irrelevant whether the guest selected door 3 and the host opened door 1 or door 2, and this lineup ("in any case the host is to open a door showing a goat") also excludes the forgettable host who will show a goat only in the piffling subset of 2/3 when by chance he just "happens" to show a goat and not the car. Once more: Neither the guest's initial choice of door nor the host's opening of a special loosing door reveals "additional information" on the actual location of the car in the famous paradox that says

– "Just only two still closed doors, one of them containing the car, the other one containing a goat, but chances by switching not 1/2 but 2/3." – Firmly assured by this lineup given in the introduction of this world famous "paradox", just have a look to the actual academic literature.

Of course, besides this explicit lineup, there could be quite other conceivable scenarios, quite other conceivable versions and variants, unsubstantial for the answer asked for in the famous paradox: That quite other variants / scenarios, not withstanding the lineup given in the introduction of this article, will give unsubstantial absurd so called "closer accuracy", depending on aberrant relative assumptions, but not affecting the famous paradox in any way, is quite another aspect. Those variants are used in maths classes on conditional probability and in numerous textbooks on conditional probability, but unnecessary and not being a tangent at all to the decision asked for in this famous paradox.

And you are right, the extremely biased host, in exceptionally opening his avoided door, shows that the chance to win by switching is 1 at max., and in opening his preferred door, the chance never can be below 1/2, so on average 2/3 though. Thus it is wise to switch in any single given game. "The more biased, the better". Not being a tangent at all and completely unsubstantial to the decision asked for in the famous paradox. See the academic literature.

But if, outside the paradox, he should offer the switching only in case the guest should have chosen the car, the odds by switching is 0.

Those other variants / scenarios are not the world famous paradox. The article should clearly show this, but doesn't. We finally should stop to higgledy-piggledy commmix the clear lineup given in the introduction of the world famous paradox with quite other optional variants, with quite other scenarios, never representing the world famous paradox. This split is not honored up to now here, and that's the long lasting and never ending conflict of the article that should stop to unnecessarily be perverted to a lesson in conditional probability theory.

Some are eagerly avoiding to show this plain and clean split, even though such clean split is necessary in respect for the readers. Thank you for your contributions, they clearly show this tohu-bohu-dilemma. Regards, Gerhardvalentin (talk) 10:42, 16 May 2012 (UTC)

Minor quibble: I don't like the language about "no new information". If there were literally no new information at all then it could be ignored and there would be no reason to switch. Although it is not new information about door #1, selectively revealing a goat behind door #3 gives us important information about door #2: in the likely event that the initial choice was wrong, that is where the car must be. ~ Ningauble (talk) 16:30, 24 May 2012 (UTC)

What are you trying to say, Ningauble. No one but you says "no new information". No one says so, but you. Please read what you are controverting: The host - if in 1/3 he should dispose of two goats, chooses one uniformly at random (Extended problem description). In this scenario the odds on the door selected by the guest remain 1/3, as - besides seeing a losing door - we did not get additional information on the actual location of the car.. Falk calls this fact "we’ve learned nothing to allow us to revise the odds on the door originally selected by the guest".

And do you really mean what you say: "Although it is not new information about door #1, selectively revealing a goat behind door #3 gives us important information about door #2: in the likely event that the initial choice was wrong, that is where the car must be." - I'm asking because you just removed/deleted this (your!) conclusion contained in the "truth table" from the article, calling this (your!) conclusion "unsourced material". So the tohu-bohu-dilemma of the article is obvious to persist. --Gerhardvalentin (talk) 00:12, 25 May 2012 (UTC)

It was you who wrote above that "no additional information (whatsoever) can arise by the host's opening of a losing door", and "we did not get additional information on the actual location of the car", reiterating that "Neither the guest's initial choice of door nor the host's opening of a special loosing door reveals 'additional information' on the actual location of the car" [emphasis in original]. In your objection immediately above, you continue to equate "the odds on the door selected by the guest remain 1/3" with "we did not get additional information on the actual location of the car" [emphasis in original]. I believe you are trying to say something that is correct, but I was quibbling about the way you said it.

In a nutshell, I mean that no new information about door #1 is not the same thing as no new information about the car. Deducing that the car is twice as likely to be behind door #2 in this scenario requires more information than if Monty never opens a door.

Yes, I meant what I said there. The so-called "truth table" was not mine, and what I removed was an unsourced derivation of the conclusion. I do not dispute the contributor's general thesis, though it would have needed a substantial rewrite for clarity and precision. Unfortunately, I am not personally aware of any citable sources on which to base a rewrite, and the contributor did not cite any. ~ Ningauble (talk) 17:57, 25 May 2012 (UTC)

Thank you for your reply, Ningauble. Do you agree that in the said scenario (if in 1/3 the Host should dispose of two goats, he then chooses one uniformly at random), by his opening of a losing door we’ve learned nothing to allow us to revise the odds on the door originally selected by the guest? -- Gerhardvalentin (talk) 09:11, 26 May 2012 (UTC)

Yes, I have said so quite plainly above, and am not sure why you think I was unclear about it.

Aside: The real challenge before us is to find a way, supported by citable sources, to help readers understand why this is so. There are several ways to understand this, but many people find them elusive. It is an obvious implication of selective information for those who appreciate it, but this is a notorious blind spot for many people. It is a simple consequence of symmetry in the scenario, but the very simplicity of the concept makes it too subtle for some to appreciate. It can be backed into as a consequence of conditional probability but, as many sources attest, this logic is poorly understood by the general public (and, I would say, even by many who have completed a college course in statistics).

It is said that the defining characteristic of a good riddle is that the answer is only obvious when you see it. Unfortunately, in this riddle many people still don't see it after it has been explained. ~ Ningauble (talk) 14:52, 27 May 2012 (UTC)

You are right, the paradox is based on the symmetry in the said scenario, this should be made clear for the reader. Quite other thinkable asymmetric scenarios, outside the famous paradox, should not be admixed but should be shown quite distinctive. I appreciate your view. And, as you agree that in the well-defined scenario we have learned nothing to allow us to revise the odds on the door originally selected by the guest, do you agree that the odds on the entity of the group of unchosen doors, as a whole, did not change likewise, in this well-defined scenario? Gerhardvalentin (talk) 09:54, 28 May 2012 (UTC)

I have said so before, and I quite fail to see the point of this line of questioning (especially insofar as you have even been questioning whether I mean what I say). What is the point you are trying to make? Does it relate to "discussing changes to the Monty Hall problem article itself"? ~ Ningauble (talk) 19:03, 28 May 2012 (UTC)

Thank you again for your reply. To repeat: If in the correct standard scenario, after the host has shown a goat, we’ve learned absolutely nothing to allow us to revise the odds on the door originally selected by the guest, then – in this well-defined scenario – we've also learned nothing to allow us to revise the odds on the entity of the group of unchosen doors, as a whole. And as to your question, I just could not believe that you, on the one hand, deleted the truth table in the article that says: Given that the guest has chosen a goat and the host opens one of his doors showing a goat then the car is to be behind the host's second door, and immediately thereafter you repeat this deleted conclusion here as your argument in the talk page ... in the likely event that the initial choice was wrong, that is where the car must be. I am focusing the miserable state of the article that actually still says "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." Different scenarios should be kept apart, just to show respect for the readers. Gerhardvalentin (talk) 21:04, 28 May 2012 (UTC)

Once more: The article still is obfuscating, not keeping different scenarios clearly apart, never clearly saying what it is just talking about. The world famous paradox forever can only exist in its unique famous clear scenario, but nowhere else. Presenting antique sources with differing stories, addressing quite differing scenarios, this has clearly to be separated from the main part on the paradox itself with its famous unique scenario. In respect for the readers. Gerhardvalentin (talk) 01:56, 19 June 2012 (UTC)

Gadfly1974's revisions

I apologize for deleting from and not just adding to the "Other Simple Solutions" section. That was rude of me. My excuse is it was my first contribution to Wikipedia, but I didn't handle it well. gadfly1974 (talk) 15:06, 2 June 2012 (UTC)

Gadfly, the content you are adding is unacceptable as well. Please do not add it to the article again. --RacerX¹¹ Talk to me^{Stalk me} 15:12, 2 June 2012 (UTC)

Psychology

Psychology should also be taken into account here, to explain why the result is Contra-Intuitive.

The contestant has chosen a door. If he switches, there are SOME chances he will be found wrong, and his initial choice was true.

Even if the chances are larger that the car is behind the other door, most contestants will stay with their initial decision, perceiving the problem as a 50-50 chance, and, for psychological reasons, holding their ground.

This is especially true in the case of three initial doors, with two left, after one has been opened. The possibility of the sting of failure - if the switch over fails and the contestants discover that they had initially chosen the correct door, but then jumped away from their own choice because of statistics, is weighed heavily in the decision making.

Also, the need for self esteem in decision making - where we lean on our decisions, so that "they must be correct", probably bring the mind to weigh this problem closer to the 50-50 and perhaps even bringing more weight to the initial choice.

This may be the reason why so many people have perceived the statistical result as a fallacy, until presented with the 'million door' example. פשוט pashute ♫ (talk) 08:33, 18 June 2012 (UTC)

DO you know of any sources which discuss this? Martin Hogbin (talk) 08:05, 20 June 2012 (UTC)

Bayes

I have added a section in which Bayes' formula is used to calculate the required probability, which no more overhead than needed. The section about Bayes, that now has been moved to the end of the article, may well be deleted. Nijdam (talk) 12:52, 20 June 2012 (UTC)

Thank you, Nijdam, I have added that same formal solution in written language for non-mathematicians.

Rick, what do you think of deleting that Bayes' section at the end of the article now, as Nijdam has suggested? Will you? Gerhardvalentin (talk) 16:00, 20 June 2012 (UTC)

The so called "overhead" was easier to read/understand and avoided the overlong font with its small font you've created now. If the aim was to create a less accessible version of the text in the end, I'd say you've succeeded, but it's beyond me what purpose that should serve.--Kmhkmh (talk) 09:21, 21 June 2012 (UTC)

The mathematics are lost. All three Bayes sections are opaque. The door numbering confuses the entire article, and it is not fixed by using detailed expressions. Bayes should not be a mysterious oracle devoid of intuition. Glrx (talk) 17:54, 22 June 2012 (UTC)

So, would it be better to show the formal solution for the genuine standard Monty Hall paradox just at the beginning of the article "Monty Hall problem" without door numbers? I suggest this variant:

Formal solution for the standard Monty Hall paradox

When selecting one of three doors, the chances on the car are 1/3 each. The possibilities to open one special door showing a goat are:

The Car being behind one of the two host's doors, that he will offer as an alternative to switch on, he must open the door with his only Goat behind: (1/3)x1 = 1/3

The Car being behind the guest's selected door, then the host, though disposing of two Goats, opens just only one of those two doors: (1/3)x(1/2) = 1/6

These are the only possibilities for the host to open a door showing a Goat. Altogether, the host opens that door with probability 1/3+1/6 = 1/2.

After the host has opened one door showing a Goat, the Car is behind his second door offered as an alternative to switch on (1/3)/(1/2) = 2/3 of the time.

Result: In the standard Monty Hall paradox, the odds for switching versus staying are exactly 2:1, irrespective of any particular actual door numbers.

Would that be easier to access for the average reader? -- Gerhardvalentin (talk) 01:18, 23 June 2012 (UTC)

It would unfortunately seem that the needs of the average reader are being ignored because of mathematical pedantry. Martin Hogbin (talk) 12:01, 23 June 2012 (UTC)

Martin, but what I just proposed here is pure mathematics indeed. Without need to conditioning on door numbers. My English not so good, please help to put it in plain English, will you? What I just proposed above is very well sourced "plain mathematics", and it even has become and nowadays fortunately already is "common knowledge", meanwhile! – It is intended just for the beginning of the famous invulnerable standard Monty Hall paradox (of identical average probability and actual probability!), that forever can be alive and exist only in its firmly arranged famous unambiguous scenario, but never anywhere else, as in the heinous scenario of the sneaky host or in the scenario of the forgetful host. Nowhere else. This invulnerable "standard Monty Hall paradox" ensures that no needless allusion could misleadingly surreptitiously be "assumed".

To condition on door numbers has been criticized and blemished by the actual academic literature as a ridiculous sleazy attempt to mislead. Blemished on the basis of the simple but strong precept named "the more biased, the better!" This precept having superior validity for the famous paradox.

So conditioning on door numbers just makes real sense in textbooks for teachers and students in studying and and in practicing conditional probability theory, say in class rooms. Without effect to "solving the world famous paradox". This should be the actual point of the article and should be shown delineated outside the counterintuitive invulnerable standard paradox with its well defined standard scenario.

And there's still a lot of work to do, to split and distinguish the various scenarios that those various partly outdated sources were addressing to, delineated from the invulnerable "standard paradox". In respect for the sources, in respect for NPOV, to tidy up with the hideous opaque mingle-mangle of the narcisstic hitherto tohobohu so called "Wikipedia article", in respect of the readers so that they can make their own clear judgement on seedy "problems". Can you help? I sincerely hope that Rick and others (?) can and will help also. You all are invited to help. Kind regards, Gerhardvalentin (talk) 12:27, 23 June 2012 (UTC)

I'm going to stick in some general comments here.

I'd like to avoid door numbers because it clouds the conventional problem.

I'm not a big fan of just plugging numbers into a formula or showing a long sequence of calculations that get a desired answer. That tactic is easy to misuse, and many editors have used it to "prove" that the remaining doors are equally likely to hold the car. (This also means I'm not a fan of recently added animated GIF.)

I lean toward opposing an odds view. Conventional Bayes is probability. Mentioning odds seems like a diversion.

CP was posed as a cure all, but it has never been exploited as such. I want Bayes formula to give insight. The MHP has the host exposing a goat; what impact does that have on the contestant's door? Look at B and ask the question what is the P that host exposes a goat? It's one. The result: if the reader accepts BF, accepts that MH can always expose a goat, then trivial math shows the selected door is still 1/3.

CP can then be used to illustrate other scenarios. If MH randomly opens one of the remaining doors, then he finds a car 1/3 of the time. 2/3 of the time he shows a goat. Plug 2/3 into BF, and you get what many intuit as the answer. That focuses the reader on the crucial issue -- Monty knows where the car is.

Glrx (talk) 20:29, 25 June 2012 (UTC)

+1. Yes to maths in a clear odds form. And yes, the Monty Hall paradox is not a puzzle of conditional probability theory only. Modern academic reliable sources show that you not even need to condition on door numbers. That door numbers are completely unnecessary to understand and to solve the famous paradox. Yes, to keep different scenarios clearly apart. And yes, not to confuse any more, but to help readers to grasp what really is the crucial difference between "still 1/3 for the door first selected" to "2/3 for the door offered as an alternative", given that the host is to always show a goat, as in the famous standard scenario.

Clearly delineated versus the quite different scenario where he just randomly opens "one" of his two doors, deleting the chance on the car in 1/3 of cases, quite outside the famous standard paradox. And, that just only "that" scenario (quite outside and inconsistent with the standard paradox) leads to the common "1/2 : 1/2" gut instinct answer. To help to grasp that this makes the very crucial difference. And, for again another scenario quite outside and inconsistent with the invulnerable standard paradox, to accent the strong dominant argument of "the more biased, the better", as per modern sources. That this other scenario is of no avail and is also quite outside and totally inconsistent with the standard paradox. So not to confuse any more by surreptitiously mingle-mangling different scenarios that different sources did address, hitherto leaving helpless readers alone. And yes to maths and to Bayes in a clear odds form. To help the readers to grasp the world famous standard paradox instead of vainly and unnecessarily teaching them conditional probability theory. To enable them to easily make their own clear judgement. Thank you once more. -- Gerhardvalentin (talk) 22:11, 25 June 2012 (UTC)

Your response confuses me. I understand you still want odds form, but I'm confused about introducing non-standard versions. Are you in favor of explaining an ignorant Monty opening a random door? Glrx (talk) 19:08, 2 July 2012 (UTC)

Yes, in a special later section to help readers grasp the paradox, the difference between chances for "staying : switching" of "1/3 : 2/3" (if the host has to show a goat) and of "1/2 : 1/2" (if the host opens one of his two doors uniformly at random, ending the show by showing the car in 1/3 of cases).

Seeing two closed doors, knowing for sure that one of those closed doors is hiding a GOAT, and knowing for sure that the other closed door is hiding the CAR, most people guess the chances are 1/2 : 1/2. You should help the readers to distinguish those two different scenarios. That should be the point of the article. Gerhardvalentin (talk) 11:57, 4 July 2012 (UTC)

Weight and structure

The real "weight" – for that world famous paradox, the real "weight" of reliable sources, proclaiming the dubious eminent helpfulness of correct analysis, based on mathematical probabilistic evaluation as a cure-all, and claiming that you indeed "could know much (?) better than the average probability" – but only under the ungiven condition that you "just knew better (!!!)" – therefore fairly admitting that – "as of course you never will know better" – accordingly yet suggesting "to just assume that you though will know better" on the basis that you just "assume to know better" is quite commendable and quite deserving, and undisputedly very relevant in teaching probability theory, but - based on the famous "standard paradox", and even in ignoring its basics, there still is the sourced policy of mathematicians, a force to be reckoned with, named "Biased? So what? Switch in each and every case after the host has shown you a distinct goat, offering the swap. - And always have in mind: The more biased, the better!!!". So, in the article, any aspect has to be clearly presented (clearly, and never more opaque) exactly there, where it belongs. But never anywhere else. So far to the desputed structure of the article. Applying the weight. In respect of the sources, and in respect of the readers. --Gerhardvalentin (talk) 23:36, 30 June 2012 (UTC)

You lost me here. I don't know what you want. The only thing I get is something should be said once in one place and no where else, but that is too general a comment. Glrx (talk) 19:12, 2 July 2012 (UTC)

I think Gerhard is saying that the simple solutions, in which we assume right from the start that (amongst other things) the host's choice of legal door is uniform and we make use of the obvious and intuitive symmetry that this implies, should dominate the article. Gerhard, please say if I have understood you correctly. Martin Hogbin (talk) 08:50, 3 July 2012 (UTC)

Yes Martin, you are quite correct. It was on *The symmetry*. And it was on how to present here the world famous counter-intuitive "paradox" and on making it intelligible, based on actual reliable sources. It was against presenting in the first line its moldy inglorious historical surrounding that could be reported about in a very later section. And it was on waste "mathematical correctness", suited for tuition. It was on the useless "idea" of any never to be known and forever irrelevant host's bias, suitable for the inglorious "history"-section only. It was against prominently presenting outdated and disproved invalid objections. It was on the sleazy retarded actual presentation of the article.

Yes, it was on symmetry. As per actual reliable sources, any host's bias will forever be completely irrelevant for this world famous paradox. Because you never can nor will "know" its existence, its extension and its direction, consequently there remains pure symmetry at last, as proven by actual reliable academic sources. To "assume" what you never will know is futile. Suited for mathematic lessons. My edit was grounded on Rick's long-ago answer that one "of course can and will detect any host's bias, its extent and its direction by watching enough shows". And it was based on actual reputable sources that say that - even if you watch trillions of shows - panta rhei. You only can base on symmetry, nothing else makes sense, and forever it will be a miscarriage to base on asymmetry, introducing an "assumed unknown user defined host's bias" calling it "c" is futile for the famous paradox. But used in maths classes not on the famous paradox, but for learning Bayes and conditional probability theory. And yes, it is on the article of this prominent counter-intuitive paradox. So "conditioning" is completely futile, also. You CAN do it, but – as per actual academic sources – never need to.

The setting "in case the host should dispose of two goats, he will choose one uniformly at random" is just only a courtesy for readers unaware of actual reliable sources. It is unnecessary for aware readers. It is the only sensible, the only possible fundament you ever can base on. Anything different is irrelevant moot for the counter-intuitive paradox, and is pure classroom subject matter in teaching maths. While the guest, as per actual sources, should trust in "switch now in this actual game, and the host is most welcome to be biased, yes, the more biased, the better!"

The article better should outline the core of the paradox: "As the host has to show a goat, the chances of staying : switching are 1/3 : 2/3" versus "If the host opens a door randomly, ending the show by showing the car in 1/3 of cases, the chances would be 1/2 : 1/2."

Once more: It is on the lated, the retarded state of the article. Glrx, please read again what I tried to say above on 30 June. Regards, Gerhardvalentin (talk) 18:32, 3 July 2012 (UTC)

Yes Martin, you are not correct. It was indeed on *The symmetry*. And it was on how to present here the world famous counter-intuitive "paradox" and on making it intelligible, based on actual reliable sources. It was on presenting in the first line its moldy inglorious historical surrounding that could be reported about in a very later section. And it was on "mathematical correctness", not only suited for tuition. It was on the useful "idea" of any never to be known and forever relevant host's bias, not only suitable for the inglorious "history"-section.

Nijdam, you say that focusing "on the forever relevant host's bias" is necessary for the standard Monty Hall paradox that is based on the well-known scenario: "if both remaining doors have goats behind them, the host chooses one [uniformly] at random". Is that really still your unsourced POV? Gerhardvalentin (talk) 11:23, 6 July 2012 (UTC)

It was on prominently presenting outdated and disproved invalid objections. It was on the sleazy retarded actual presentation of the article.

Yes, yes, it was on symmetry. As per actual reliable sources, any host's bias will forever be completely irrelevant for this world famous counter-intuitive "paradox". Because you never can nor will "know" its existence, its extension and its direction, consequently there remains pure symmetry at last, as proven by actual reliable and even unreliable academic sources. To "assume" what you never will know is not only futile, it is part of good academic practice. Suited for mathematic lessons, as well as for this article. The solution is based on actual reputable sources that say that - even if you live long enough a glorious life, and watch trillions of the famous well known shows on the telly with the famous well known host who's name is for ever and ever and for always connected to this world famous counter-intuitive "paradox" - nothing will add to your understanding. You only can base on Bayes or symmetry, nothing else makes sense, and forever and ever and always it will be a miscarriage to base on unreliable sources, even if they are academic and well published in famous magazines througout the world, asymmetry, it is futile for the famous paradox. Used in maths classes on the famous paradox, shows the use of Bayes and conditional probability theory. And yes, it is the only way to for the article to treat this prominent counter-intuitive paradox. So "conditioning" is absolute necessary. You MUST do it, as per actual academic sources.

"You MUST do it, as per actual academic sources"? Nijdam, you did not mention any source that says that for the standard Monty Hall paradox that is based on the well-known scenario: "if both remaining doors have goats behind them, the host chooses one [uniformly] at random", you MUST use conditional probability based on door numbers, to make the correct decision: to stay or to switch to the door offered as an alternative. Please name that one source. Because conditioning on door numbers in teaching conditional probability theory, is quite another thing. Gerhardvalentin (talk) 09:54, 6 July 2012 (UTC)

The setting "in case the host should eat both goats, he will at first choose one uniformly at random" is just only a courtesy for readers who doesn't like the taste of goat meat. It is unnecessary for e-readers. It is the only sensible, the only possible fundament you ever can base this world famous counter-intuitive "paradox" on. Anything different is not to be considered as oppose to the opponents of the controversial opinion of those who considered this world famous counter-intuitive "paradox" as pure classroom subject matter in teaching maths. While the guest, as per actual most reliable and distinguished sources, should trust their own intuition more than exact mathematical reasoning in considering this world famous counter-intuitive "paradox", and the host is most welcome to be blindfolded, yes, the more blindfolded, the better!"

The article better should emphsize on the core of the this world famous counter-intuitive "paradox": "As the host has to show a goat,he definitely will show a goat, or else will be severely punished. In fact the odds of staying : switching are 1 : 2" , showing the values of the conditional probabilities.

Have fun, Nijdam (talk) 08:21, 6 July 2012 (UTC)

Formal solution

Yes Nijdam, in fact the odds of "staying : switching are 1 : 2" , but you need no conditional probabilities to show that. Above I suggested on 23 June to use this following

Formal solution for the standard Monty Hall paradox, just at the beginning of the article, and to forget on unnecessary conditional probabilities in the very first section:

When selecting one of three doors, the chances on the car are 1/3 each. The possibilities to open one special door showing a goat are:

The Car being behind one of the host's doors, he for sure must open his door with the only Goat behind: (1/3)x1 = 1/3
The Car being behind the guest's door, the host opens uniformly at random only one of his two doors, although he disposes of two goats: (1/3)x(1/2) = 1/6

These are the only possibilities for the host to open a door showing a Goat. Altogether, the host opens that door with probability 1/3+1/6 = 1/2.

After the host has opened one door showing a Goat, the Car is behind his still closed door that he offers as an alternative to switch on (1/3)/(1/2) = 2/3 of the time.

Result: In the standard Monty Hall paradox, the odds for "staying versus switching" are exactly "1 : 2", irrespective of any particular actual door numbers.

As this will be easier to access for the average reader I will be going to write it in the article, if no-one opposes. Gerhardvalentin (talk) 14:33, 6 July 2012 (UTC)

Well, well, I would never have figured this out, amazing! Why don't you public this in one of the world's most famous statistical journals. Nijdam (talk) 18:14, 6 July 2012 (UTC)

Other host behaviors

1.

If the player picks Door 1 and the host's preference for Door 3 is q, then in the case where the host opens Door 3 switching wins with probability 1/3 if the car is behind Door 2 and loses with probability (1/3)q if the car is behind Door 1.

Is the following meant?

If the player picks Door 1 and the host's preference for Door 3 is q, then the probability that the car is behind Door 2 and the host opens Door 3 is (1/3)*1 = 1/3, and the probability that the car is behind Door 1 and the host opens Door 3 is (1/3)*q.

2.

...so the player cannot attribute a probability other than the 2/3 that vos Savant assumed was implicit.

I think here 2/3 should be changed to "q = 1/2".

(By the way: The correctly (which does not mean pedantically) formulated problem has a nice simple (although maybe somewhat abstract) solution: Before the game starts the candidate knowing the rules thinks: I'll choose door 1 which has a probability of p1=1/3 of winning. Then the host has to open door 2 or 3 with a goat, and then, before my second choice, I have to determine the new probability p2 for door 1. What I can say about p2 is that it will not depend on the door which the host will open. For I know already that the host will open door 2 or door 3 with a goat; and the method to determine p2 and therefore the result is the same whether I only think in my mind that the host opens door 2 or 3 or whether I wait till he opens a door really. So I can calculate p2 before the game starts. So p2=p1=1/3 because I cannot change the probability for door 1 only by thinking.)

--Albtal (talk) 10:43, 10 July 2012 (UTC)

It is almost correct, and the correct formulation of this solution is the one using symmetry. What's missing is that your p2 is not well-defined. Do you mean the probability that door 1 hides the car in the case the host has opened another door? Then it is not the probability you're looking for, then that is the probability for door 1 hiding the car in the case the host opens door 3. Nijdam (talk) 13:09, 10 July 2012 (UTC)

I just planned to supplement my comment by the remark that this "solution by just thinking" strongly depends on the statement that p2 is the same independent of the door the host opens (which mathematically means q = 1/2 (see above)). I think if this statement is true ("symmetry") then the whole solution is true.

To your comment:

p2 is the probability that door 1 hides the car in the case the host has opened another door. If q = 1/2 (by definition of task or by explicit supplemental assumption) p2 is "by symmetry" the same, whether the host opens door 2 or door 3. So p2 is well-defined.

Here you see why being precise is apparently difficult. If p2=P(Car is behind 1|host opens 2 or host opens 3), then to state p2 is the same whether door 2 is opened by the host or door 3 is opened, is nonsense, because there is only one p2, and what is the meaning of it being the same. Something like "my left leg is just as long as". Nijdam (talk) 22:05, 11 July 2012 (UTC)

The question what p2 really is I left to the reader.--Albtal (talk) 23:36, 11 July 2012 (UTC)

Really?Nijdam (talk) 09:04, 12 July 2012 (UTC)

Yes. It is called the p2-paradox.--Albtal (talk) 09:58, 12 July 2012 (UTC)

Furthermore I don't think that we have to consider the version "with door numbers" as different from the version without. And the question of the task setting is undoubtedly, whether the contestant should switch just before the second choice. The version with door numbers is only a specialisation of the problem w.l.o.g.:

If q = 1/2, p = 2/3 for switching in the general case; so, of course, in the special case too.

And as I wrote earlier, the treatment is equivalent if we have to assume q != 1/2, maybe when given an opportunity to pick between two losing doors, Monty will open the one on the right. In this case q = 1 for the rightmost of the not chosen doors, q = 0 for the leftmost. We have to distinguish two cases: 1. The host will open the right most of the not chosen doors: p = 1/2 for switching; 2. The host will open the left most of the not chosen doors: p = 1 for switching. So we can solve the complete problem without knowing any door numbers. And if, in addition, a specialised question with door numbers is posed, we can apply the general solution to solve this problem.

The difference is not whether there are door numbers in the task setting but whether the question is for the probability at the beginning of the game or for the situation just before the second choice.

So if Monty will open the one on the right the answer to the question "What is the probability at the beginning of the game to win by switching?" is 2/3; and the answer for the situation just before the second choice I gave above.

And as I mentioned: Everybody knows that in the MHP the question is for the probability just before the second choice. But often there seems to be the (unproven) assumption that the probabilities at the beginning of the game are the same.

(And it is obvious that the door numbers in Marilyn vos Savant's Parade problem only served as a concretisation of the problem thought without loss of generality, which means that they assumed that the solution does not depend on the door numbers (especially that q = 1/2 (if they had been asked) ...)).

--Albtal (talk) 15:43, 10 July 2012 (UTC)

Yes. As to the standard version, it is obvious that the value of the unconditional probability cannot change in the slightest, regardless of which door will be opened by the host. The value of the conditional probability will exactly be the same then, regardless of which door has been opened by the host. The article should make clear when it is talking on completely other scenarios, quite outside the standard version of the famous paradox that it pretends to present. Gerhardvalentin (talk) 17:00, 10 July 2012 (UTC)

And, as to "any host's bias" quite outside the standard version, and diametrically contrary to the standard version of the article, irrespective of assuming "q" to be 0 or 1, or any value in between, differing from 1/2:

Then the probability to win by switching "pws" in 2/3 of cases will be 1"diff" (min. 0, max. 1/6) below 2/3, giving pws a value from 1/2 to 2/3, but in the remaining 1/3, pws will inevitably be 2"diff" (min. 0, max 1/3) above 2/3 giving pws a value from 2/3 to 1, so on average pws will be exactly 2/3 though, as a factual inevitability, and valid for every "q". And you surely are right about MvS's assumption. Gerhardvalentin (talk) 03:16, 11 July 2012 (UTC)

Although a little reluctant, I'll give it a try. You definitely have problems with formulating what you mean, or you formulate things you do not understand. For instance: it is obvious that the value of the unconditional probability cannot change in the slights is a meaningless statement. Any probability does not change (not even in the slightest). May be you yourself have an idea of what you mean to express, but you hopelessly fail. When you say: ' The value of the conditional probability will exactly be the same then, regardless of which door has been opened by the host", you may wish to say that all the conditional probabilities (on the car by switching) for different doors opened by the host has the same value. And yes, that's exactly what for instance Rick and I are telling you all along. But it needs some proof, it is not sufficient to just mention this. Nijdam (talk) 09:04, 12 July 2012 (UTC)

The Final Solution

After literally years of talking, the editors of this page are no closer to agreement than they were a year ago. Nor has there developed any easily-countable everybody-but-the-one-holdout-agrees consensus, as usually happens in these situations.

I have a bold solution that I am absolutely sure will result in a better article with a lot less time and effort expended.

Put an indefinite topic block -- no editing of the MHP page or MHP talk page -- on every single person who has ever edited either. (This of course includes me.) Then turn the article into a stub, ripping out and discarding everything anybody has done. In other words, nuke everything and ban everybody.

Wikipedia has 17,093,564 registered users who have made 543,655,915 edits so far. Plenty of them are capable and willing to create a new page on this topic without all of this drama. Within a few months they will create a new article that is far superior to the one we have now, and they will do it without any major conflicts. Giving the boot to a couple of dozen editors who, collectively, have completely failed to figure out what should be in the article will, in the long run, have a positive effect.

Ban and nuke. It's an idea who's time has come. --Guy Macon (talk) 10:51, 9 July 2012 (UTC)

Although I understand your sentiment, that idea is not really compatible with the concept of 'Wikipedia, the encyclopedia anyone can edit'. Despite fundamental disagreements there has been no incivility or edit warring here nor anything else that would justify the wholesale banning of editors. Martin Hogbin (talk) 14:51, 9 July 2012 (UTC)

I fully agree that it would topic ban those who are not part of the problem (including me!) but in my opinion, this would be a good place to invoke the oft-misused WP:IAR rule. If the "encyclopedia anyone can edit" rule prevents us from improving Wikipedia, ignore it. Clearly what we are doing now is not working, and all attempts to resolve the issue have failed miserably. Topic banning and stubifying will fix the problem. We would, of course, carefully explain that the topic ban is not based upon any bad behavior. --Guy Macon (talk) 17:13, 9 July 2012 (UTC)

Okay! Guy Macon, d'accord! I agree, for obviously there's no way out. Gerhardvalentin (talk) 17:42, 9 July 2012 (UTC)

Since I believe we are on the verge of initiating an RFC, this proposal seems at best premature. -- Rick Block (talk) 02:30, 10 July 2012 (UTC)

I agree, if the dispute gets settled in the next month or so, and will happily report that I jumped the gun, giving up on a years-long process right before success. Please do prove me wrong. Nothing would make me happier. --Guy Macon (talk) 03:57, 10 July 2012 (UTC)

Who will decide which of these editors will be banned? Martin Hogbin (talk) 08:06, 10 July 2012 (UTC)

Under my proposal, nobody will decide. They would all get nice polite notices saying that administrators X and Y are asking everyone who has contributed to walk away and give the topic a fresh start with new editors, and that they are getting the notice even though they have not contributed to the page in quite some time.

An alternative would be to only topic ban those who have contributed in the last six months. What I do not want to do is to require someone to decide that user A is causing the problem and user B is not. Arbcom already did that. It did not resolve the issue.

As I said, I really hope that you will prove me wrong and come up with a solution by the end of July. As Bullwinkle J. Moose so often says; " This Time For Sure!. " --Guy Macon (talk) 17:16, 10 July 2012 (UTC)

I think there is no prospect of agreement, even on an RfC, in the near future. Rick and I agreed the wording but now he wants to add some stuff to it. Martin Hogbin (talk) 18:40, 10 July 2012 (UTC)

Martin and I agreed on the wording of the description of the dispute, but have not agreed on the exact phrasing of the question to be asked following this description. I'm not suggesting we "add some stuff to it" ("it" meaning the agreed description). I have solicited opinions on the phrasing of the question from 3rd parties which have been less than clear, and am currently pursuing other alternatives. -- Rick Block (talk) 19:07, 10 July 2012 (UTC)

Wow. Disagreement about what the disagreement is about. What a shock. Never saw that one coming... --Guy Macon (talk) 19:19, 10 July 2012 (UTC)

Well, it has been an additional twelve days. How is that "doing what we have been doing for the last few years again and again is sure to get results Real Soon Now" plan working out for you? --Guy Macon (talk) 20:04, 21 July 2012 (UTC)

Critizism of the simple solution

I do not agree with the late edit of Martin, I have no idea what he is referring at when pointing to a retraction by the authors. Nijdam (talk) 15:13, 9 July 2012 (UTC)

I removed '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 '. This was based on the paper by Morgan but, as you know, the authors said, in response to our letter, 'had we adopted conditions implicit in the problem, the answer is 2/3, period'. That seems pretty clear to me. Martin Hogbin (talk) 21:13, 9 July 2012 (UTC)

They just said "the answer is 2/3, period." Answer to what? To the conditional probability. What else? Nijdam (talk) 10:27, 10 July 2012 (UTC)

This whole section was a mess. I've rewritten it. -- Rick Block (talk) 04:15, 10 July 2012 (UTC)

Why have you missed out any reference to the Morgan quote above 'had we adopted conditions implicit in the problem, the answer is 2/3, period'? What do you suppose Morgan mean by 'conditions implicit in the problem'? Martin Hogbin (talk) 08:09, 10 July 2012 (UTC)

I'm not sure what the point here is. This section does not say anything about the 1/(1+q) answer from the Morgan et al. paper, which is obviously what they're referring to in the quote above. I think a much more relevant quote is this one (from their rejoinder to vos Savant's comments [1]) One of the ideas put forth in our article, and one of the few that directly concerns her responses, is that even if one accepts the restrictions that she places on the reader's question, it is still a conditional probability problem. One may argue that the information necessary to use the conditional solution is not available to the player, or that given natural symmetry conditions, the unconditional approach necessarily leads to the same result, but this does not change the aforementioned fact. -- Rick Block (talk) 16:00, 10 July 2012 (UTC)

It is inappropriate to criticize a simple solution by changing the problem it purports to solve. Glrx (talk) 15:03, 10 July 2012 (UTC)

Who are you suggesting is doing that? Martin Hogbin (talk) 15:30, 10 July 2012 (UTC)

More specifically, are you suggesting this section is not accurately representing what is said by the many sources to which it refers, or, are you making a personal statement that you disagree with the approach used by these sources? -- Rick Block (talk) 16:00, 10 July 2012 (UTC)

Martin, the article says that Morgan et al. "are" doing it. Yes Glrx, you are right again: the excellent mingle-mangle article, owned for years now, never straightforward clearly did say nor says straightforward and clearly on which scenario it is just actually talking about. The reader never will know: is it actually talking on the standard version, or is it actually talking on quite other conceivable but diametrically opposed versions, completely outside the scenario of the standard version. That's the crux of that owned article, the crux of the perpetual dilemma here. Gerhardvalentin (talk) 16:13, 10 July 2012 (UTC)

As I wrote earlier Critizism of the simple solution is a strange section. Someone in the "literature" came up with the difference between a problem with and without door numbers, and in addition the Wikipedia reader is surprised by the disagreement in the "literature" whether the question he can read at the beginning of the article is the "first or second question" (see my remarks below to door numbers and so on ...). And what is the difference between "switching" and "always switching"? The probabilities for both in the concrete game are the same ... If "always switching" means "whatever door the host opens" it should be made clear. And here my suggestion for a new article: 1/2, 1/2 (or 1/3, 1/3); 1/3, 1/6; 1/3, 1/3*q --Albtal (talk) 16:46, 11 July 2012 (UTC)

This is fully correct, Albtal. Modern academic mathematical literature proofs that the best decision in each and every game is to switch and never to stay, any other decision is throwing you back. This is applicable to the symmetric standard variant, but to any assumed host's bias (whatsoever) as well. The conclusion of this modern academical paper: You cannot do better than to "switch right away in every game, and the host is most welcome to be biased: The more biased, the better!". As said: any staying throws you back. Gerhardvalentin (talk) 17:17, 11 July 2012 (UTC)

Albtal - The difference is between switching whatever door the host opens (effectively, deciding to switch BEFORE seeing which door the host opens or, if you prefer, after the host opens a door but without knowing which door) as opposed to switching with knowledge of which door you picked and which door the host opened. The door numbers are irrelevant in either case (even without numbers there would be a door on the left, a door in the middle, and a door on the right - so the doors are inherently distinguishable to both the contestant and the host). What these sources are saying is that in the latter case the probability of winning by switching (assuming you picked Door 1 and the host opened Door 3) is P(car behind door 2|player picks door 1 and host opens door 3) while in the former case it is P(car behind door 2 or door 3|player picks door 1). These are different probabilities - a solution telling you one is not telling you the other even if they have the same numeric value unless you also show why they must be same. The clearest way to see these probabilities are different is to vary the problem so that they have different values - which is what these sources do, demonstrating both that there indeed is a difference and that solutions determining one don't determine the other. As you say below "Everybody knows that in the MHP the question is for the probability just before the second choice". Indeed. What these sources are saying is the "simple" solutions do not determine this probability and make no attempt to connect the probability they do determine (before the host has opened a door) with what everybody knows the question is asking about, i.e. they're "incomplete" or "misleading". -- Rick Block (talk) 20:35, 11 July 2012 (UTC)

Yes. As I wrote, there is no need for questions, one without and the other with door numbers, but simply ask for the probability of winning by switching

1. before the game starts (or before the host opens a door)

2. after the host has opened a door

One of the "simple" solutions is: Switching wins in two of three cases, so p = 2/3. This is true, before the host has opened a door. This is not a "complete proof" (for 2.), but a key step for understanding the solution, and in no way "misleading". Going further on the simple way just before reaching the goal there is the interesting question: If the probability 2/3 has changed after the host has revealed a goat: Where did it disappear? The only possibility for this is that there are (at least) two cases: One with a probability higher than 2/3, and one with a lower. But where might they come from? The only place for an asymmetry in the game is that the host prefers a door if he has a choice. If he does so I can't know how, and I cannot solve the problem (with exact probabilities). But why should he prefer any door? There is no reason. So the solution is 2/3. (See also the simple solution below: If the solution is independent of the door which the host opens, it is 2/3.)

Another remark: There are no "conditional" and "unconditional" problems or questions. There are only problems and questions, which sometimes may be solved by using conditional probabilities - or not. This whole "conditional salad" seems to come from a source whose authors wrote about the one-million-doors-analogy of Marilyn vos Savant: She then went on to give a dubious analogy to explain the choice.--Albtal (talk) 23:09, 11 July 2012 (UTC)

I should also remark here that it is my opinion that the ongoing argument is only a small side problem within MHP which doesn't really interest anybody but surely should be mentioned for completeness and correctness. And overloading the task setting pedantically will cause a yawn before the reader understands the essential conditions (I cannot prove this exactly, even if applying conditional probabilities). As Martin Hogbin wrote in a discussion with me in May 2012 (I temporarily had the user name Scharzwald) both sides of the ongoing argument don't agree in my view of the main problem. I agree with Marilyn vos Savant that the most significant condition is that the host always opens a loosing door on purpose. And she herself writes in her book (p. 15): When I read the original question as it was sent by my reader, I felt it didn't emphasize enough that the host always opens a door with a goat behind it. I myself would formulate this central condition as follows: The contestant now determines two doors, of which the host has to open one with a goat. For every understanding of the problem which is not equivalent to this has no 2/3-solution. But there is a crucial difference to Marilyn: I would formulate this condition in the question, not in the answer. But so the world is full of strange answers for a question which does not really have a 2/3 solution. I understood Martin Hogbin well. Surely my suggestions for a better article persist. Maybe sometime the majority changes.--Albtal (talk) 09:28, 12 July 2012 (UTC)

Gerhard - no one is saying don't switch. Always switch is the answer Morgan et al. came up with 20 years ago. Nothing "modern" about it. -- Rick Block (talk) 20:35, 11 July 2012 (UTC)

Self-deception? Rick, the simple solutions here in the article are explicitly based on full symmetry, as per the standard version by Krauss and Wang. Presenting the explicitly symmetric standard version where "different numeric values" are banned and are impossible from the outset, by "simultaneously" confusingly using sources that show optional different numeric values though, by irresponsibly ignoring and violently removing the given symmetry??? It is the charade of this messy article to deceive, and to misguide and to mislead. Here are your own words you just wrote above:

"a way to see these probabilities are different is to vary the problem so that they have different values - which is what these sources do".

These are your own words. And you do not hesitate to name the simple solutions of the fully symmetric standard version to be shaky, incomplete, misleading, bluntly false and not answering the "right" question etc. And that is what for years you have been doing here. Unbearable. The fully symmetric version does not need to condition on door characteristics, as per modern sources.

Rick, please distinguish the famous paradox with its strict conditions of full symmetry as per the standard version (Krauss and Wang), keeping it apart from quite other contradictory scenarios that a priori are forbidden by those well defined strict conditions. Please clearly distinguish those contradictory versions and show them after a clear headline saying that the following sections are different to the famous standard version and do no more concern that famous standard version. You can show there conditional probability theorems, also. To show the conditional probability is fine, but its a bold lie that it is academical consent that you MUST use it to "solve" the symmetrical paradox.

Once more: you CAN do it, but modern academic sources agree that you do not need it to solve the symmetrical paradox. I hope you can an will help to muck out the crap. Gerhardvalentin (talk) 22:17, 11 July 2012 (UTC)

Gerhard - please distinguish Truth from TRUTH. I'm merely saying what it is that these sources say. Also note that they are numerous and keep appearing (1991, 1992, 2001, 2005, 2009 - and this is not even a complete list). Whether you, I, or anyone else agrees is completely beside the point. The POV they express certainly belongs in the article. -- Rick Block (talk) 02:08, 12 July 2012 (UTC)

Rick, we have recently established on Sunray's page that very few, if any, reliable sources say that, considering only the case that the host must choose an unchosen goat-hiding door uniformly, the simple solutions are wrong or incomplete. Martin Hogbin (talk) 09:35, 12 July 2012 (UTC)

As anyone following the provided link can plainly see, what we have established is that you think this, I don't, and you are apparently willing to argue tendentiously that these sources don't say what they clearly say. -- Rick Block (talk) 15:13, 12 July 2012 (UTC)

As anyone can plainly see, this article on the world famous counter-intuitive paradox is well-founded on its only correct famous "standard version" of full symmetry, firmly excluding any "asymmetric host's bias" (modern sources proof that you only MAY base on symmetry). The famous "standard version" presented here, does firmly exclude any "closer results" than the average probability to win by switching (pws) of 2/3. Each and every "simple solution" shown here is / are based on that famous standard version only, all "simple solutions" here are based on strict symmetry, so they are correct as per modern sources. In an enduring confusing way, Rick Block insists on citing especially one source that – quite contrary to the standard version presented here – explicitly addresses one quite other scenario of unknown asymmetry and – these are Rick Block's own words above on his impurity source: – they "vary the problem so that they have different values - which is what these sources do" (once more: these are Rick Block's words above). This is impure as it only makes real sense if (contradictory to the standard version that the article is about) one can assume that the host's asymmetric bias (!) can reveal additional information on the whereabouts of the prize, giving "closer results" of pws from 1/2 to full 1. The article should meticulously keep apart the fully symmetric "standard version" that the article is about, from quite other scenarios that a dated source had in mind. And it should clearly be shown that you "CAN" use conditional probability theory, but - as actual sources clearly proof - that this never is "NECESSARY" in any way to make the only correct decision. The article is tattered in indistinctly using sources, and needs to be repaired. Gerhardvalentin (talk) 09:31, 18 July 2012 (UTC)

Extended problem description

I suggest to make the article more transparent by renaming this headline to

"Extended description of the standard version"

Any valid objections? Gerhardvalentin (talk) 18:04, 10 July 2012 (UTC)

Please cite the ISO number, or official designation of another recognized standards setting organization. ;-)

More seriously, there are two substantial changes introduced in this version and I don't think they are equally "standard". The first change is to stipulate that the host must open an unchosen door revealing a goat. This convention is used so universally that it may be regarded as standard. The second change is to provide explicit frequency distributions (for the host's choice of goat and for initial placement of the car). Although this convention is common in sources that use MHP to explain conditional probabilities, it is by no means prevalent in popular literature that does not employ the same formalism. The latter often use, at least implicitly, a Bayesian perspective rather than a frequentist one. (Cf. Richard Gill's paper,[2] §3)

As I remarked elsewhere, I think it is confusing to interpose a frequentist formulation of the question before giving "simple" answers that do not employ frequency distributions. I think it casts those solutions in a poor light, and I do not think labeling that formulation as "standard" represents a neutral point of view. ~ Ningauble (talk) 17:07, 21 July 2012 (UTC)

P.S. – I would be grateful if anyone could locate additional sources that expressly explain the Bayesian perspective on MHP. An accessible explanation of the logical basis might help to frame the different solutions as complementary approaches, and reduce the article's tendency to give undue weight to apparent controversy. (Gill's paper for Statistica Neerlandica assumes that his audience understands the epistemological difference, and the popular literature that tends to employ Bayesian reasoning tends not to trouble its audience with abstract underpinnings.) ~ Ningauble (talk) 17:14, 21 July 2012 (UTC)

;-) The article already said "A fully unambiguous, mathematically explicit version of the standard problem is ..." — So the headline should say what is to follow ... ;-) Gerhardvalentin (talk) 23:15, 21 July 2012 (UTC)

* Also Henze, Norbert (1997). Stochastik für Einsteiger: Eine Einführung in die faszinierende Welt des Zufalls, 9th edition 2011, pp. 50-51, 105-107, Springer, ISBN 9783834818454, (restricted online copy at Google Books) says:

Der Standhafte gewinnt dann den Hauptgewinn, wenn sich dieser hinter der ursprünglich gewählten Tür befindet, und die Wahrscheinlichkeit hierfür ist 1/3. Ein Wechsler hingegen gewinnt das Auto dann, wenn wenn er zuerst auf eine der beiden "Ziegentüren" zeigt (die Wahrscheinlichkeit hierfür ist 2/3), denn nach dem Öffnen der anderen Ziegentür durch den Moderator führt die Wechselstrategie in diesem Fall automatisch zur "Autotür". Bei allen diesen Betrachtungen ist natürlich entscheidend, dass der Moderator die Autotür geheimhalten muss, aber auch verpflichtet ist, eine Ziegentür zu öffnen. Wer dieser Argumentation nicht traut und lieber praktische Erfahrung sammeln möchte, lasse sich unter der Internet-Adresse [3] überraschen.

"For all of these considerations it is of course essential that the host is obliged to maintain secrecy regarding the car hiding door, and is also obliged to open a door with a goat behind." — That's what I just added to the section "Other simple solutions based on the standard version of the paradox".

Krauss&Wang defined the only senseful valid scenario, and Norbert Henze (Stochastik für Einsteiger: Eine Einführung in die faszinierende Welt des Zufalls) even is utterly ignoring any other improper scenario. He says

of course you can refine/narrow down the scenario, e.g. if the host has two goats to choose from, he could open the door with the smaller number with probability q

and he hurries to instantly add:

Similar examples can be found in educational books on stochastics. Their sole purpose is to schematically practice conditional probability arithmetic.

But he does NOT say that this could be of any relevance for the world famous paradox. — So it's up to you to decide what the world famous paradox (1/3 : 2/3, and not 1/2 : 1/2) indeed is, and what a mess this article still is. Gerhardvalentin (talk) 23:15, 21 July 2012 (UTC)

Apart from apparently suggesting that Wikipedia is the official standards setting organization, I don't see how the foregoing remarks have any bearing on my objection to labeling a frequentist formulation of the problem as "standard". ~ Ningauble (talk) 13:20, 22 July 2012 (UTC)

As a 'strong Bayesian' I too would love to see a solution based on a Bayesian interpretation of probability but, unfortunately, it seems that most serious commentators prefer what looks like a frequentist (or realist as Richard might call it) approach to the problem. Martin Hogbin (talk) 17:29, 22 July 2012 (UTC)

In Barbeau's literature overview, cited in the article, he says the "standard analysis" is based on both assumptions (host always opens an unselected door to reveal a goat, choosing randomly if both conceal goats). Selvin, in his second letter (not his first as Richard inaccurately claims in his paper), says the same thing - making these not just the "standard" assumptions but also the "original" assumptions. Lacking a source claiming that most popular sources (who don't say what they're doing) are taking a "Bayesian" perspective, I think the most reasonable approach is to include both of these as part of the "standard" problem. Doing this conveniently makes the answer, whatever your interpretation of probability and whether you interpret the question to be about a strategy of switching vs. staying or about a particular case where the player knows which door she picked and which door the host opened, 2/3 chance of winning by switching. Distinguishing among these is clearly something the article should do at some point - but forcing all answers to be the same gives us no reason to prefer one answer over another, which rather than not being neutral seems precisely neutral. -- Rick Block (talk) 00:41, 23 July 2012 (UTC)

If I understand it correctly, Ninguable's point is that all the analyses of the problem seem to take a frequentist approach to the problem in which it is necessary to assume distributions for those that are unknown. Many people find a Bayesian approach, in which probability is defined as a state of knowledge, to be more natural. As I said, years ago, right at the start of this discussion, using a Bayesian approach we are obliged to take probability to be evenly distributed between possibilities about which we have absolutely no knowledge. The Bayesian approach thus gives the same result as the frequentist (realist, probability theory) result where uniform distributions are assumed for the initial car placement, the player's initial choice, and the host's choice of legal door but there is no need for these assumptions to be explicitly stated because they are an integral part of the Bayesian perspective. Martin Hogbin (talk) 08:53, 23 July 2012 (UTC)

More specifically, my point is that those who address the problem using frequentist principles have redefined the question in terms of frequency distributions in order to facilitate thier analysis.

Taking this as the canonical problem can lead to a vague and unfounded impression that logical solutions using inferences that do not refer to frequency distributions are somehow lacking, and hence to the POV that only conditional probability gives the correct answer to life, the universe, and everything. However, life does not always hand us neat frequency distributions on a silver platter: "Imagine you are on a game show..." and the host goes off-script with an unprecedented scenario. ~ Ningauble (talk) 18:32, 23 July 2012 (UTC)

Why delete my contribution?

I added a few days ago this explanation:

Alternatively one can change the game slightly to understand the solution.^{[citation needed]} Suppose that there are two players: the original contestant who does not change his first choice after Monty shows the goat; and a second contestant who comes in at this stage to choose the remaining door. Obviously because there are two players and only two doors which might contain the car, between them the two players have a 100% chance of winning the car. The first player has the 1/3 chance he started with: nothing has changed -- so the second player has a 2/3 chance of winning. This is the same probability of the single player winning the car if he changes his choice.

This a simple explanation in the vein of many solutions to mathematics or logic problems. It is very clear and I have used it dozens of times to explain the MH problem to those who disbelieve it. I don't want to get into an edit war but it is a little high-handed (Glrx) to remove it with the comment that it adds nothing.Cross Reference (talk) 03:01, 19 July 2012 (UTC)

If the "first" player has a 1/3 chance with his first choice after Monty shows the goat, he has a 2/3 chance with the other remaining door. You don't need a second player.

But more important: You write The first player has the 1/3 chance he started with: nothing has changed. If so we hadn't a MHP at all. Indeed your argument seems to be based on the widespread error that the probability for the first door cannot change if the host opens another door with a goat. But the 1/3 chance for the first door only "stays" if the host is obligated by the rules of the game to open a not selected door with a goat. And it is exactly this obligation which leads to probabilities of 1/3 and 2/3 instead of 1/2 and 1/2; not that he opens the door with a goat for some other reason. And it is exactly this obligation which was missing in the original problem setting. As I wrote earlier the crucial condition in the problem set should be formulated as: The contestant now determines two doors, of which the host has to open one with a goat. And to those who disbelieve the 2/3 solution you can now say: If I choose door 1 I will win the game by switching if the car is behind door 2 and if it is behind door 3. For if the host opens door 2 I shall take door 3, and if he opens door 3 I shall take door 2. This is a first important step to understand the 2/3-solution; and you can demonstrate this easily with three playing cards.

If the host is free in his decision but acts exactly like in the original problem set (that is opening an unchosen door with a goat and offering a switch) the contestant cannot know whether the host wants to help him or if he wants to disabuse him of the right door. So the answer p=1/2 for each of the remaining doors is reasonable. --Albtal (talk) 09:16, 19 July 2012 (UTC)

@Cross Reference. You "Added another simple solution".[4] At least one other editor questioned the addition because Ningauble tagged it with citation needed and commented "I don't see how this elaboration is any clearer than the preceding paragraphs".[5] I also questioned its addition, so I removed it. It is appropriate to debate adding the explanation back on this talk page. WP:BRD

To me, your explanation jars by saying "change the game slightly" and then tightly associates the chosen and remaining doors with the old and new contestants. The explanation adds more complexity with the additional contestant, but that contestant does not add clarity. It was the longest paragraph in the section.

To be clear, I did not say the addition "adds nothing" (and Ningauble did not say that either). My summary states "Changing game explanation adds little".[6] There is something to your explanation. It prohibits the initial contestant from switching. At the start, we know the initial contestant had a 1/3 chance, so there's a notion that no matter what Monty does with the remaining two doors, a stick-no-matter-what contestant will end up with the car 1/3 of the time, so a switch-no-matter-what strategy will win 2/3 of the time. We can reach that result without changing the game or adding a second contestant -- which is essentially the explanation in the first paragraph of Monty Hall problem#Other simple solutions and the thrust of the entire section.

Glrx (talk) 16:05, 19 July 2012 (UTC)

@Glrx -- sorry for misquoting you. The explanation (it's is not really another solution) is in the vein of other questions such as the well known one about the pilgrim who goes up a mountain. There is a single winding path up the slope, and he leaves approximately at dawn and arrives at the top more or less at sundown. He never deviates from the narrow path, but he does take time on the way up for eating, resting and playing his flute. Next day he comes back down the hill, again stopping randomly and and arriving at the bottom as the sun goes down. Was there any time where he was at the same point on the path to the nearest millisecond of time as he was the day before? That can be proven by some reasonably heavy duty maths, but the easiest way to satisfy yourself is to imagine him doing it in both directions on the same day and meeting himself at some point. Change the question and it become obvious. The solution (2/3 if you switch, 1/3 if you stick) is obvious to me after a little thought (and I have modeled it in Excel to 10,000 lines) but I have rarely successfully explained it to the doubters until I started using the second player concept.Cross Reference (talk) 02:33, 20 July 2012 (UTC)

This is conceptually similar to what I suggested a while ago (see discussion now archived at [7]). Explicitly changing the problem makes the 1/3 vs. 2/3 answer easier to understand. What I suggested was meticulously sourced, but even so at least Martin and Glrx didn't like it. Are there other opinions about this? -- Rick Block (talk) 15:37, 21 July 2012 (UTC)

Very good and illustrative example, Cross Reference. The crux here is that, when showing illustrative examples, some editors say that "this is changing the problem". Gerhardvalentin (talk) 18:11, 21 July 2012 (UTC)

No Rick, you are disarranging again. This article is about the famous symmetrical standard version of the paradox, that ALL simple solutions here are based on. You have to observe that it is inappropriate to criticize them by changing the problem, in drudgingly observing which one of his two doors the host just has opened. And to call it "changing the problem" if you talk of the original problem again. In forgetting what the article is about. Yes, in talking of the symmetrical version, you call that "explicitly changing the problem". The article is, in the first line, on the "symmetrical standard version" of this counter-intuitive paradox. In this standard version, the probability to win by switching is 2/3, full stop. 2/3 "before", and 2/3 "after".

In this standard version, the "difference" of "simple" probability and of "conditional" probability is irrelevant. So please help the reader to grasp, without still trying to confuse him/her. Quite different scenarios of a host, who will only offer to switch in case the guest should first have selected the car, are to be shown quite "outside" the famous counter-intuitive standard paradox, in a later and clearly separated section. As well as the host, who in some certain scale will forget the location of the objects and subsequently will risk to be showing the car instead of his goat, and by that ruining the chance to win by switching to a certain extent. As well as the forever only to be "assumed" host who shows, by exceptionally opening the door that he usually is "assumed" to strictly avoid to open, that the probability to win by switching will be "1" in that case, being subject matter of maths teachers in probability theory. Every "simple solution" presented in THIS article is based on the symmetric standard version, so no need to "explicitly changing the problem". Once more: All "simple solutions" here are based on the famous symmetric standard version. And please stop to inappropriately criticize a simple solution here, by changing the problem it purports to solve. Please stop to disarrange. Once more: please help the reader to grasp, without still confusing him/her. Gerhardvalentin (talk) 18:11, 21 July 2012 (UTC)

Time for The Final Solution?— Preceding unsigned comment added by Guy Macon (talk • contribs) 01:27, 22 July 2012

With the help of Norbert Henze ("For all of these considerations it is essential of course that the location of the car behind the doors has to be kept secret by the host, and that he is also obliged to open a door showing a goat", or: "For all of these considerations it is of course essential that the host is obliged to maintain secrecy regarding the car hiding door, and is also obliged to open a door with a goat behind." and: "Of course you can refine/narrow down the scenario, e.g. if the host has two goats to choose from, he could open the door with the smaller number with probability q. Similar examples can be found in educational books on stochastics. Their sole purpose is to schematically practice conditional probability arithmetic."), and with the help of other modern sources, and with the help of Rick Block and of Martin, of Glrx, Guy Macon, of Kmhkmh and Ningauble and others it should be possible to get to a far better version without further health warnings aimed in the wrong direction, and without lessons in practicing conditional probability theory, within the irrevocable fixed period, I guess ;-) Gerhardvalentin (talk) 10:45, 22 July 2012 (UTC)

I support you and wish you luck Gerhard but the article is still dominated by editors who cannot see the simple fact that you (and yet one more new editor above - Albtal) plainly can, namely that in the symmetrical case the host's choice of unchosen goat-hiding door is unimportant and need form no part of the solution. Martin Hogbin (talk) 15:05, 23 July 2012 (UTC)

Meaning?

What is the meaning of: "Other simple solutions based on the standard version of the paradox", and why is the so called "combining doors solution", which is wrong, as Devlin himself admits, still in the article? Nijdam (talk) 08:55, 23 July 2012 (UTC)

All "simple solutions" are based on symmetry. After citing the mathematically explicit Krauss and Wang version of the standard problem (symmetry), the article says "Solutions" and presents as one first solution the table by vos Savant (Parade) saying that staying has a chance to win the car of 1/3, while switching has a chance of 2/3.

After vos Savant's solution, before presenting Carlton, Henze and many others, there is a new headline titled "Other simple solutions ...".

And as to the entities of 1, of 2 and of 3 doors: The entity comprising only one single door has 1/3 chance to hide the prize, while the entity of all three doors (for example door#1 + door#2 + door#3 altogether) have 3/3 chance ("1") to hide the prize. Gerhardvalentin (talk) 13:37, 23 July 2012 (UTC)

Nijdam, can you show me where Devlin admits that he was wrong. Martin Hogbin (talk) 15:06, 23 July 2012 (UTC)

+1: [citation needed] ~ Ningauble (talk) 17:53, 23 July 2012 (UTC)

We should not be calling MPH a paradox in the section heading, because a so-called "veridical paradox" is not a genuine logical paradox. It is not a case of two sound lines of argument leading to incompatible conclusions, it is simply a case of common sense being wrong. (There would be a real paradox if Bayesian and frequentist reasoning gave conflicting conclusions; but they don't, and it is not merely a coincidence, as some here seem to believe, that they both give the same odds.) The whole "based on..." clause is superfluous: context should indicate what question the solutions answer. ~ Ningauble (talk) 17:53, 23 July 2012 (UTC)

+1, Ningauble. The context should show it. Yes! But for years, after presenting the Krauss+Wang definition of symmetry as "the basics", followed by simple solutions obviously based on those "basics", especially Nijdam ("that has to be proven, you need a proof") insisted in citing health warnings in an opaque manner, saying that simple solutions do not answer the question put, that they solve the wrong problem, that they are incomplete etc. etc., in a very common way, WITHOUT mentioning that those sources either explicitly address asymmetry (changing the problem) – or just are mathematical cure-all approaches that try to solve both, the symmetrical AND the asymmetrical variants also. But, as more and more modern sources are emphasizing, regarding the paradox it is indispensable ("OF COURSE"!) to base on symmetry, whereas "assumed" deviations by "assumed" host's bias, giving "assumed" results, are a matter of learning maths, their purpose is to schematically exercise conditional probability, as per numerous convenient textbooks, serving exactly that educational purpose. And, at least in the past, Rick had permanently pointed up that this educational material for maths classes is quite numerous, indeed. Yes, it is, but nevertheless the clear and distinct structure of the article should help the reader to grasp what the paragraph he just reads actually is talking about. Gerhardvalentin (talk) 18:52, 23 July 2012 (UTC)

Request for explanation

Let's suppose I'm out of the room during the initial phase of the event, and walk in at the last moment. All I see are two closed doors, and presumably one prize. Why should I assume that knowing any history prior to this moment would change the "two doors, one prize" moment? The article didn't clearly explain, at least to me, why the history should have any significance. I guess I (incorrectly) think of it like the gambler's fallacy... prior events having no impact on the current ("two doors, one prize") moment. JeramieHicks (talk) 22:06, 1 August 2012 (UTC)

If the show proceeds given the normal assumptions (car is randomly placed, player picks a door, host opens a different door choosing randomly if the player's initial pick hides the car), the probability the car is behind the door the player picks is 1/3 while the probability the car is behind the other door is 2/3. By "probability" I mean the limit of the relative frequencies of these outcomes if you repeat this same set up numerous times (see "frequentist probability"). This is also the probability you would deduce in a Bayesian analysis of this situation (which allows an analysis of events you might not repeat using information available to an observer). You might have noticed some of the folks on this page don't seem to get along - the difference between "frequentist" and "Bayesian" is possibly the root of much of the conflict.

Back to your question. The frequentist and Bayesian probability the car is behind the door the player initially picked is 1/3, while both of these are 2/3 it is behind the other door. If you arrive late, we can now see the difference between these two interpretations of probability. The frequentist probability is and will forever remain the same. The Bayesian probability however is a function of the information you know. Rather than measure the probability of "what is" (in the sense of what we'll see if we repeat the same thing over and over again) it really measures the probability that someone knowing a certain amount of information can make a correct choice. The difference is subtle but very real. If you don't know the setup, to pick the car you must effectively pick randomly between the two remaining doors. A random choice between two alternatives, regardless of any "real" (frequentist) probability has a 50/50 chance of being correct. So, in a Bayesian sense, the "probability" the car is behind each door for you (not knowing the set up) is 1/2 - while at the same time the frequentist (and, for those in the audience, Bayesian) probability the car is behind the player's initial pick is 1/3 (and 2/3 for the other door).

A little algebra may (or may not) make this more clear. Let's say there are two alternatives and the frequentist probability between them is p and 1-p (where p is anything between 0 and 1 - i.e. we hide a car between two doors using any algorithm ranging from "always hide it behind door 1", to "flip a coin to pick which door to hide it behind", to "always hide it behind door 2", to "hide it randomly behind one of three doors, let a player pick one door and have the host open another door picking randomly if the player's door hides the car"). You arrive on the scene not knowing the algorithm we used. The frequentist probability that the car is behind door 1 is p, and that it is behind door 2 is (1-p) - meaning if we do this over and over again these will be the limits of the relative frequencies of where the car is. You don't know this. If we do this 100 times (where, whatever the algorithm, we've ended up with door 1 and door 2), then roughly 100*p times the car will be behind door 1 and 100*(1-p) times the car will be behind door 2. If you pick randomly, of the 100*p times the car is behind door 1 you'll pick door 1 half of these, so you'll be right about 50*p times. Similarly, of the 100*(1-p) times the car is behind door 2 you'll be right about 50*(1-p) times. Altogether, you're right 50*p + 50*(1-p) times, which is 50 times, which is 1/2 of the time.

The point here is that to someone who doesn't know the "actual" (frequentist) probability, picking randomly between two choices makes the result a 50/50 choice. Another way to say this is if you don't know the history and there are only two choices left, you have a 50/50 chance of making a correct choice. On the other hand, this definitely doesn't mean that if you're choosing between, say door 1 and door 2, if you repeat whatever the set up was it will converge to 50% of the time the correct choice is door 1 while 50% of the time the correct choice is door 2. -- Rick Block (talk) 05:16, 2 August 2012 (UTC)

Oh dear! Martin Hogbin (talk) 12:39, 2 August 2012 (UTC)

Or simply put: walking in after the initial phase, you do not know which one of the remaining closed doors is the one originally picked by the player. Repeating will show you also the cases where door 2 is initially picked and door 3 opened by the host. The relative frequencies for both doors to hide the car will both tend to 50%. Nijdam (talk) 11:10, 2 August 2012 (UTC)

Oh dear, yes! Gerhardvalentin (talk) 00:45, 3 August 2012 (UTC)

A little clearer. If only the article concentrated on issues that people are actually interested in, like this one. Martin Hogbin (talk) 12:39, 2 August 2012 (UTC)

The fact that many people wade through the whole article and are left asking the same questions does seem to indicate that the article is wanting something. On the other hand, it may be a certain something that defies any possible appeal to common sense. ~ Ningauble (talk) 17:51, 2 August 2012 (UTC)

My answer to JeramieHicks's original question is this: What the person arriving in the middle does not realize is that the visible goat does not just reveal information about one of three doors; but specifically reveals information about one of two doors not initially chosen. The narrower scope makes it much more informative: 1/2 is more than 1/3. (This is a Bayesian perspective, which does not correspond to what frequentists define as probability analysis.) The gambler's fallacy does not apply here: it pertains to independent events (e.g. separate rolls of the dice), not to incremental information about one thing (i.e. the location of the car). ~ Ningauble (talk) 17:51, 2 August 2012 (UTC)

@JeramieHicks: First, as I wrote above: I myself would formulate the central condition of the task set as follows: The contestant now determines two doors, of which the host has to open one with a goat. For every understanding of the problem which is not equivalent to this has no 2/3-solution.

Then, if you "choose" say door 1 - what means that the host has to open door 2 or door 3 with a goat - you know before the host opens a door that you will win the game by switching if the car is behind door 2 or door 3. For if the host opens door 2 you will take door 3, and if he opens door 3, you will take door 2. So you have a 2/3 chance by switching before the host has opened a door. The question is now whether the chance to win by switching is still 2/3 if the host has opened door 2 or 3 with a goat. Suppose the chance is now 1/2. Then there would be a strange situation: You have invented a game which you always start with a 2/3 chance which will always change to a 1/2 chance during the game.

Surely the "history" matters. You know many examples of this. The essential mathematical consideration of the MHP is, that the probability for the host to open his door is twice as big if the contestant has chosen a goat (p=1) as if he has chosen the car (p=1/2). (This simple fact is hidden in the section Bayes' theorem.)

You may also consider the following two cases, where the contestant chooses door c, the host opens door h, and the other remaining door is s:

1. Car behind c and host opens h: p1 = 1/3 * 1/2 = 1/6

2. Car behind s and host opens h: p2 = 1/3 * 1 = 1/3

So the probability to win by switching is p = (1/3) / (1/3 + 1/6) = 2/3

The person who does not know the rules of the game or the history simply doesn't know the chances.--Albtal (talk) 19:43, 2 August 2012 (UTC)

JeramieHicks, that's the shortfall, the still lasting deficit of that article. Read above what I wrote at The Monty Hall Paradox or The Monty Hall "Problem" ?.

The striking point is the well defined scenario, the scenario that this famous paradox is based on. The symmetry of this scenario secures that the door first selected by the guest (one out of three) has and retains exactly a 1/3 chance to win the car, while the set of those two unchosen doors (with a chance of 1/3 each), as "an entity", has and retains a unite 2/3 chance to win. In spite you know for sure that – because there is only one car – that those two doors on average hide 1 1/3 goats, so are to hide "one goat at least", their unite chance to win the car is and retains 2/3. Given by the symmetry of the scenario (if two goats are behind the host's doors, then he will choose one "uniformly at random"!), these chances (1/3 resp. 2/3) cannot change, even after the host has opened one of his two doors showing a goat. Whatever door the host opens showing a goat,
be it #2 or #3, the door first selected by the guest retains its 1/3 chance, and so the host's still closed second door retains unite 2/3 chance.

Now imagine that someone, who comes in after the host has opened one door, not knowing about the scenario (rules) nor which door has been chosen by the guest. He has a 1/2 chance to pick the guest's door [say #1] and a 1/2 chance to pick the still closed host's door [be it say #2 or #3]. He does not know that [say] door #1 has a 1/3 chance only, but the still closed host's door #2 resp. #3 has a 2/3 chance. If you are a frequentist, you could argue that he should observe that only in 1/3 of cases he will win by picking #1, while in 2/3 of cases he will win by picking the still closed host's door #2 resp. #3. But as he is unable to distinguish between those doors (not knowing that [say] door #1 is the door first selected by the guest), that does not help him in any way. His chance to win by randomly picking one of the two still closed doors will exactly be only 1/2 therefore,
as (1/3 x 1/2) + (2/3 x 1/2) = 1/2.

And, as Marilyn vS said: Imagine there is a million doors with only one car and else just goats. The guest selects one door with a 1/millionths chance. Then the host, disposing of 999'999 doors, shows 999'998 goats, leaving only one of his doors closed. What would you think of the chance to win by switching to the host's only one still closed door?

But it is the still lasting deficit of the article to fob the readers, leaving them alone. Gerhardvalentin (talk) 00:35, 3 August 2012 (UTC)

Slight bias

I currently believe that there is a bias on the page that the 2/3-1/3 probability solution is the right one. However, let it be noted that Wikipedia is an UNBIASED encyclopedia. Therefore, a new section, with the reasoning for the other argument, should be created, and any bias hints found in the first should be edited out.

Thanks,

Newellington (talk) 14:06, 4 August 2012 (UTC)

Under the rules of the fully unambiguous, mathematically explicit version of the standard problem (Krauss and Wang description from this page as well as Henze, 1997 and many others), the 1/3 - 2/3 probability solution is the correct one, indeed. Sources that show probability solutions of differing values were addressing quite different scenarios (see above The Monty Hall Paradox or The Monty Hall "Problem" ?). You are right, those differing scenarios, used in lessons on probability theory, using also a host who is not bound to maintain secrecy regarding the car hiding door are used to schematically practice conditional probability theory (Henze, 1997). Yes, those quite other conceivable but diametrically opposed versions on some unknown asymmetry, completely outside the scenario of the standard version, have to be treated in a quite separate section, but never in the main section that is based on the standard problem. Gerhardvalentin (talk) 17:14, 4 August 2012 (UTC)

WP is based on what is said in reliable sources and they all agree that, given the usual assumptions, the probability of winning by switching is 2/3.

The problem with this article is that, rather than concentrating on explaining that fact, it is drawn into academic sideshows. Martin Hogbin (talk) 17:18, 4 August 2012 (UTC)

In response to Gerhardvalentin, I know that this version is correct. Everyone knows that. However, what I am saying is that a section on the wrong explanation should be created, without saying that it is wrong, unless adequate sources cannot be cited for this method. You're right that a standard problem does indeed exist, but we still should display the unstandard problem.

Newellington (talk) 00:58, 5 August 2012 (UTC)

In response to Martin Hogbin, you are absolutely right. The article repeats the explanation of the 2/3-1/3 paradox, using a selection of various charts that all say the same thing. Thank you for your added complaint in this section. This page really needs to be improved.

Newellington (talk) 14:08, 5 August 2012 (UTC)

Every citable source on the Monty Hall problem mentions the naïve 50:50 answer and says that it is wrong. It doesn't even qualify as a fringe theory: it is just a mistake. The whole point of the puzzle is that the common sense first impression is wrong. ~ Ningauble (talk) 18:36, 5 August 2012 (UTC)

Newellington, just to be clear, is your complaint that the answer might not be 2/3 or that the article fails to explain well why 2/3 is the correct answer?— Preceding unsigned comment added by Martin Hogbin (talk • contribs) 21:34, 5 August 2012

Neither of the above. My complaint is that I believe there isn't enough coverage of the 1/2-1/2 theory, and why it is or isn't wrong. I agree that 2/3 is the correct answer, but I don't agree that the 1/2-1/2 theory should be ignored completely. It should be given coverage in the article, regardless of its accuracy.

Newellington (talk) 17:44, 7 August 2012 (UTC)

There used to be a section explaining why the answer is not 1/2 (see, for example, [8]). I haven't chased down exactly when this went away - but is this sort of what you're talking about? -- Rick Block (talk) 18:52, 7 August 2012 (UTC)

Newellington, the problem is that this article is a mess. It does not clearly distinguish between the world famous "PARADOX" where there are two still closed doors, one of them is the door originally selected by the guest, and the second one is the still closed host's door that he just has offered as an alternative, to switch on. You know that one of these two doors for sure contains a goat (the second goat!), while the other door for sure contains the prize. No one knows which door hides the prize and which door hides the goat. So most people say the chances are 50:50. But in the only correct scenario of the world famous "PARADOX" the chances for staying:switching are exactly 1/3:2/3. As said, this is the world famous "PARADOX".

The article should clearly treat this paradox, making it intelligible and then, in later sections, should show quite other scenarios, where the "PARADOX" cannot exist at all. See what I wrote above: The Monty Hall Paradox or The Monty Hall "Problem" ?.

There are some scenarios, outside the "PARADOX", where the chances are 50:50 indeed. For example in a scenario where the host does not know the location of the car. In 1/3 of cases, if he has two goats to show (chances staying:switching 1:0), this is no problem. And in another 1/3 of cases , when he disposes of a goat and the car or vice-versa (chances staying:switching 0:1), and he just by chance opens the door hiding the goat, this is no problem, too. But in the rest of 1/3 of cases, disposing of a goat and a car (chances staying:switching 0:1), he inevitably will unfortunately open the door with the car, and in this 1/3 of cases was eliminating the guest's chance to win. By eliminating 1/3 of the chances to win reduces the probability to win by switching from 1/3:2/3 to exactly 1/2:1/2. And there are other variants where the score can be 50:50. For example if the host is extremely biased to open just the door with the smaller number. He can do that if he got two goats (chances of staying:switching 1:0), and he can do that if he got the car and a goat, given the goat is behind his preferred door (chances staying:switching 0:1). So even here, whenever he opens his preferred door, the chances to win by switching are 1/2:1/2.

The article should clearly show what the clean "PARADOX" really is, with it's unambiguous mathematically fully correct symmetric scenario, and then, in later sections, the article should very clearly show what NEVER CAN BE the famous paradox at all: The paradox cannot exist in deviant scenarios. The article is a mess in not explicitly and clearly distinguishing between the only correct and clear scenario representing the paradox, and other scenarios where the paradox never existed and never can exist. Gerhardvalentin (talk) 20:30, 7 August 2012 (UTC)

In response to Rick Block, I understand now. That is what I was talking about. In response to Gerhardvalentin, I understand that the article is a mess. That's why I came here. Thanks for the info, my issue can be considered resolved. Thanks.

Newellington (talk) 23:04, 7 August 2012 (UTC)

Newellington, you are welcome to help making the article on the world famous PARADOX what it should be: no more misty and opaque dissimulation, but clear and intelligible for the readers. Gerhardvalentin (talk) 10:04, 8 August 2012 (UTC)

Ten Years And A Million Words

Here is what we have "accomplished" so far:

STATISTICS:
1,269,228 talk page words

Total Edits: 14,463 edits (10,610 talk page, 3,853 article)

Average: 120 words per talk page edit.

Individual Edit Counts:
Rick Block         = 2,304 edits (1,799 talk, 505  article) 
Martin Hogbin      = 2,058 edits (1,981 talk,  77  article)
Blocked User G.    = 1,428 edits (1,363 talk,  65  article)
Gill110951         =   790 edits   (602 talk, 188  article)
Nijdam             =   672 edits   (602 talk,  70  article)
Gerhardvalentin    =   498 edits   (447 talk,  51  article)
Heptalogos         =   398 edits   (398 talk,   0* article)
Glopk              =   366 edits   (234 talk, 132  article)
Kmhkmh             =   352 edits   (352 talk,   0* article)
Guy Macon          =   304 edits   (304 talk,   0* article)
Blocked IP Sock R. =   182 edits   (182 talk,   0* article)
Dicklyon           =   164 edits   (123 talk,  41  article)
Father Goose       =   143 edits   (107 talk,  36  article)
Glrx               =   135 edits    (70 talk,  65  article)
Ningauble          =   106 edits   (106 talk,   0* article)

Notes:

Article created in Feb 2002.

"Talk page" is defined as Talk:Monty Hall problem plus Talk:Monty Hall problem/Arguments. It does not include any discussions at noticeboards or on user talk pages.

"0*" means "16 or lower" (limitation of counting tool).

Word count was done by cutting and pasting the text from all the archives into one text document and using UltraEdit to get a word count. If there are duplicates in the archives, someone wanting to read them all would still have to slog through well over a million words. Other statistics came from WikiChecker. I welcome anyone who wants to do their own count.

By comparison:

Harry Potter and the Philosopher's Stone  =  76,944 words
Harry Potter and the Chamber of Secrets   =  85,141 words
Harry Potter and the Prisoner of Azkaban  = 107,253 words
Harry Potter and the Goblet of Fire       = 190,637 words
Harry Potter and the Order of the Phoenix = 257,045 words
Harry Potter and the Half-Blood Prince    = 168,923 words
Harry Potter and the Deathly Hallows      = 198,227 words
All Harry Potter books                  = 1,084,170 words

One thing that struck me while skimming through the archives was that again and again someone would claim that all the issues are settled or that they soon would be. You can respond to this with Yet Another Claim that this will be solved Real Soon Now, but I won't believe you.

I propose the following solution. Yes, this is a serious proposal, and yes, I do realize that it cannot be decided here.

[A]: Apply a one year topic ban on everyone who has made over 100 edits, (which includes me) with it made clear that we are banning everyone and that this does not imply any wrongdoing.

[B]: Reduce the article to a stub.

[C]: Let a new set of editors expand it.

If anyone tries to invoke a policy that would not allow this, invoke WP:IAR. Nothing else has worked. I don't believe that anything else is going to work. Our best mediators, arbcom members etc. have utterly failed to solve this problem. My proposal will solve it.

(Puts on asbestos suit) Comments? --Guy Macon (talk) 21:51, 4 August 2012 (UTC)

What exact problem would this be the fix for? -- Rick Block (talk) 00:16, 5 August 2012 (UTC)

Ten years and a million talk page words without any agreement or any indication that you will reach an agreement if we give you another ten years. Article quality declining as different editors thrash trying to shape it into conflicting visions of what they think it should be. That's a significant problem. --Guy Macon (talk) 11:58, 5 August 2012 (UTC)

Most 'good ideas' aren't and this one certainly isn't. You efforts to help are appreciated but this is not the way, do you have any other suggestions Guy? Martin Hogbin (talk) 12:47, 5 August 2012 (UTC)

I have thought and thought, and I really can not see anything else that will prevent this going on for twenty years and two million words without any resolution. Unless, of course, one of you is 97 years old, in which case nature will resolve the dispute...

At the very least, can we all agree that the above numbers, demonstrate a real problem? --Guy Macon (talk) 13:31, 5 August 2012 (UTC)

New guy to the scene chiming in here. Surely the article has improved from 10 years aago, hasn't it? One could make the same argument against the Wikipedia project as a whole. --RacerX¹¹ Talk to me^{Stalk me} 14:11, 5 August 2012 (UTC)

If you go to the top of this page and click on the "Show" next to "Article milestones", I think you will agree that any of the three points where it was decided that it was good enough to be a featured article are better than it is now. Especially interesting are some of the arguments made when it was demoted:

"There are major, long-term disagreements among a number of editors on how to present the topic [which] have been going on for years [...] those disagreements have negatively impacted the end product: the article itself."

and

"I agree that the article appears a bit like patchwork (as a result of the 'eternal disagreements'. Imho the best solution would be if all old editors voluntarily withdraw from the article (other than commenting) and a few new qualified editors attempt a complete overhaul."

If my proposal comes to pass, there is nothing to stop you or any other editor who has not been involved in this fiasco from looking at old versions and cutting and pasting anything you find to be an improvement into the article. Basically I am advocating that you (Racerx11) and other uninvolved editors be given a free rein to improve the article as you see fit while I and a handful of others with over 100 edits be ordered to not interfere in any way. And even though I know nothing about you, I am pretty sure that you can do a better job. --Guy Macon (talk) 17:27, 5 August 2012 (UTC)

These statistics are sadly amusing, but I am not laughing. This is the second time Guy has proposed a blanket ban (cf. "The Final Solution", 9 July 2012), and this time he is including me by name. Since he says he is serious about banning me, I must ask a couple serious questions:

Is there any precedent for this sort of blanket ban affecting persons not found to be culpable?
Conversely, is there any precedent for Arbcom providing injunctive relief from indiscriminate sanction proceedings?

It might be wise to reinforce the asbestos suit with Kevlar. ~ Ningauble (talk) 18:10, 5 August 2012 (UTC)

No precedent at all. Nothing even close has ever been tried. I would be shocked if my proposal was accepted as written. Plus, of course, it can't happen based on a proposal here; it would almost certainly have to be an Arbcom decision. BTW, I pulled "100 edits" out of a hat. You could use a cutoff of 200, 500, or 1,000, any of which would exclude you. I sure would not want to try to make a case that out of those million words and 14,463 edits your 106 edits are a significant part of the problem. --Guy Macon (talk) 18:23, 5 August 2012 (UTC)

Do counts by themselves indicate a problem? What are the similar counts for, say, Intelligent Design (a featured article about a controversial topic)? I agree the article's quality has declined, but I can't see how this particular "solution" addresses this issue. -- Rick Block (talk) 18:35, 5 August 2012 (UTC)

That's an excellent question. I will do a count at Intelligent Design later today and post the results here so we can compare. Good thinking! --Guy Macon (talk) 19:07, 5 August 2012 (UTC)

Got the statistics. Comparing with Intelligent design was very good thinking on Rick's part.

Statistics for Intelligent design and Talk:Intelligent design

2002 - 2012 -- ten years, just like MHP.

Total Edits: 37.084 edits (24,551 talk page, 12,533 article)
MHP has      14,463 edits (10,610 talk page,  3,853 article)

Top Five Individual Edit Counts:
K             = 4192 (3339 talk page,  853 article)
FeloniousMonk = 2479 (1317 talk page, 1162 article)
Dave souza    = 1558 (1236 talk page,  322 article)
Jim62sch      = 1317 (1140 talk page,  177 article)
Hrafn         = 1023  (883 talk page,  140 article)
...
(MHP has counts of 2,304, 2,058, 1,428, 790 and 672)

ID has 68 archive pages and archives every 15 days
(MHP has 28 and archives every 14 days)

I am not going to count every word, but 5 archives chosen at 
random had counts of 56,996, 25,588, 30,119, 9,743 and 28,712.
30,000 x 68 = 2,040,000, so roughly two million talk page words.
(MHP has 1.3 million talk page words)

Conclusion: by most metrics, Intelligent design has about twice the volume of MHP. --Guy Macon (talk) 12:27, 6 August 2012 (UTC)

A more meaningful statistic

Surely this is a more meaningful statistic. Anyone who doubts my summary is welcome to check the facts for themselves to ensure that I have not missed anyone out or misrepresented anyone. Martin Hogbin (talk) 21:42, 5 August 2012 (UTC)

An alternative

In the discussion between myself and Rick Block in Sunray's user space, Rick and I agreed on the following wording for an RfC.

The aim of this RfC is to resolve a longstanding and ongoing conflict involving many editors concerning the relative importance and prominence within the article of the 'simple' and the more complex 'conditional' solutions to the problem. The 'simple' solutions do not consider which specific door the host opens to reveal a goat (see examples [23] [24]). The 'conditional' solutions use conditional probability to solve the problem in the case that the host has opened a specific door to reveal a goat (see examples [25]).

One group of editors considers that the 'simple' solutions are perfectly correct and easier to understand and that the, more complex, 'conditional' solutions are an unimportant academic extension to the problem.

The other group believes that the 'simple' solutions are essentially incomplete or do not answer the question as posed and that the 'conditional' solutions are necessary to solve the problem. Both sides claim sources support their views.

That argument is unlikely to ever be resolved but two proposals have been made to resolve the dispute. Both proposals aim to give equal prominence and weight to the two types of solution.

One proposal is for the initial sections including 'Solution' and 'Aids to understanding' to be based exclusively on 'simple' solutions (with no disclaimers that they do not solve the right problem or are incomplete) then to follow that, for those interested, with a section at the same heading level giving a full and scholarly exposition of the 'conditional' solutions.

The other proposal is for the article to include in the initial 'Solution' section both one or more 'simple' solutions and an approachable 'conditional' solution (showing the conditional probability the car is behind Door 2 given the player picks Door 1 and the host opens Door 3 is 2/3) with neither presented as "more correct" than the other, and to include in some later section of the article a discussion of the criticism of the 'simple' solutions.

The discussion stalled on the proposed addition of introductory wording. If there is a consensus here, how about you start an RfC using this agreed wording with/without any additional introductory wording that you consider necessary. Martin Hogbin (talk) 21:59, 5 August 2012 (UTC)

I emailed medcom (3 days ago), asking for a volunteer to finalize the question and suggest a decision rule (after individually asking Sunray, Andrevan, AGK, and WGFinley). -- Rick Block (talk) 22:20, 5 August 2012 (UTC)

Would you be happy to leave the finalising completely up to the volunteer? Would you be happy for Guy to do this, if willing? Martin Hogbin (talk) 23:25, 5 August 2012 (UTC)

Yes. -- Rick Block (talk) 03:01, 6 August 2012 (UTC)

And you? Just FYI, Guy (WGFinley) has suggested we submit a mediation request to agree on the RFC. I think if we both agree to let a third party finalize what we have there's no need to do this. BTW, if you don't agree, I will be proceeding with an RFC on my own (with or without a 3rd party finalizing anything). -- Rick Block (talk) 05:35, 6 August 2012 (UTC)

I would be happy for Guy Macon to do it if he is willing.Martin Hogbin (talk) 18:18, 6 August 2012 (UTC)

I am willing to do anything that I can to help. As an aside (not in response to the comment above), given the fact that there is another editor in this thread with "Guy" in his name, it is probably best to call me "Guy Macon", at least until I achieve my goal of having the Wikipedia community award me the title of Dalek Supreme... --Guy Macon (talk) 21:57, 6 August 2012 (UTC)

Thanks Guy. Rick and I agree the wording above. Perhaps you could start an RfC with any additional wording you think necessary. If neither proposal is accepted than we can continue looking for other solutions. I think it would be worth making an effort to get the maximum number of users to respond to the RfC.

I think we both agreed to limit the number of words used by Rick and myself to support our proposals and respond to others. It also might be a good idea to ask other regulars here to (voluntarily) limit their contributions also. Martin Hogbin (talk) 08:22, 8 August 2012 (UTC)

Rather than trying to hammer out such details, I suggest that one or both of you hand me the text you prefer, and if you know it, a brief note about what you think the other one prefers. I will put it into RfC format with what I think are a good set of ground rules (subject to what is and isn't allowed in an RfC). I will run it by everyone who follows this talk page for comments and suggestions (but not approval or veto), tweak it as needed, and ask "is there one editor who is willing to be the submitter of this RfC?" If no, in the dustbin it goes. If yes, (only one editor needs to agree) I will submit the RfC and advertise it in the appropriate places.

So what happens if some other editor objects to the RfC? Nothing. It only takes one person to submit an RfC. I will, of course try very hard to write it so that both sides are willing to put their name on it as submitter, but agreement is not required. I really have have no preference as to the outcome and I have the same generally positive feelings about everyone involved, so I should be able to do a reasonable job of avoiding obvious bias. --Guy Macon (talk) 15:08, 8 August 2012 (UTC)

We've agreed to everything in the first show/hide box at http://en.wikipedia.org/wiki/User:Sunray/Discussion_of_Monty_Hall_RfC#Outside_comments_on_including_NPOV_in_the_question except the five words Martin objects highlighted in yellow. Discussion about a consensus rule, without agreement, is at http://en.wikipedia.org/wiki/User:Sunray/Discussion_of_Monty_Hall_RfC#Where_are_we.3F. -- Rick Block (talk) 15:21, 8 August 2012 (UTC)

As Rick confirms we have agreed almost everything and are happy for you to use your discretion and judgement to decide the rest. As you say, it does not require the agreement of anyone to have an RfC. If either proposal is accepted that will be a consensus decision. If neither proposal is accepted then anyone else is quite free to make their own proposal and have an RfC on it. Martin Hogbin (talk) 23:40, 8 August 2012 (UTC)

The Monty Hall Paradox or The Monty Hall "Problem" ?

The world famous "paradox", in a given scenario, is absolutely correct in saying:

Just only two still closed doors in a given scenario, one of them containing the car, the other one containing a goat,
but chance to win the car by swapping from the door initially chosen by the guest to the offered still closed door of the host is not 1/2, but exactly 2/3.

This is quite counterintuitive but absolutely correct in a well defined given standard scenario. As said, this is the world-famous paradox.

But the Monty Hall "Problem" is trying to show that, in quite other assumable scenarios, offside the world famous paradox, additional information could be cogitable by some host who is NOT obliged to maintain secrecy regarding the car hiding door, so that – in those quite other scenarios, and derogating from the standard scenario of the world famous paradox

the chance to win by swapping could be 0 if the host should offer the swap only if the guest should have chosen the prize,
or the chance could be 1/2 if the host, not knowing the actual location of the car, eliminates the chance on the car in 1/3 and only in a subset of 2/3 just "happens" to coincidentally open a door hiding a goat and not hiding the car,
or the chance has to be within the fixed range of at least 1/2 to full 1/1, so on average exactly 2/3 though, depending on a completely unknown but supposed asymmetry in the special behavior of the host, if in 1/3 he opens one of his two losing doors, NOT being obliged to maintain secrecy regarding the car hiding door.

As said, all of this is outside the standard scenario of the world famous paradox that the article pretends to present and that should be the point of the article, just in respect of the readers.

The actual article is a confusing mingle-mangle of all of that, unnecessarily presenting just lessons in conditional probability theory, making it very hard for the reader to grasp what the smoke-screen-article in fact is all about. That intolerable unclear mingle-mangle finally should have an end. The structure of the article should be clear and intelligible, and the title of the lemma should read again "The Monty Hall Paradox", as it was before the sanctioned page ownership appeared. Gerhardvalentin (talk) 09:54, 28 May 2012 (UTC)

Um, what? The page was created as Monty Hall problem on Sept 22, 2001 with this edit. It remained at that name until March 23, 2006, when it was moved to Monty Hall Paradox and then quickly moved back on March 26, 2006 (discussion concerning this is in the archives here). It has remained at this name since. -- Rick Block (talk) 19:04, 28 May 2012 (UTC)

It's interesting to note that in Russian, Ukrainian, Polish and Hungarian WPs this is called a "paradox" in the article titles. In all the Western European languages that I can read (tennish) they go for "problem". Can't read the Asian ones :) Malick78 (talk) 21:27, 28 May 2012 (UTC)

I think Gerhard is actually making a point about the irrelevance of the 'conditional' solutions to this famous puzzle rather than quibbling about the name of the article. Martin Hogbin (talk) 21:41, 28 May 2012 (UTC)

Exactly, but not against a conditional solution per se but for a clear structure of the article showing the "clean paradox" that does not need lessons in conditional probability theory. Showing later that quite other scenarios do not affect the "clean paradox". And any conditional formula – if any – just in the form of clear odds. Gerhardvalentin (talk) 22:08, 28 May 2012 (UTC)

JeramieHicks, please read the above also. Gerhardvalentin (talk) 00:35, 3 August 2012 (UTC)

The best paradoxes raise questions about what kinds of contradictions can occur – what species of impossibilities are possible.

William Poundstone, Labyrinths of Reason (1988), p. 19

There is a vernacular sense in which riddles like this are sometimes called paradoxes but, as a lover of true logical paradoxes, I am not happy with labeling the Monty Hall problem this way (and certainly not with all capitals in discussion below, as if this were a technically defined usage). The conflict between the 50:50 and 1/3:2/3 answers does not involve a logical contradiction, it is simply a case where logic contradicts naiveté. ~ Ningauble (talk) 15:12, 13 August 2012 (UTC)

Not happy with naming the paradoxical situation to be a paradox? – But many academic sources call it exactly that way, and also a page of university of California, San Diego, department of mathematics calls it to be a "paradox", despite your reservation.
They even explicitly say: This problem was given the name "The Monty Hall Paradox",
and – in contrasting the paradoxical probability to win by switching (pws) of 2/3 with the non-paradoxical pws of 1/2 in the scenario of some "unaware host",
they explicitly say: The change in the statement of the problem is so slight that this might be the reason this problem is such a "paradox."

Contrasting exactly those two (slightly) different scenarios would be beneficial for the still retarded article, to help the readers to grasp the cause of that veridical paradox. I fear the cloudy article might stay to be retarded for the next some years. Gerhardvalentin (talk) 11:25, 15 August 2012 (UTC)

It is not a paradox, that's why we call it the MH Problem. Nijdam (talk) 13:12, 15 August 2012 (UTC)

What is the conflict about

This world famous paradox explicitly says

that the car and the Goats were placed randomly behind the doors before the show and
in any case the Host is to open a door showing a Goat, offering a switch to his second still closed door,
and nothing can ever be known about any possible Host's preference, if in 1/3 he should dispose of two Goats, as he then chooses one uniformly at random, being bound to maintain absolute secrecy regarding the car hiding door.

This well defined scenario of the world famous counterintuitive paradox ensures that, neither by the Guest's initial choice of door nor by the Host's opening of a special loosing door we could have learned anything to allow us to revise the odds on the door originally selected by the Guest, nor compared to the entity of the group of all unchosen doors, as a whole. Absolutely nothing. No additional information whatsoever, as per the unambiguous and mathematically fully correct scenario of Krauss&Wang, Henze and others.

Once more: In this scenario the odds on the door originally selected by the Guest remain unchanged 1/3, as we have learned nothing to allow us to revise the odds on the door originally selected by the Guest.

And this scenario ("in any case the Host is to open a door showing a Goat") also firmly excludes any forgettable Host who in 1/3 of cases disrupts the chance on the car, and will show a Goat only in the subset of 2/3 when, by chance, he just "happens" to show not the Car but a Goat, reducing herewith the chance on the car by swapping from 2/3 to 1/2.

In this scenario the only correct answer forever only can be based on the average probability of 1/3 by staying vs. 2/3 by switching, as no better knowledge will ever be available.

Actually, the article still is not on the famous paradox, but is about teaching conditional probability theory, staring on door numbers.

Once more: The article, based on actual academic sources, should well arranged report about an extremely counterintuitive but nevertheless clear and very exactly defined world famous paradox with its understood basic standard parameters. So, with respect of the readers, should in the first line pay respect to that clear and well defined standard scenario. That should be the point of the article, just in respect of the readers. But actually the article is an obfuscating and confusing mingle-mangle, not even trying to correctly and clearly address the world famous paradox with its distinct scenario and its well-defined and exact certain information content. As said, those basic parameters of the standard version show a clear scenario, with a firm information content, yes they clearly define the exact information content of this paradox. They firmly and emphatically are excluding, from the outset, in the first place a priori and explicitly, and in a very assertive way that, after an unselected door – irrespective of its door number – has been shown to be a loser, any eventual connected "possibly to be gathered additional information" on the chance of the door originally selected by the Guest could ever be supposed to exist. The standard version clearly shows that such additional information has firmly been excluded and never can be given nor can be expected. And, as a consequence, also definitely excluding any eventual connected possibly to be gathered additional information on the odds of the group of all unselected doors altogether, as a whole entity likewise. – But, quite contrary, and quite outside the famous paradox and its firm scenario, the article is still telling in a misty and treacherous way:

"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."

Cheekily and hideously ignoring the clear and unambiguous scenario the paradox is based on. This article actually still is a lesson in teaching conditional probability theory in a classroom, in an educational way eagerly distinguishing "before" and "after", but never about the core of this world famous paradox.

Woolgathering of teachers in maths may belong to an article on Bayes, but not to this counterintuitive paradox. So a clear structure is indispensable, in any case such woolgathering in later sections at the end of the article, titled "All kinds of quite different other scenarios that, outside the paradox, are used in classrooms to teach conditional probability theory".

But never within the main section about this explicit well defined and clear scenario that this world famous paradox is based on. Just to show respect for the readers.

Rick, we all know that maths-teachers see it from an entirely different view, not addressing the paradox, but addressing lessens in maths. Please help to address the paradox instead, to make this article intelligible and beneficial for the readers. --Gerhardvalentin (talk) 08:51, 31 May 2012 (UTC)

Thank you for your opinion about what the true paradox is. However, I am much less interested in your opinion than in the opinions of reliable sources, whose views must (per WP:NPOV) be represented "fairly, proportionately, and as far as possible without bias". Whether we agree with what reliable sources have to say or not, editing MUST be done from a neutral point of view. Another way to say this is that we must not inject our own views into the article. I'm happy to work toward that goal. Are you? -- Rick Block (talk) 19:25, 31 May 2012 (UTC)

Rick, you have to present what reliable sources really say, and you are not to higgledy-piggledy mix incoherent multifocal perspective. There are sources on different scenarios, and you have to keep different scenarios apart. No source says that in the standard scenario, where the car and the goats were placed randomly behind the doors before the show, and in any case the host is to open a door showing a goat, offering a switch to his second still closed door, and nothing will ever be known about any possible host's preference, if in 1/3 he should dispose of two goats, as he then chooses one uniformly at random, that in this standard scenario only a player that always switches will have a chance to win of 2/3, but that "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."

To obfuscate by confusingly mixing and not distinguishing what different sources – based on their respective subject – really say, and not to clearly distinguish those various specific views, is not the basic of an article that should be intelligible and beneficial for the reader. We all, and you too should pay respect for the reader. So please do not call that incoherent misty higgledy-piggledy mix of multifocal scenarios the article actually shows (or hides?) to be "what the sources say". Please help to put things together, but correctly structured and intelligible, and not nebulous convoluted. -- Gerhardvalentin (talk) 22:28, 31 May 2012 (UTC)

Gerhard, you are not alone. Even Morgan et al, who created this whole mess, have now retracted their argument that the answer is anything other than 2/3. You are also quite right that there are no sources which state that the conditional solutions are required when the host is known to choose evenly. This nonsense belongs is a section on academic extensions to the problem. Martin Hogbin (talk) 22:50, 31 May 2012 (UTC)

Gerhard - is there some change you're suggesting to the article? If so, what is it? -- Rick Block (talk) 18:37, 1 June 2012 (UTC)

Rick, so you really don't see the hideous state of that article? An article that perkily pretends to present a world famous counterintuitive paradox that, under the strict rules of a very special appropriate and well known standard scenario, provides just only one correct answer to the question

"In this given scenario, is it to your advantage"?

A scenario, that explicitly excludes that the Host offers a swap only if the Guest should have selected the prize, and that explicitly excludes that the host will ever be showing the car, a scenario that explicitly excludes that the Host's special behavior in selecting one of two goats, e.g. in opening of his "strictly avoided door" ever could signalize that the odds on the door offered to swap on could be beyond average? A scenario that explicitly excludes that those odds can ever differ from average? In this given world famous standard-scenario, there only can be one correct answer.

This standard scenario comprising every single one of the following possible constellations likewise:

After the Guest selected door #1, the Host has opened door #2

After the Guest selected door #1, the Host has opened door #3

After the Guest selected door #2, the Host has opened door #1

After the Guest selected door #2, the Host has opened door #3

After the Guest selected door #3, the Host has opened door #1

After the Guest selected door #3, the Host has opened door #2

Every single one of those possible constellations is / are comprised likewise in this standard scenario.

But the article impudently says that this is incomplete or solves the wrong problem, for It does not consider the question: given that the contestant has chosen Door 1 and given that the host has opened Door 3, revealing a goat, what is now the probability that the car is behind Door 2? And the article says 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.

Yes, I said impudently. Because the obfuscating, nebulous and indistinct article does not distinguish what scenario it is reporting on.

Another comment: Martin has just proposed today to delete this error from the article. He says "In the light of Morgan's retraction I suggest that we remove that error", and I do support that proposal. -- Gerhardvalentin (talk) 10:00, 9 July 2012 (UTC)

There should be a clear and clean split between the world famous paradox and its standard scenario (say 80 percent), and later other sections on quite other, on quite different scenarios that some sources are talking about (say 20 percent).

The article should be 80 percent on the famous counterintuitive paradox, trying to make it intelligible for the readers that the chances are not "1/2 : 1/2" but that they are "1/3 : 2/3". This section could also present a short conditional formula in odds form. And max. 20 percent of the article should be on quite other, on quite DIFFERING scenarios that some out-of-date sources had in mind. Showing that door numbers could only be of relevance in case that some Host's bias could be suspected, otherwise meaningless and just interesting in teaching probability theory.

And all of that Bayes' formula back to the article of Bayes, and just a link to that article. The article is a mess and should clearly present that counterintuitive paradox, and should stop to be indistinct, nebulous and obfuscating. I hope you can help. With respect for the sources and what they really say and address to, and with respect for the readers. -- Gerhardvalentin (talk) 08:58, 2 June 2012 (UTC)

Rather than move the formal derivation to the Bayes' theorem article (the derivation of this problem has no particular significance in the context of that article), how about moving it to the end of the article as it was when the article was a featured article (for example, this version). Would that help? Is there anyone who would be opposed to this? -- Rick Block (talk) 19:01, 2 June 2012 (UTC)

No Rick, the "formal derivation" does not really need Bayes'. Just present a short proof in odds form right at the beginning, or just write the following:

If for example the Guest initially selects Door 1, and the Host opens Door 3, the probability of winning by switching is 2/3:

P(C_{=2}|H_{=3},S_{=1})={\frac {{\frac {1}{3}}\times 1}{{\frac {1}{3}}\times ({\frac {1}{2}}+1+0)}}={\tfrac {2}{3}}.

This is valid for any door the Guest should select, and valid for any other door the Host should open.

And you can show that clearly arranged table also: (new headline added 13 August 2012) Gerhardvalentin (talk) 12:04, 13 August 2012 (UTC)

Table

Say, Guest selected D1	Door opened by Host
Initial arrangement (probability)	Open D1 (probabilty)	Open D2 (probability)	Open D3 (probability)	Joint probability	Win by staying	Win by switching
Car Goat Goat (1/3)	No	Yes (1/2)	No	1/3 x 1/2	Yes (1/6)	No
Car Goat Goat (1/3)	No	No	Yes (1/2)	1/3 x 1/2	Yes (1/6)	No
Goat Car Goat (1/3)	No	No	Yes (1)	1/3 x 1	No	Yes (1/3)
Goat Goat Car (1/3)	No	Yes (1)	No	1/3 x 1	No	Yes (1/3)

We observe that the player who switches wins the car 2/3 of the time. We also see that Door 3 is opened by the Host 1/2 = 1/6+1/3 of the time (row 2 plus row 3), as must also be the case by the symmetry of the problem with regard to the door numbers — either Door 2 or Door 3 must be opened and the chance of each must be the same, by symmetry.

And please do not cite sources that say "incomplete" or that say "that does not mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the Host opens" without mentioning that those sources clearly are ignoring the only correct standard scenario that shows the world famous paradox, and that the paradox is based on, but that those sources report on quite different scenarios, where the world famous paradox never existed. On quite different scenarios, not just on "variants". The article should be clear and intelligible for the readers. Regards, -- Gerhardvalentin (talk) 15:01, 3 June 2012 (UTC)

The Bayes-based formal derivation has been in the article for a very long time, including two versions that were approved by the entire community during featued article reviews [9] [10]. I don't understand how moving it to the end of the article so that it is effectively an appendix does not address your concern. Including this does not mean any formal derivation needs Bayes. -- Rick Block (talk) 16:41, 3 June 2012 (UTC)

Rick, you correctly say ... showing ... the Bayes-based formal derivation ... at the end of the article... does not mean any formal derivation needs Bayes." So why are you of the opinion that it is still a necessity, though?

But the really important point is not to obfuscate. To present the clear MHP with it's only correct decision: to swap doors. And to clearly show that correct scenario of symmetry. Only this scenario can give the famous answer to the famous question, based on the chance to win the door by staying in only 1/3, and to double it to 2/3 by swapping. Only the correct scenario forever firmly excludes that the Host could be forgetful and firmly excludes that the Host could be biased. Only the correct scenario gives the correct probability that has forever to be the exact average probability. Show that in this scenario the conditional probability for door #2 and for door#3 is identical if the Guest should have chosen door #1, and that this is true for all permutations. Show it by Bayes' factor or by Bayes' rule in odds-form:

The odds on the car being behind Door 1 given the player chose Door 1 are 1:2 against. If the car is behind Door 1 (the door chosen by the player) the probability the host opens Door 3 is 0.5 (no host-bias). If on the other hand the car is not behind Door 1 (the door chosen by the player) the probability the host opens Door 3 is 0.50, because the car is now equally likely behind Doors 2 and 3, and he is forced to choose the other one. The Bayes' factor or likelihood ratio is therefore 0.5/0.5 = 1, and the posterior odds remain at 1:2 against.

And, in citing sources that address quite different, quite other scenarios, make clear that they do not address the usual standard scenario. Try to help the readers to grasp what that famous paradox is about, and what it isn't about. - Gerhardvalentin (talk) 20:46, 3 June 2012 (UTC)

I'm not saying it's necessary, I'm saying it should not be deleted from the article. The reason I think it should be in the article is because it or something very much like it is in pretty much any serious source about the Monty Hall problem, including Selvin's second letter, Devlin's follow up column, Krauss & Wang (which is fundamentally a psychology article, not a math article!), etc etc etc - so the article would not be complete without it. You apparently don't like using Bayes' Theorem. No one is saying you or anyone else has to. -- Rick Block (talk) 21:26, 3 June 2012 (UTC)

Regrettably you do not answer to what I say, but focusing on the unnecessary and secondary Bayes' formula only. I repeatedly said that the article is obfuscating for the (new) reader, in not distinguishing on what hell of scenarios it is talking about, mixing different scenarios with quite different characteristics without clearly to distinguish what that means. The Bayes' formula is not the main concern, although the article should not unnecessarily be messed into a lesson on conditional probability theory. You can say things clearly, and to show probability by Bayes' rule in odds-form would be much more suitable for this article. But please address to the main aspects I mentioned, are you willing to follow those aspects? What do you approve, and what are you disapproving? Can you help to make the article on this world famous paradox clear and intelligible as it should be, even for the (new) reader? As to me, Bayes is just one small side aspect. Regards, -- Gerhardvalentin (talk) 22:35, 3 June 2012 (UTC)

If you're looking for me to in some way pledge allegiance to The Truth as you see it, you're out of luck. My allegiance is to what reliable sources have to say. I don't care even the teeniest tiniest bit what you think The Truth is, so if this is your goal here just stop. Rather than argue about The Truth, I'm perfectly willing to talk about specific changes you're interested in. You've suggested deleting the section presenting a formal proof using Bayes' Theorem showing that the conditional probability of switching to Door 2 given the player initially picks Door 1 and the host opens Door 3 (which also applies to any other combination of initial player pick and door the host opens) is 2/3. My response has been a suggestion to move this to the end of the article so it is effectively an appendix rather than delete it, since it (or something like it) appears in many reliable sources. If this is not the the main change you're interested in, what is? -- Rick Block (talk) 05:08, 4 June 2012 (UTC)

Rick, you should at least show respect for the sources, they are going from different points of view. The article is confusingly decomposed, never clear what it is just talking about. The editor's duty is to represent the sources in their respective context, intelligible and not confusingly mingling different aspects. Otherwise the article is crap. It is a pity that you do not want to respect the sources. It is the whole article that shows that narcissistic dilemma. Please help to make it clear and intelligible, with respect for the readers. -- Gerhardvalentin (talk) 10:47, 4 June 2012 (UTC)

Again, if there are any specific changes you're interested in by all means bring them up, or just fix it. If we can return to the formal Bayes' Theorem section, would moving this section to the end of the article help address any of your concerns? -- Rick Block (talk) 15:18, 4 June 2012 (UTC)

Okay, the explanation after the first table in "Solutions" reads "A player who stays with the initial choice wins in only one out of three [...] while a player who switches wins in two out of three.", so I changed columns accordingly (not "switching <> staying", but "staying <> switching"), according to that explanation. Okay? -- Gerhardvalentin (talk) 09:09, 5 June 2012 (UTC)

Rick, you have moved the Bayes' Theorem (Bayes-based formal derivation) to the end of the article, saying "Thus, if the player initially selects Door 1, and the host opens Door 3, the probability of winning by switching is [...] 2/3". But unfortunately you forgot to mention there what you have written here on the talk page, just a few lines above:

"... which also applies to any other combination of initial player pick and door the host opens) is 2/3."

Please add these (your) words there, because that (otherwise unnecessary) formal derivation should be cleared away completely, yes replace it by a formal derivation in clear odds-form, showing that door numbers are completely irrelevant for the standard scenario. And please make clear there in the article that antique sources that prominently stick on door numbers are not addressing the (since decades) world famous paradox with its unique famous standard scenario, that never can exist anywhere else, but are talking of quite different stories, not addressing the world famous counterintuitive paradox itself. And you should say there that such differing scenarios are prominently used in maths classes to teach probability theory. Please keep different scenarios clearly apart. Thank you. -- Gerhardvalentin (talk) 01:41, 19 June 2012 (UTC)

Does the source that is referenced (Gill 2002) say this? If so, why do you ask me to do this rather than doing it yourself? If not, then to what source would you attribute this? I simply moved the section without changing it. It is not in any sense of the word "mine". -- Rick Block (talk) 02:15, 19 June 2012 (UTC)

Rick, the counterintuitive "paradox" (not "1/2 vs. 1/2" but counterintuitive "1/3 vs. 2/3") is valid for each and any single game based on its unique famous well defined standard scenario called the "standard problem". This standard scenario explicitly ensures that, after the host has shown a goat, we have learned absolutely nothing to allow us to revise the odds on the door first selected by the guest (Falk) and hence – in this well-defined scenario – we've also learned absolutely nothing to allow us to revise the odds on the entity of the group of all unchosen doors, as a whole. Absolutely nothing. This shows and assures that door numbers are irrelevant for the standard problem.

The invulnerable "standard problem" assures that conditioning on door numbers is an unimportant side-show there, in this unique famous and well defined "standard problem". Nevertheless you can show the chances by switching in a clear mathematical odds-form, just at the beginning.

Please help to show what the famous paradox is about, in a well organized article. Any differing, any deviating scenario that does not concern the standard problem, should be presented clearly delineated, for the benefit of the (average or new) reader. In later and special sections you can present one deviant scenario, allowing the chance for switching to vary from (assumed) 1 at max. to (assumed) 1/2 at min., but never less, so on average exactly 2/3, though, affirming that it is wise to switch in each and any given game, even in that one deviant scenario: "the more biased, the better!". And you can show there another special deviant scenario of the forgetful host, who will show the car instead of a goat in 1/3 of cases, disrupting the chance on the car in that 1/3 of cases, and who will show a goat only in 2/3 of cases, so reducing the chance to win by switching from 2/3 to 1/2.

And in another section you can show there the scenario of the sneaky host also, who offers a switch only if the guest, by chance, should have selected the car and not one of the two goats.

And you can show maths there in those later sections, where it belongs, with just assumed differing scenarios outside the "standard problem" that will just lead to assumed probability of 0 to 1, for the apprentice (outside the "standard problem"). Nevertheless this article should not be degraded to a nonrelevant maths lesson in conditional probability theory.

We should stop to disconcert and stop to disorient. A clearly structured outline will help the reader to grasp what quite differing and opposing scenarios are talking about. From the clean and invulnerable "standard problem" with its firm answer/decision to other and quite differing stories. -- Gerhardvalentin (talk) 14:32, 19 June 2012 (UTC)

I agree with much of what Gerhardvalentin says. Glrx (talk) 17:38, 22 June 2012 (UTC)

The never ending conflict could easily be settled: stop fobbing the readers. Gerhardvalentin (talk) 16:53, 11 July 2012 (UTC)

JeramieHicks, please read the above also. Gerhardvalentin (talk) 00:35, 3 August 2012 (UTC)

The article still fobs the reader

No progress, it is still the same. The article pretends to present the world famous "standard version" of full symmetry, citing the famous "extended description of the standard version" by Krauss and Wang (2003), and it shows correct solutions that explain the paradox and give the correct answer to that fully symmetrical "Krauss and Wang STANDARD version": This famous version secures that the technical value of unconditional probability and of conditional probability unavoidably must in any case be fully identical (pws:2/3) and in accordance with the average pws. The host is choosing uniformly at random if (in 1 of 3) he should have two goats, so any host's bias "q" is a priori excluded, from the outset. No necessity at all to condition on door numbers, their color or their placement. Everyone is free to do so, but this forever will be completely effectless. This is the charm of that world famous counter-intuitive veridical paradox.

But note: Quite another thing is that a large number of textbooks on probability theory like to pay special attention to undue "still assumed" host's bias "q" for the purpose of teaching and learning conditional probability theory. But this is being quite outside the world famous counter-intuitive standard version of the paradox with its only correct decision to switch to the door offered, based on the only to ever be known average probability.

But the article, instead of helping the reader to grasp what all is about, still likes to inspire the reader with distrust, skepticism, suspicion and disbelief. The owners of the article for years liked to use one special trick: by first of all reporting on sources that criticize quite other scenarios outside and diametrically contrary to the symmetrical standard version presented in the article. You can show conditional probability and the forbidden "q", yes. But where it belongs, in a section that tells about devious scenarios.

Without saying in a "new distinct headline" that they are leaving now the field of the standard version, they like to abruptly and explicitly argue about halve-baked confusion, citing sources saying that any "simple solution", not using conditional probability and not using the variable "q" to express the illegal unfounded and excluded and therefore inapplicable "assumed host's bias", are shaky, incomplete, misleading, bluntly false and not answering the "right" question. Without a clear headline that they already have left the symmetrical standard version, and not saying that they report now on quite other scenarios, beneficial for teaching conditional probability theory only. The old special trick is still alive.

Rick Block has just edited a section now where all of that applies. Although we have been fighting for a clear structure of the article, he still mixes different scenarios, different versions without a clear headline to distinguish what he just is reporting about. Nopeless. Gerhardvalentin (talk) 16:53, 11 July 2012 (UTC)

JeramieHicks, please read the above also. Gerhardvalentin (talk) 00:35, 3 August 2012 (UTC)

Edit war

I am perplexed by a recent sequence of reverts:

On July 21-23 Gerhardvalentin rewrote portions of the article.
On July 24 Nijdam reverted the changes with the comment "revert to by most editors approved version".
On July 26 & August 5 an IP sock attempted to restore the changes, and was reverted by Nijdam and Guy Macon.
On August 5 Martin Hogbin reverted to the change by Gerhardvalentin with the comment "These changes have not been discussed and there is certainly no consensus for them."

Note that the final revert (4) actually restored the recent changes (1). I agree with the two editors (2, 3) who attempted to revert these changes, and there does not appear to be a consensus for keeping them. I think the changes can be criticized for injecting awkward or inappropriate language, personal opinion, and polemical tone into the article.

Do we need to discuss the changes in detail, or can we get a quick consensus on keeping or removing them? ~ Ningauble (talk) 15:59, 5 August 2012 (UTC)

My revert had nothing to do with content. If a change needs to be made it needs to be made by a non-banned editor -- if a sockpuppet of a banned editor makes an edit, I don't even look at it before reverting it, at which point the real editors can continue editing the article as if the sockpuppet had never been there. --Guy Macon (talk) 17:03, 5 August 2012 (UTC)

To wit: My edit of July 21-23 was fully based on the sources that Kmhkmh had just mentioned. Nijdam reverted with an inapplicable comment. As to awkward or inappropriate language, please help to clean it. Tone should not be polemical (?). Please help the article to separate the clean effectual paradox, existing in a well defined unambiguous and mathematical correct scenario, from other scenarios that also should be discussed. But please help to get rid of the mingle-mangle state of the article. Stop to fob the readers. Gerhardvalentin (talk) 17:19, 5 August 2012 (UTC)

And as to the "polemical tone": It's what reliable sources say: "Beispielsweise könnte der Moderator für den Fall, dass er eine Wahlmöglichkeit zwischen zwei Ziegentüren besitzt, mit einer bestimmten Wahrscheinlichkeit q die Tür mit der kleineren Nummer wählen (s. Übungsaufgabe 15.3). Beispiele wie 15.3 finden sich häufig in Lehrbüchern zur Stochastik. Ihr einziger Zweck besteht darin, mit bedingten Wahrscheinlichkeit schematisch rechnen zu üben." It is not enough to dislike what modern reliable sources say, to keep them out of the article. Gerhardvalentin (talk) 18:30, 5 August 2012 (UTC)

I agree with Ningauble that these changes are mostly not for the better. -- Rick Block (talk) 17:56, 5 August 2012 (UTC)

Please regard WP:NPOV and WP:Technical. Once more: First of all, your POV of disliking what modern reliable academic sources explicitly and almost bluntly say and are commenting on this very matter, is not enough to permanently and intolerably trying to blanking them out from the article. And observe WP:Weight Gerhardvalentin (talk) 19:56, 5 August 2012 (UTC)

I reverted only for the reasons stated in my edit summary. Maybe I misread the changes. It looked to me as though Nijdam had removed the, long standing, 'Combining doors' solution from the article.

I also agree with Gerhardvalentin's comments above but accept that his changes could do with some rewriting by a native English speaker and some reorganisation. With the current impass however, collaborative editing is impossible. Martin Hogbin (talk) 20:35, 5 August 2012 (UTC)

Thanks. Please help to get the article what it is expected to be by actual reliable academic sources and their weight and, first of all, by Wikipedia policy and by the readers. Please help. Everyone is invited to forget about any unnecessary conflict, in favor of the article. Gerhardvalentin (talk) 21:02, 5 August 2012 (UTC)

Nijdam has reverted Gerhard's changes again, so it seems like we may want to discuss them. I think there are basically three.
1) Change all instances of "standard problem" to "symmetric problem", including changing the heading "Other simple solutions" to "Other simple solutions based on symmetry".
2) Add a paragraph in the "simple solution" section comparing a strategy of staying with a strategy of switching (sourced to Henze).
3) Move the paragraph about Falk's "equal probability" assumption from the "Sources of confusion" section to the "Variant" section, renaming this section "Variants of asymmetry, assuming a non-confidential host".
If we're going to discuss these, I think it may be helpful for someone not strongly vested in the topic (Guy Macon?) to help structure the discussion. Per the thread above, we may be on the verge of a related RFC, so another possibility is that this discussion is premature. In any event, I strongly suggest all edit warring stop immediately. -- Rick Block (talk) 05:58, 6 August 2012 (UTC)

To make clear what the text of the article actually is talking about, we should explicitly say that

the "standard version of the problem" is a "symmetric standard version of the problem", and that all "simple solutions", without exception, are clearly based on this "symmetric standard version", so on the standard version of symmetry.
Henze is explicitly basing on the symmetric standard version, and he shows it by Bayes, and then he shows it by "strategy" what the actual chances" staying : switching" are "1/3 : 2/3" in any actual case.
Falk is clearly excluding any difference of "before : after" if we do not have "knowledge" of some host's bias (otherwise we "would have learned nothing to revise the odds on the door first selected by the guest").
As she clearly addresses "known host's bias", and not to confuse the readers, these words of Falk belong to the section "Variants of asymmetry by assuming a non-confidential host", yes.

We should go back to my proposal and then discuss these items one by one, making the article clearer for the reader, step by step. Gerhardvalentin (talk) 11:09, 6 August 2012 (UTC)

I'm lost. Nijdam (talk) 11:25, 6 August 2012 (UTC)

Rick, will you help to make the article on the world famous PARADOX clear and intelligible, by distinguishing the distinct PARADOX that only can exist in its world famous symmetric scenario, separating it from quite other scenarios where the PARADOX never existed nor ever can nor will exist (biased host has NOT to keep secrecy regarding the door hiding the car, etc. etc.)

The article - after Nijdam's revert - again is a mingle-mangle of PARADOX and NON-PARADOXICALS. Please let us discuss how the PARADOX can be separated and clearly distinguished from quite other scenarios, scenarios that the paradox has never been based on? I would like to go back to my last version, and we could discuss item per item. Gerhardvalentin (talk) 21:04, 7 August 2012 (UTC)

Gerhard - 3 editors (Nijdam, Ningauble, and I) have indicated a preference against at least the totality of your changes. Rather than "go back" to your last version I think it would be more appropriate to discuss them individually item per item leaving the article as is for now. Is my list above accurate? If so, how about taking them one by one (one at a time) creating a new section to discuss each. Note that if the RfC being discussed just above actually happens, it might be better to wait until it is resolved. -- Rick Block (talk) 15:30, 8 August 2012 (UTC)

Thank you Rick for your proposal. Your interpretation of my concern is correct, let me repeat my answer from above:

Separating the valid clean PARADOX from asymmetric scenarios

To make clear what the article actually is talking about, the reader should be informed that

the "standard version of the problem" is a fully symmetric standard version of the paradox, and that all "simple solutions", without exception, are clearly based on this standard version of full symmetry.
Henze, not being a mulish "frequentist", and others are "of course" explicitly basing the clean paradox on the symmetric standard version of a host who is bound to maintain absolute secrecy regarding the car hiding door. Show Henze's "strategy" that explains why the actual chances of "staying : switching" are exactly "1/3 : 2/3" in any actual case.
Falk is clearly excluding any difference of "before : after" if we do not have "knowledge" of some host's bias (she says "and you know about that bias"). In absence of a "known bias", "we've learned nothing to allow us to revise the odds" (on the door first selected by the guest).
As she clearly addresses some "known host's bias", and not to confuse the readers, these words of Falk belong to the section "Variants of asymmetry by assuming a non-confidential host".

Which item do you prefer to begin the discussion? And as to your proposed RfC: I'm going to ask Guy Macon myself to help me with my own proposal to clearly distinguish from now on the clean valid paradox from any other asymmetric scenario, and to stop indistinctly mingle-mangle different sources of quite different vantage point in indistinctly hiding such discrepancy. I am asking you to help me, also. Regards, Gerhardvalentin (talk) 17:56, 8 August 2012 (UTC)

Gerhard this is exactly what my proposal on the impending RfC will do. Both sides will gain. The simple, notable, unintuitive probability puzzle will be first covered as exactly what it is, a simple, notable, unintuitive probability puzzle with no distractions. When that is done, we will be free to discuss the finer points of probability, problem variations, the 'right' and 'wrong' ways of solving the problem in a calm and scholarly manner based on what the sources have to say on the subject. Martin Hogbin (talk) 23:47, 8 August 2012 (UTC)

Yes, Martin. The world famous PARADOX only "exists" in – and consequently "is" based on its fully unambiguous and mathematically correct defined scenario. (Uni of San Diego calls it "senario"). This "paradox" never existed and never can nor will "exist" anywhere else.

And – after the paradox has completely and exhaustively been presented – the "core reason" for the deficient and faulty 50:50 intuition should be presented in the first section of showing "Other scenarios outside the paradox", of leaving the correct scenario. It is the "Monty does not know"-version, where in a full "1 out of 3" the chance to win by switching will be wasted and destructed and so the chance to win by switching can only be 1/2.

Only thereafter quite other unfounded and inapplicable scenarios of some assumed host who is NOT bound to keep secrecy regarding the car hiding door, outside the PARADOX, should be shown as examples used in classrooms of schematically training probability & statistics, based on relevant textbooks. Such clear structure of the article is indispensable to finally stop fobbing the readers. Will you sign my RfC, also? Gerhardvalentin (talk) 10:18, 11 August 2012 (UTC)

What is 'a fully symmetric standard version of the paradox'? Nijdam (talk) 07:48, 9 August 2012 (UTC)

And what is there meaning of 'the clean valid paradox'?Nijdam (talk) 07:55, 9 August 2012 (UTC)

The fully symmetric scenario is is what MvS said to have been the basics of her answer: the objects are supposed to have been placed uniformly at random before the show (as we have no better evidence), the guest is supposed to have selected her initial door uniformly at random (as we have no better evidence), and the host is obliged to maintain absolute secrecy regarding the car hiding door (as we have no evidence of defraud). Thats the fully symmetric standard version of th paradox, as per Krauss and Wang and others. And the paradox is depicted as follows: Just only two still closed doors, one of them containing the car, the other one containing a goat, but chance to win the car by swapping from the door initially chosen by the guest to the still closed host's door that he just offered as an alternative, is not 1/2, but exactly 2/3. Henze refers to http://math.ucsd.edu/~crypto/Monty/monty.html, e.g. Gerhardvalentin (talk) 08:12, 9 August 2012 (UTC)

Well, you haven't got a lucky hand in explaining things. Let me ask it this way: What is the difference between 'the standard version of the paradox" and the 'fully symmetric standard version of the paradox'? And the question: What is the meaning of 'the clean valid paradox', still stands. Nijdam (talk) 15:20, 9 August 2012 (UTC)

I don't think so. Do you know, for example, what the department of mathematics of the University of California, San Diego says? Did you read Richard Gill? He once invited you on a cup of coffee, but you denied. Just read the actual relevant sources. Gerhardvalentin (talk) 16:17, 9 August 2012 (UTC)

Is this an answer to the above questions?? (BTW I never denied to drink coffee with Richard, don't write things you do not know about.)Nijdam (talk) 09:15, 10 August 2012 (UTC)

Yes. Gerhardvalentin (talk) 12:48, 10 August 2012 (UTC)

Comment

In don't really want to participate in the discussion, but since Gerhard Valentin refers to "my source", I'd like to point out, that his quote from the source may be somewhat misleading and out of context, as he ommits important context and actually combines lines from different sections without indicating it. The correct full quote can he found be found here: Henze p. 106. Note that the section from which the first part of the quote was taken comes with some introductory commentary on modeling and assumptions, which was omitted in the quote. Furthermore the final bold part actually stems from the next chapter.--Kmhkmh (talk) 07:21, 6 August 2012 (UTC)

Out of context? Full text: In chapter "7 Laplace Modelle", middle of page 50, Henze says:

7.5 Zwei Ziegen und ein Auto
In der Spielshow Let’s make a deal! befindet sich hinter einer von drei (rein zufällig
ausgewählten) Türen ein Auto, hinter den beiden anderen jeweils eine Ziege. Der
Kandidat wählt eine der Türen, etwa Tür 1, aus; diese bleibt aber vorerst verschlossen.
Der Spielleiter, der weiß, hinter welcher Tür das Auto steht, öffnet daraufhin eine der
beiden anderen Türen, z.B. Tür 3, und es zeigt sich eine Ziege. Der Kandidat kann nun
bei seiner ursprünglichen Wahl bleiben oder die andere verschlossene Tür (in unserem
Fall Nr. 2) wählen. Er erhält dann den Preis hinter der von ihm zuletzt gewählten Tür.
In der Kolumne Ask Marilyn des amerikanischen Wochenmagazins Parade erklärte
die Journalistin Marilyn vos Savant, dass ein Wechsel des Kandidaten zu Tür 2
dessen Chancen im Vergleich zum Festhalten an Tür 1 verdoppeln würde. Das war
die Geburtsstunde des sog. Ziegenproblems im Jahre 1991, denn die Antwort von Frau
Marilyn löste eine Flut von Leserbriefen mit gegenteiligen Meinungen aus. Es gab
Standhafte, die ihre ursprüngliche Wahl beibehalten wollten, Randomisierer, die sich
nur mittels eines Münzwurfes zwischen den verbleibenden Türen entscheiden mochten,
und Wechsler, die ihre ursprüngliche Wahl verwerfen wollten.
Etwa 90% der Zuschriften an Frau Marilyn spiegelten die Meinung wider, die Chancen
auf den Hauptgewinn hinter Tür 1 hätten sich durch den Hinweis des Moderators von 1
zu 2 auf 1 zu 1 erhöht, da jetzt zwei gleichwahrscheinliche Türen übrig geblieben seien.
Frau Marilyn blieb jedoch bei ihrer Empfehlung und führte die folgende Argumentation
ins Feld: Das Auto ist mit Wahrscheinlichkeit 1/3 hinter Tür 1 und folglich mit
Wahrscheinlichkeit 2/3 hinter einer der anderen Türen. Öffnet der Moderator eine dieser
beiden Türen, so steht die Tür fest, hinter welcher das Auto mit Wahrscheinlichkeit 2/3
verborgen ist. Folglich verdoppelt Wechseln die Chancen auf den Hauptgewinn.
Dieses Argument tritt noch klarer hervor, wenn man die Erfolgsaussichten der Strategien
eines Wechslers und eines Standhaften gegenüberstellt. Der Standhafte gewinnt dann
den Hauptgewinn, wenn sich dieser hinter der ursprünglich gewählten Tür befindet,

and he continues on page 51:

und die Wahrscheinlichkeit hierfür ist 1/3. Ein Wechsler hingegen gewinnt das Auto
dann, wenn er zuerst auf eine der beiden ”Ziegentüren“ zeigt (die Wahrscheinlichkeit
hierfür ist 2/3), denn nach dem Öffnen der anderen Ziegentür durch den Moderator
führt die Wechselstrategie in diesem Fall automatisch zur ”Autotür“ . Bei allen diesen
Betrachtungen ist natürlich entscheidend, dass der Moderator die Autotür geheimhalten
muss, aber auch verpflichtet ist, eine Ziegentür zu öffnen. Wer dieser Argumentation
nicht traut und lieber praktische Erfahrungen sammeln möchte, lasse sich unter der Internet-
Adresse http://math.ucsd.edu/~crypto/Monty/monty.html überraschen.

And then some examples for training purpose are following: "Trainingsaufgaben".

Page 99 begins with the headline "15 Bedingte Wahrscheinlichkeiten" - chapter 15 Conditional probabilities, and page 105 shows "15.9 Das Ziegenproblem". He says that the price has been placed uniformly at random, and the candidate chooses one door uniformly at random. He says that the host has no choice in case the guest should have chosen a goat. But if the guest should have chosen the prize, then we assume that the host choses one of his two goats uniformly at random. He shows by Bayes that the guest doubles his chance by switching. And then on page 106 he adds:

Die oben erfolgte Modellierung soll die Situation des Kandidaten vor dessen Wahlmöglichkeit
so objektiv wie möglich wiedergeben. Man mache sich klar, dass ohne konkrete
Annahmen wie z.B. die rein zufällige Auswahl der zu öffnenden Ziegentür im Falle
einer Übereinstimmung von Autotür und Wahl des Kandidaten eine Anwendung der
Bayes-Formel nicht möglich ist. Natürlich sind Verfeinerungen des Modells denkbar.
Beispielsweise könnte der Moderator für den Fall, dass er eine Wahlmöglichkeit zwischen
zwei Ziegentüren besitzt, mit einer bestimmten Wahrscheinlichkeit q die Tür mit der
kleineren Nummer wählen (s. Übungsaufgabe 15.3).
15.10 Beispiel (Fortsetzung von Beispiel 15.3)
Beispiele wie 15.3 finden sich häufig in Lehrbüchern zur Stochastik. Ihr einziger Zweck
besteht darin, mit bedingten Wahrscheinlichkeit schematisch rechnen zu üben.

and he adds some decision trees, evidently for training purpose.

English: "Examples like 15.3 are often found in educational textbooks on stochastic. "Their only purpose is to schematically training to calculate with conditional probability"

I repeat:

Bei allen diesen Betrachtungen ist natürlich entscheidend, dass der Moderator die Autotür geheimhalten muss, aber auch verpflichtet ist, eine Ziegentür zu öffnen (the confidential host has to keep secrecy regarding the door hiding the car) and

Ihr einziger Zweck besteht darin, mit bedingten Wahrscheinlichkeit schematisch rechnen zu üben.

Their only purpose is to schematically train calculations of conditional probability.

Not a matter of the paradox, but a matter of maths lessons.

So, not out of context. Read it. Henze means what he says regarding the paradox, and he says it bluntly clear. The article needs to factor this view of modern academic sources of Henze and many others. Rick, please help to shape the article accordingly. Gerhardvalentin (talk) 09:42, 6 August 2012 (UTC)

I only point to the fact that the referred example 15.3 is about two dice thrown out of sight. So, Gerhard, what is your point? Nijdam (talk) 10:55, 6 August 2012 (UTC)

Furthermore Henze quotes Vos Savant with her erroneous way of arguing. It seems he follows here there in her reasoning (the combined doors "solution"), making him a bad mathematician. Nijdam (talk) 11:06, 6 August 2012 (UTC)

TRUTH? What is your reputable source for this disqualification of Norbert Henze? You clearly highlight the dilemma of this messy article and of the never ending conflict. No respect to WP, no respect to the academic sources, no respect to the readers. Gerhardvalentin (talk) 11:30, 6 August 2012 (UTC)

@nijdam: Calling Henze "bad mathematician" because you don't like his approach is frankly nonsense. For all we know, he's most likely a better mathematician than you are.

Norbert Henze

@Gerhardvalentin: First I'm not sure why you are posting here long quotes (again in a slightly different way) which can been seen much better with the original sectioning in the link I've provided. And yes your original quote was somewhat out of context as it lacked the most important part being: "Die oben erfolgte Modellierung soll die Situation des Kandidaten vor dessen Wahlmöglichkeit so objektiv wie möglich wiedergeben. Man mache sich klar, dass ohne konkrete Annahmen wie z.B. die rein zufällige Auswahl der zu öffnenden Ziegentür im Falle einer Übereinstimmung von Autotür und Wahl des Kandidaten eine Anwendung der Bayes-Formel nicht möglich ist. Natürlich sind Verfeinerungen des Modells denkbar." (The modeling above should reflect the candidate's situation before his choice as objectively as possible. One needs to beware however, that without [additional] concrete assumptions as for instance the random choice of goat door in case of the candidate's [first] choice and the car door coinciding the application of the Bayes' formula won't be possible. Of course one could consider refinements of the model.) This came before your originally quoted text but without the bold print part. In essence he says, that additional assumptions are required and that introducing the parameter q is valid refinement. Now you've added the bold printed part from the next section, which talks about a complete different problem, that is example 15.3 (a dice problem). And the bold printed line that you've (falsely) added to your quote (Beispiele wie 15.3 finden sich häufig in Lehrbüchern zur Stochastik. Ihr einziger Zweck besteht darin, mit bedingten Wahrscheinlichkeit schematisch rechnen zu üben.) refers to the dice problem of example 15.3 (Beispiel 15.3). Note in the earlier section about the MHP in your quote Henze talks about exercise problem 15.3 (Übungaufgabe 15.3) and not example 15.3. So the bold line you've added to Henze's comment on the MHP is from another section and refers to another problem (the dice problem). I assume that aside from quoting somewhat sloppily you missed the difference between example 15.3 and exercise problem 15.3. This would not have happened if you had read whole section 15.10 rather than just its first few lines (context !) as then it becomes apparent to which problem he was referring in the first lines of that section. In any case you clearly misquoted Henze and by that creating claim, that Henze never stated. Henze never stated the MHP refinement with q (or the conditional treatment as such) have as their sole purpose the schematical training of calculating conditional probabilities, instead he stated that the dice problem (and alike) have as their sole purpose to schematically train the calculation of conditional probabilities. --Kmhkmh (talk) 12:35, 6 August 2012 (UTC)

You surely are right. Just to see what Henze says: Page 99 starts with chapter "15 Conditional probabilities", and at the beginning he shows some examples. In section

15.1 Example (5 lines) he mentiones an urn with 2 red, 2 black and 2 blue balls. After some balls have been taken out, he asks what is the probability that the two balls that had first been taken out were red?, continuing straight with section 15.2 (three lines). He mentions again the "Ziegenproblem" (goat problem) that was subject of chapter 7.5, and he says:

15.2 Example

In the situation of the "goat problem" (see section 7.5), door #1 had been pointed at, and the host has disclosed door #3 to show a goat. What is the probability that the guest wins the car by switching to door #2? and he straight continues

15.3 Example

In the neighboring room two dices are rolled ... and he again asks What is the probability that at least one dice shows a six?

And after those three examples of coloured balls, of three doors with two goats and a car, and of two dices, and after his three perspective questions "What is the probability that ...", he immediately continues, still in section 15.3, to treat these three examples a little closer. He outlines (still in the beginning of section 15.3 !) that all three examples have something in common. They have in common, that there exists a result (though unknown to us) of some stochastic event.

The whole section 15.3 is on these three examples: coloured balls, doors and dices. After the later section 15.9, treating again the MHP, in section 15.10 he refers again to section 15.3 (please remember what section 15.3 is about: It is the summary on the coloured balls, the three doors and on the dices, and Henze says that "such examples as in 15.3" (imo he evidently means conditional probability examples) can be found in textbooks for training conditional probability calulus. "Their sole purpose is to schematically train to calculate conditional probability." So I did not cite "out of context", but I said exactly what Henze says. Thank you for your efforts, your objections are helpful to clearing what the matter is. The clearer, the better. Kind regards, Gerhardvalentin (talk) 19:21, 6 August 2012 (UTC)

Yes, you did cite out of context for the reason explained above and by that (accidentally) constructed a completely fictious statement of Henze. When he is talking about the about the common feature of all 3 examples, he is talking about a typical feature of conditional probabilities, which is present is all 3 examples 15.1-15.3. But he is not talking about all 3 examples having just the purpose of schematically training conditional probabilities. That statement is just made about example 15.3, when it is detailed and extended in example 15.10. Note that Henze treats all 3 introductory examples 15.1-15.3 in greater detail in the later sections, but only for the treatment of example 15.3 he states that its sole purpose is to schematically train calculating conditional probabilities. Literally he states: "Beispiele wie 15.3 finden sich häufig in Lehrbüchern zur Stochastik. Ihr einziger Zweck besteht darin mit bedingenten Wahrscheinlichkeiten schematisch rechnen zu üben." (Examples likes 15.3 are often found in probability & statistics textbooks. Their sole purpose is to train the schematically computation of conditional probabilities.). Note he states examples like 15.3, he does not state examples like 15.1-15.3. Furthermore if you carefully go through detailed treatment of example example 15.3 in example 15.10, then you see, that he explains in detail, why he sees example 15.3 as (problematic) schematical training. This is specific to example 15.3 (and alike) and has nothing to do with example 15.1 or 15.2. There is absolutely nothing in Henze's text that suggests, that "examples like 15.3" is to be understood to include examples 15.1 and 15.2, on the contrary of the 3 given examples he considers 15.1 and 15.2 as valid and sees only 15.3 as problematic and constructed for the (sole) purpose of schematical training. So to be clear here, rather than "doing exactly what Henze says", you did not understand at all what Henze was saying. But be that as it may I'm not on a mission to ensure your proper understanding of Henze's book and I'm not interested in turning this into yet another neverending subthread on MHP. If you insist on reading something into it, which is not there, that's your business. Alright before I'm finally going to shut up for good one more piece of information. The German noun "Stochastik" can't really be translated with "stochastic(s)", the latter is more or less identical to the German adjective "stochastisch", which has a somewhat different meaning. The former is correctly translated as "probability [theory] & statistics" (see Henze p. 1)--Kmhkmh (talk) 20:47, 6 August 2012 (UTC)

Thank you, Kmhkmh. Gerhardvalentin (talk) 10:04, 8 August 2012 (UTC)

Update

I had to deal with some personal issues, but now I am starting on the RfC. I see that Gerhardvalentin has some concerns, so here is what I am going to do about that; first I am going to file the RfC as discussed with Martin / Rick. Hopefully that will give us a clear picture of the consensus of a larger group of editors and one party will graciously accept not getting his way. Then I am going to go back and compare the results with what Gerhardvalentin wants. If what he wants is a lot like one side of the RfC, then we are done. If it is quite a bit different, and we cannot reach a consensus here as to whether to adopt it, I will run another RfC. --Guy Macon (talk) 15:42, 11 August 2012 (UTC)

Why would the voice of GerharhValentin have such weight? Nijdam (talk) 20:04, 11 August 2012 (UTC)

It doesn't. If you have objections with unclear consensus that appear to require an RfD to resolve, You can file an RfD, either alone or with my help. --Guy Macon (talk) 21:27, 11 August 2012 (UTC)

What does RfD have to do with resolving this issue? hydnjo (talk) 01:44, 12 August 2012 (UTC)

RfD has nothing to do with dispute resolution, but RfC does. If you look at Wikipedia:Dispute resolution, there are two places where RfCs are mentioned. The one we are talking about is in the RfC through article talk pages section. Also see Wikipedia:Requests for comment and Wikipedia:Requests for comment#Request comment through talk pages. --Guy Macon (talk) 03:12, 12 August 2012 (UTC)

Update?

Any news Guy? Martin Hogbin (talk) 09:52, 27 August 2012 (UTC)

Adams

The quote from "Adams" among the simple solutions is taken out of its context, and not meant to be a solution to the original problem. Nijdam (talk) 12:36, 18 August 2012 (UTC)

Carlton

I cannot find the used wording of Carlton's solution in the referred paper. On the contrary: Carlton gives the conditional probabilities. Nijdam (talk) 12:44, 18 August 2012 (UTC)

We had an extended discussion about this during mediation, which ended up (see [11]) with this suggested wording An intuitive explanation is to reason that a player whose strategy is to switch loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3 (Carlton 2005). This refers to section 5 of Carlton's paper, and is worded to reflect that Carlton does not present this as a "Solution". The discussion was sufficiently disrupted that we never got to the point of talking about exactly where in the article this sentence should be placed. I think there's a pretty good argument that this should not be in the "Solution" section referenced to Carlton since Carlton presents something else as his solution and he specifically does not call this a "solution" but rather an "intuitive explanation". Whether it's in the "Solution" section or not, I think this wording is better, and if it's in the "Solution" section we might want to further clarify that this is not Carlton's solution (implying it is Carlton's solution is significantly misrepresenting this source). -- Rick Block (talk) 04:17, 20 August 2012 (UTC)

Okay Rick, I see you changed Carlton's simple reasoning into what it is according to Carlton himself: an intuitive explanation. Nijdam (talk) 08:55, 28 August 2012 (UTC)

What if 2 players are playing it?

I wonder what happens if 2 players are in the game. One chooses door #1, and the other chooses door #2. They both switch every time, and the host opens door #3 every time. This is where I can't understand the solution - if each of them has a 66% chance, it means that after a million rounds, each of them should have about 666K wins. Obviously it can't happen, because there is only one car. Does anyone have a good explanation for that?

Yes, in your scenario the host must always choose door 3. That means that the car can never be placed behind door 3 so it is a different game. Martin Hogbin (talk) 12:29, 30 August 2012 (UTC)

Another answer is to follow through what actually happens in this game. Assuming the car is placed randomly, then 1/3 of the time the host opens door 3 and the car is there. Perhaps we say both players lose in this case. Then we're left with the 1/3 of the time the car is behind door 1 (and the player who originally picked door 2 wins if they both switch every time), and the 1/3 of the time the car is behind door 2 (the player who originally picked door 1 wins). They each win only 1/3 of the time overall, which is 50% of the time they don't both immediately lose (because the car was behind door 3). If we do this a million times, they both immediately lose 333K times when there's a car behind door 3 and then they each win 333K times of the remaining 666K times there was a goat behind door 3.

In the regular Monty Hall problem there can only be one player since the host cannot open the door where the car is (and if there are two players the car will be behind the 3rd door 1/3 of the time). If the player always picks door 1 the host can't always open door 3, so either you have to look at what happens if the host opens door 3 OR if the host opens door 2 (some people prefer to look at it this way - more or less ignoring the example case mentioned), or you have to look only at the subset of cases (the conditional probabilities) where the host opens door 3. Either way you look at it, you win 1/3 of the time if you stay with your original choice and 2/3 of the time if you switch. If we do this a million times and you always pick door 1 and you always switch, you win by switching to door 2 333K times out of the 500K times the host opens door 3 (this is every time the car is behind door 2 and half the time the car is behind door 1) and another 333K times by switching to door 3 in the 500K times the host opens door 2 (every time the car is behind door 3 and the other half the time the car is behind door 1). Overall you win 666K out of a million times - divided into 333K out of 500K times the host opens door 3 and 333K out of 500K times the host opens door 2. -- Rick Block (talk) 04:47, 31 August 2012 (UTC)

Another explanation:

The crucial rule which leads to the 2/3 solution is that the host is obligated to open a not chosen door with a goat and to offer a switch. Surely the host cannot open door 3 with a goat "every time", because sometimes there is the car. But you might say: "If player 1 has chosen door 1 and player 2 has chosen door 2 and the host opens door 3 with a goat and offers a switch to both players, he has fulfilled the rules completely; but the chances for both players must be 1/2 if switching or not."

First we consider the game with only one player. The host fulfilling the rules of the game then opens door d with probability 1 if the player has chosen a goat, and with probability 1/2 if the player has chosen the car. The final chance for the switching player is (1/3 * 1)/(1/3 * 1 + 1/3 * 1/2) = 2/3.

Now we consider the case with two players: The host fulfilling the rules of the game then opens door d with probability 1 if player 2 has chosen the car, and with probability 1, too, if player 1 has chosen the car. The final chance for both switching players is (1/3 * 1)/(1/3 * 1 + 1/3 * 1) = 1/2. (We are not interested in the case where both players have chosen a goat ...)--Albtal (talk) 08:44, 3 September 2012 (UTC)

"Almost" correct, Albtal, but only "almost" correct. You said "If player 1 has chosen door 1 and player 2 has chosen door 2 and the host opens door 3 with a goat and offers a switch to both players, he has fulfilled the rules completely; but the chances for both players must be 1/2 if switching or not."

You can compare this situation to the forgetful host who, by showing the car in 1/3, and by that eliminating the chance to win by switching, has reduced the chance to win by switching from full 2/3 to 1/2 only. So in that scenario the host "has NOT complied with the rules completely", and so this is NO MORE the famous MHP where the chance to win by switching is exactly 2/3 in each and every game. Similar to the "biased host" who also does NOT comply with the rules, and by that might generate some difference between his two doors. In the only correct MHP there is NO difference at all. No matter which one of his two doors he opens, in the correct famous MHP – as per modern reliable sources – the chance to win by switching is exactly 2/3 in each and every game, no matter which one of his two doors the host opens, and that makes the charm of the world famous MHP. But there are variants, "en masse". Gerhardvalentin (talk) 11:54, 3 September 2012 (UTC)

Of course the game with two players is different from the game with one player, and it is in no way my opinion that the two players variant "contradicts" the 2/3 solution for the one player variant; and I intended to show this by answering a "fictive" question. (You may even consider my straightforward different solutions as a "proof" that the task sets themselves are different.) I think it is a good template for the different variants to consider the relation between the two probabilities with which the host opens his door if the player has chosen the car or not. This ratio leads directly to the solution. (And the "variants" show that it is important to formulate the correct problem and to argue precisely.)--Albtal (talk) 13:24, 3 September 2012 (UTC)

I was just adding the following remark to my posting above (08:44; "Edit conflict" with Gerhardvalentin):

(If, as in the section "Extended description of the standard version" of the article, it is an explicit condition in the task set that the host has to open each of the two possible doors with probability 1/2 if the player has chosen the car, two players are not possible from the start.)--Albtal (talk) 12:24, 3 September 2012 (UTC)

Exactly, Albtal, and moreover: As per modern reliable sources, the host might be biased in any order, no matter in which direction and no matter to which degree. For if you allow "q" to differ from 1/2 in asymmetry, then you necessarily have to allow "q" to vary from 0 to 1 to any degree, and by the law of total probability, for any actual given game, it necessarily will be 1/2 again, no way out. Morgan's hasty error in reasoning was to completely forget about that coerciveness. So the principle "The host has to open each of his two doors with probability 1/2 if the player has chosen the car" is an inevitable implicitness and, since long ago, is state of the art, as per modern reliable academic sources. That ensures that it never can matter which one of his two doors has been opened by the host. Regards, Gerhardvalentin (talk) 13:18, 3 September 2012 (UTC)

For the solution of the MHP and its explanation we don't need any sources, according to the Wikipedia rules: For example: the statement: "Paris is the capital of France" needs no source. But there should be a self-evident rule always to make clear which problem we are talking about.--Albtal (talk) 16:08, 3 September 2012 (UTC)

Actually we do, a simple well known statement about Geography is different from a statement about probabilities and the computation of them. Moreover Paris being the capital is not a subject of dispute, whereas the MHP is very much subject of dispute inside and outside of WP, consequentially the WP rulers require sources. While it is true that most/some variations of MHP may be elementary enough to argue that (at least without a dispute) sources are not strictly required, the main issue is not with the derivation of those solutions but the ambiguity of the original problem and which implicit assumptions might be appropriate or not. Exactly that requires sources. We cannot write, that the following implicit assumption is "natural" because WP editor xy says so, but it must be a reputable external source that says so. However the requirement of sources is no obstacle in clearly separating the variations of the problem, that is something the article should do anyhow completely independent of the question of sources.--Kmhkmh (talk) 09:40, 4 September 2012 (UTC)

+++ Yes, that's exactly the point. Since years the article never says which "problem" (scenario / game) it just deals with, paragraph by paragraph. And when the reader finally is to expect strawberries, the article just mixes apples with pears, without saying so. Gerhardvalentin (talk) 18:40, 3 September 2012 (UTC)

+1 Yes, Kmhkmh: "Clearly separating the variations of the problem, that is something the article should do anyhow". But since years it doesn't, that's the dilemma of this article, and that dilemma finally should have an end. By goodness, let's hope the stereotypical dissimulation of this mingle-mangle article will come to an end, soon. Gerhardvalentin (talk) 11:15, 4 September 2012 (UTC)