Talk:Monty Hall problem/Archive 28

Other simple solution - John Tierney
John Tierney in the New York times has an explanation - not sure if it's fancy enough to qualfy as a 'solution'.
 * When you stick with Door 1, you’ll win only if your original choice was correct, which happens only 1 in 3 times on average.
 * If you switch, you’ll win whenever your original choice was wrong, which happens 2 out of 3 times.

Works for me ! Anyone else find that helpful ? --195.137.93.171 (talk) 17:31, 6 January 2012 (UTC)
 * OK Tierney is mentioned throughout the article and that NYT page is linked, but I feel it would be worth adding the above wording somewhere.
 * I feel it would also be worth adding a section on 'Consequences and applications'. I am particularly interested to find out how I can make money out of it ! --195.137.93.171 (talk) 17:37, 6 January 2012 (UTC)


 * Is this substantially different from the wording that starts the "Other simple solutions" section:
 * An even simpler solution is to reason that switching loses if and only if the player initially picks the car, which happens with probability 1/3, so switching must win with probability 2/3 (Carlton 2005).
 * Just curious how you think Tierney's wording differs. -- Rick Block (talk) 19:46, 6 January 2012 (UTC)

This is a typical case of overcomplicating a problem. One of the doors is always removed. It is always a goat. It becomes a non-factor. This means that there is NO DIFFERENCE if the door is removed BEFORE OR AFTER THE CHOICE! It leaves the contestant with a 50/50 chance of having picked the car on the first try. — Preceding unsigned comment added by MrShemp (talk • contribs) 00:26, 12 January 2012 (UTC)


 * Well - every single reliable source says the chances are 1/3:2/3. I'd be happy to try to help you see why, but this is not the place.  If you're interested, please start a thread on the talk:Monty Hall problem/Arguments subpage. -- Rick Block (talk) 01:38, 12 January 2012 (UTC)


 * MrShemp, quite simply your logic and math are wrong. if door 2 contains a goat, and this is revealed to you before you make your choice, you definitely would not choose door 2. if door 2 contains a goat but this is not revealed to you, you may still pick door 2. — Preceding unsigned comment added by Bpjoyce10 (talk • contribs) 06:37, 19 January 2012

Totally overthought this problem
There's no need for all these mathematical proofs. If you stick, you have a 1/3 chance of picking the prize. If you switch, however, you are trying NOT to pick the prize. By switching when you do not win the prize, you will win every time. This means a 2/3 chance of winning. — Preceding unsigned comment added by 94.7.196.157 (talk) 01:23, 21 February 2012 (UTC)
 * You are quite right, of course, and you are not the first person to point this out. Martin Hogbin (talk) 09:38, 21 February 2012 (UTC)
 * Yep, it really is very simple. Yet the majority of people get it wrong when they first encounter it. That is what makes a good riddle: the answer is only obvious when you see it. And this is what makes the literature, and this article, so overwrought: explaining the obvious is no simple matter. ~ Ningauble (talk) 20:14, 21 February 2012 (UTC)

Risen from death
My ban has expired, and I wonder (wondered for the last year) that the simple solution Devlin gave (the combined doors reasoning), is still present. Even knowing Devlin himself has come to the insight it's flawed. Nijdam (talk) 22:04, 27 March 2012 (UTC)


 * In what way is it flawed? We can discuss this on my talk page if you wish. Martin Hogbin (talk) 22:26, 27 March 2012 (UTC)

Maybe it suffices to mention that, as you know, every door has the same probability 1/3 on the car (in the standard version). Hence it is impossible to reach any conclusion that says different. Nijdam (talk) 22:36, 28 March 2012 (UTC)


 * The flaw is that there is no law in probability that lets you group two doors together such that their combined probability before and after the host opens one of them remains constant. Assuming the car is located at the outset randomly (uniformly distributed), then p(door 1) = p(door 2) = p(door 3) = 1/3.  Any two of these sum to 2/3.  After the host opens a door there are now a new set of conditional probabilities.  If the host opens door 3 the new probabilities are p(door 1|host opens door 3) = ?, p(door 2|host opens door 3) = ?, and p(door 3|host opens door 3) = ?.  We indeed know p(door 3|host opens door 3) is 0, but nothing we've said (so far) says anything about the sum of this and p(door 2|host opens door 3).  p(door 2) + p(door 3) is 2/3, but this has nothing do to with p(door 2|host opens door3) + p(door 3|host opens door 3).


 * Examining Devlin's quote in detail (Devlin calls the doors A,B, and C with A being the one the player picks and C being the one the host opens):


 * 1) There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3.


 * Right. p(door B) + p(door C) = 2/3


 * 2) I'll help you by using my knowledge of where the prize is to open one of those two doors to show you that it does not hide the prize. You can now take advantage of this additional information. Your choice of door A has a chance of 1 in 3 of being the winner. 


 * Right. p(door A) = 1/3, in the same way that p(door B) = p(door C) = 1/3


 * 3) I have not changed that.


 * Well, actually, you have. You've changed p(door A) into p(door A|host opens door C).  The value of this conditional probability is the whole point of the problem.  If what is meant here is that p(door A) is the same as p(door A|host opens door C) some sort of argument is needed since this is neither obvious nor even necessarily true given what has been said so far.  My assumption is most people when they read this (perhaps even Devlin when he wrote it) are thinking it is talking about p(door A), i.e. door A's probability of hiding the car before the host opens any door.  This is indeed still 1/3, but p(door B) and p(door C) are also still 1/3.  If we want to talk about probabilities after the host opens door C, we must be talking about one of two sets of conditional probabilities.  One set conditioned on the host opening door B (which could have, but didn't, happen from our initial starting point), the other conditioned on the host opening door C.  By opening door C the host has changed each door's probability into a conditional probability (conditioned on the host opening door C).


 * So, the probability that door A hides the car at the start is 1/3. We all agree about that.


 * What value does it change to after the host has opened door C?


 * Or, if you prefer: P(Car=A)=1/3. What is P(Car=A|host opens door C|the host chooses uniformly between legal doors)?   1/3?  More than 1/3?   Less than 1/3? Martin Hogbin (talk) 18:07, 5 April 2012 (UTC)


 * Where, pray tell, did "|host chooses uniformly between legal doors" come from? You seem to be saying there is no distinction between P(Car=A) and P(Car=A|host opens door C) if they happen to have the same numeric value.  Or perhaps you're saying it doesn't matter if the logic is confused as long as it ends up with the correct numeric result.  Again, the whole question is the relationship between P(Car=A) and P(Car=A|host opens door C), or (equivalently) between P(Car=B) and P(Car=B|host opens door C).  Per Falk, there are two very strong, competing, intuitions:
 * 1) The host cannot change the initial probability the car is behind the door the player picks, i.e. P(Car=A) must have the same value as P(Car=A|host opens door C), which implies P(Car=B) must be different than P(Car=B|host opens door C), or
 * 2) Since the car was equally likely to be behind each door at the beginning, with only 2 doors left it must be equally likely to be behind the remaining doors, i.e. P(Car=A) = P(Car=B) implies P(Car=A|host opens door C) = P(Car=B|host opens door C)
 * IMO, glossing over the distinction between P(Car=A) and P(Car=A|host opens door C) is glossing over the entire problem. Most people's initial intuition is #2.  A different intuition leads to #1.  Relying on either of these results in the wrong answer depending on the exact problem set up.  -- Rick Block (talk) 19:55, 5 April 2012 (UTC)


 * "host chooses uniformly between legal doors" is part of the standard setup.


 * Perhaps now you could answer my question is the new (conditional) probability 1/3, >1/3, or < 1/3? Martin Hogbin (talk) 20:03, 5 April 2012 (UTC)


 * Part of the standard setup according to whom? The answer to your question is (of course) 1/3, but this doesn't make P(Car=A) the same as P(Car=A|host opens door C) any more than 2 apples is the same as 2 oranges.  I am really not in the least interested in arguing with you about this (we're long past the point where any discussion between us is likely to be productive).  Although I agree they have the same numeric value, you are clearly not going to convince me P(Car=A) is mathematically the same as P(Car=A|host opens door C) and I am clearly not going to convince you that they're mathematically different.  -- Rick Block (talk) 06:37, 6 April 2012 (UTC)
 * The host chooses a legal door uniformly according according to K&W, the article itself, Nijdam and probably you. It is universally accepted that in the standard version of the problem the host is taken to choose uniformly between legal doors. If you like, I can spend a few paragraphs on the arguments page going through old ground again to make this clear but I suspect you are just trying to avoid the obvious conclusion shown below.


 * So you agree that the probability that door A hides the car given that the host has opened door C is 1/3. You will note that this value has (obviously) not changed from its initial value, thus Devlin's argument is fine.


 * I fully understand that P(car=A) is conceptually different from P(car=A|host opens door C) which is conceptually different from P(car=A|host opens door C|host chooses uniformly between legal doors) which is conceptually different from the actual conditional probability that we wish to calculate, which is P(car=A|host opens door C|host chooses uniformly between legal doors|host says the word 'pick'|loads of other stuff that clearly does not affect the numerical value of the probability that we wish to calculate).  Martin Hogbin (talk) 08:42, 6 April 2012 (UTC)


 * So you agree that P(car=A) is conceptually different from P(car=A|host opens door C), but so long as they have the same numeric value (which is forced by additional constraints on the host in the explicit, "standard" version often omitted in popular versions) you see this difference as being no more important than conditioning on the specific words the host uses and have no problem with arguments that mathematically replace one with the other (if we call these X1 and X2, the "combining doors" argument is effectively X1=1/3, X2+Y2+0=1, so [???] 1/3+Y2=1, so Y2 must be 2/3 QED)? I have a question I'd like you to ponder (don't feel compelled to respond here) - what exactly is the critical difference between the situations where everyone completely understands the car is behind door 1 or door 2 with probability 1/3 (i.e. before the player makes her initial pick, and after the player makes her initial pick but before the host opens anything) and the situation nearly all people have trouble with?   To be more specific, why is it obvious that P(car=A) = P(car=B) = P(car=A|player picks A) = P(car=B|player picks A) = 1/3, but not obvious what the value is of P(car=A|player picks A|host opens C) or P(car=B|player picks A|host opens C)?  Hint: it has something to do with the sample space. -- Rick Block (talk) 16:34, 6 April 2012 (UTC)
 * The fact is that when Devlin says that the host opening door C does not affect the probability (obviously meaning its numerical value) that the car is behind A he is correct. I find that obvious, you may not, but it is in a reliable source and it is factually correct. There is therefore no reason to remove it from the article.
 * 4) But by eliminating door C, I have shown you that the probability that door B hides the prize is 2 in 3.


 * No. By opening door C he's shown you the conditional probability of door C is 0.  You can't take this 0 and substitute it for p(door C) in statement #1 since p(door C) and p(door C|host opens door C) are completely different things (apples and oranges).


 * We've been over this so many times I find it hard to believe anyone does not understand it. -- Rick Block (talk) 15:01, 5 April 2012 (UTC)


 * [citation needed] Where does Devlin himself report that the insight of combining the un-chosen doors is flawed, and in what respect does he say that it is flawed? ~ Ningauble (talk) 18:31, 5 April 2012 (UTC)


 * See http://www.maa.org/devlin/devlin_12_05.html. After some somewhat still confused attempts at providing an "intuitive" explanation (distinguishing the standard problem and what amounts to the "host forgets" variant), he says "Confused? As sometimes arises in mathematics, when you find yourself in a confusing situation, it may be easier to find the relevant mathematical formula and simply plug in the appropriate values without worrying what it all means."  He then proceeds to show how to find the probabilities in both variants using Bayes' formula. -- Rick Block (talk) 19:55, 5 April 2012 (UTC)
 * Not quite retracting his simple explanation is it? More like saying a more complicated solution is required for a more complicated problem. Martin Hogbin (talk) 20:05, 5 April 2012 (UTC)
 * He's contrasting two simple versions of the problem, where the "intuitive" reasoning that produces the "correct" answer for the standard version fails for the other version (for reasons that are not obvious). Is he retracting his simple explanation?  No.  But he's definitely admitting it's flawed ("confusing") and that an easier approach to the correct answer may be to use Bayes' formula.  -- Rick Block (talk) 06:37, 6 April 2012 (UTC)
 * We could argue forever about what Devlin was thinking but, whatever you might be taking him to mean, it is not a clear enough retraction of his explanation to justify the removal of a section closely based on a reliable source. Martin Hogbin (talk) 08:26, 6 April 2012 (UTC)
 * Ironically, a plug-and-grind approach "without worrying what it all means" may be persuasive, but it neither explains nor justifies anything. Anything becomes less "confusing" if one accepts an invitation to take it on faith. The application of Bayes' formula (or other formalisms) is correct, and the combined doors approach is correct; but in both cases the underlying concepts, and the method of their correct application, elude most people. Hence, I think Devlin is very apt in his closing remark: "So there you have it. Whether you believe it is another matter." ~ Ningauble (talk) 16:03, 6 April 2012 (UTC)
 * "Correct" is actually not the issue - combining doors is a published solution. The question is whether we should editorially choose to include it or not, and (if so) whether we should also include various published caveats or criticisms and (again, if so) whether these caveats or criticisms should be presented essentially inline.  Call it what you will, Devlin says his intuitive solution appears to apply to the commonly posed "host forgets" variant but is not valid in this case.  It's perhaps interesting to note that the difference between the "host forgets" (or "host doesn't know where the car is") version and the so-called "standard" version was the original point of Whitaker's letter to vos Savant. -- Rick Block (talk) 16:34, 6 April 2012 (UTC)
 * That he suggests combining doors may be confusing, in that it does not answer a different problem, is not a substantive caveat or criticism. That he chose not to remove potential confusion by explaining how to apply the method correctly is regrettable. A little explanation of what it means might have been in order, but his agenda was to present a different method. The point of my post above was that his non-explanation is not only regrettable, it is risible when he asks us ignore the meaning of the method he presents. His exercise in meaninglessness is not a serious caveat or criticism. ~ Ningauble (talk) 19:53, 10 April 2012 (UTC)

Is this the source article you are all talking about? If so, I don't see anything here that could be considered a "retraction". I interpret it as Devlin simply using the problem's alternate version of a third party randomly chosing another door to illustrate the key to understanding the problem; namely, the fact that Monty knows which door the car is behind and always opens a door that reveals a goat in the standard version. Devlin goes on to explain in detail how the alternate version results in the naively intuitive reasoning of "makes no difference if I switch" is actually correct...for the alt version only. There is nothing earth-shattering, surprising or confusing about this to me. It is crucial to completely understanding the original problem in the first place. On a related note regarding the orginal problem, I don't think Devlin did a very good job intuitively explaining the original problem. In my opinion, the best way is to point out to the reader that by switching, the only way you can lose is if you happen to select the door with the car behind it initially. Probability 1/3 of course. -- Racer X11 Talk to me Stalk me  17:06, 6 April 2012 (UTC)
 * Exactly, although I would point out that some people do find Devlin's solution intuitive and convincing. Martin Hogbin (talk) 20:55, 6 April 2012 (UTC)


 * I even will go further interpreting Devlin's words. Devlin and many others are fully mistaken in their simple combined doors idea. And it is my opinion Devlin has come to this insight himself too, but is not willing to plainly admit this, so gives some misty formulation to cover up. Nijdam (talk) 08:12, 7 April 2012 (UTC)


 * Nijdam, that is an interesting point you make about people not wanting to admit their mistakes, and yes it does happen. I would say that the Morgan paper is a good example.  My guess is that at least one author of the paper got the answer wrong (in fact 1/2) and, rather than admit their mistake, they decided to concoct a formulation is which 1/2 could be a correct answer.  This ruined a simple and unintuitive problem that most people get wrong.  To their credit, they have almost admitted this in response to our letter. Martin Hogbin (talk) 08:56, 7 April 2012 (UTC)


 * Can't follow you here. Nijdam (talk) 06:15, 8 April 2012 (UTC)


 * Let me explain, I fully admit that this is idle speculation on my part. Most people, when asked for the probability that a player who swaps will win the car, will give the incorrect answer of  1/2.  My suggestion is that at least one of the Morgan authors gave this wrong answer to one of their colleagues. Having had their mistake pointed out to them and not wishing to admit that they were wrong, they then tried to come up with a plausible formulation in which the correct answer was indeed 1/2.  This, of course, is the contrived case where the player chooses door 1 and the host always opens door 3 when permitted by the rules.  This contrived host bias forms the basis of their paper. In their response to our letter they agree that 'the answer' was, in fact 2/3. Martin Hogbin (talk) 09:30, 8 April 2012 (UTC)
 * I can't say it isn't true, but I very much doubt it, as in that case this possibility would be emphasized more. And from what you wrote before, I got the impression you thought the article was meant to criticize vos Savant and the simple solution, which you didn't like. But anyway, one is entitled to change their opinion. As you'll know I value the article as it makes clear the distinction between the "unconditional" interpretation of the problem and the standard form, and the respective solutions. Nijdam (talk) 18:40, 8 April 2012 (UTC)


 * I would add that the common emphasis on a specific example case ("You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?'), and the fact that this changes the sample space in an unexpected way (in the reduced sample space door 1's conditional probability retains the same numeric value as its unconditional probability while door 2's conditional probability doubles) is an integral part of this problem. Of course, it can be posed in an unconditional kind of way - for example by suppressing the knowledge of which door the host opens (which is extremely artificial given the gameshow context), or changing the question from what would you do in a specific case to what is the best strategy to pre-select before going on the show - but this is not in the least typical, and (per Krauss&Wang) is not how nearly everyone interprets the common problem statement.  Although they certainly could have presented it in a less hostile way (as others have, for example Gillman, and Grinstead&Snell, and Rosenthal, etc.), I think Morgan et al.'s criticism of vos Savant (for using an unconditional solution for what at least seems to be intended to be, and most people interpret as, a conditional problem) is spot on.  Regarding their supposed "retraction" - please read Coffer2theorems reply about this in the archives.  I completely agree.-- Rick Block (talk) 06:06, 9 April 2012 (UTC)

Sockpuppet Investigation
2 more IPs blocked. Drop me a line on my talk page if any more IP editors show up who just happen to have strong and biased views on a year-old arbcom case. --Guy Macon (talk) 22:26, 7 April 2012 (UTC)


 * Guy, is there any reason that Glkanter should not be allowed to return to editing WP now that the other's bans have expired? Martin Hogbin (talk) 16:44, 10 April 2012 (UTC)


 * If you look at Arbitration/Requests/Case/Monty Hall problem you will notice that most remedies are for a period of one year but one - a topic ban on MHP for Glkanter - is indefinite.  In addition, If you look at User:Glkanter you will see that he is indefinitely blocked from all editing (post-arbcom admin decision, not arbcom ruling). He is, of course, free to request that the block be removed, but to do that he will have to convince an uninvolved admin that he will comply with Wikipedia's policies. --Guy Macon (talk) 22:51, 10 April 2012 (UTC)

Section 2.6 ("Alternative derivations")
The last sentence, "The conditional probability is ... 2/3," is ambiguous (which conditional probability?). I originally interpreted it as the conditional probability of the host opening Door 3 (the most recent one mentioned), and confused myself because that probability is obviously 1/2.

Also, the section appears to be copied from Gill's paper with a few modifications. The modifications, incidentally, make the section much less clear; the corresponding section in Gill's paper is unambiguous, saying that "the conditional probability of winning by switching ... equals the unconditional probability of winning by switching, 2/3." I'm not sure what to do here: I don't know if I should just fix the problem, or remove the section entirely for being a copyright violation. Eyu100(t 01:05, 13 April 2012 (UTC)


 * I do not think that there is a copyright problem as it is probably Gill who added the section, it is a short excerpt from a longer paper which I think is acceptable in any case. Martin Hogbin (talk) 08:39, 13 April 2012 (UTC)

Proposed new section
Just to make it perfectly clear what I'm talking about, here's a draft of the new section (in its entirety) I'm proposing be added, between the existing "Extended problem description" and "Solutions" sections.

[R] Related, simpler problem

Before addressing the solution to the standard problem, Carlton (2005) suggests it helps to consider the following simpler problem also presented in Grinstead and Snell (2006:137). Imagine a contestant plans to play Let's Make a Deal and will employ either the "switching strategy" (under this strategy the player always switches) or the "staying strategy" (under this strategy the player never switches). Which of these strategies is better? As long as a contestant using the "switching strategy" initially picks a door hiding a goat, the contestant will win the car - since the host must reveal the other goat and the contestant will switch to the only remaining door, which must hide the car. A player using this strategy loses if and only if the player happens to initially pick the door hiding the car. For example, if the player initially picks door 1, this strategy wins the car if it is behind either door 2 or door 3 (2 chances out of 3) and loses only if the car is behind door 1 (1 chance out of 3). Conversely, a player who employs the "staying strategy" wins only if the car is behind the initially chosen door (1 chance out of 3).

If a pre-selected staying strategy wins with probability 1/3 and a pre-selected switching strategy wins with probability 2/3, how can the probabilities of winning by staying and winning by switching after the host opens a door be different? The answer, as shown below by numerous solutions to the standard problem, is that they're not! These solutions show the highly unintuitive result that if a player picks door 1 and the host opens door 3, the probability the car is now behind door 1 is 1/3 (not 1/2!) while the probability it is behind door 2 is 2/3 (not 1/2!).

Martin - seeing the actual text do you still object to this? Is there anything about it you like? Is there any way we can modify it to address whatever concerns you might have? -- Rick Block (talk) 05:42, 14 April 2012 (UTC)


 * It is just a pointlessly confusing, irrelevant, and long winded way of saying this (which is equally correct):

[M]
 * As long as a contestant initially picks a door hiding a goat, and then switches the contestant will win the car - since the host must reveal the other goat and the contestant will switch to the only remaining door, which must hide the car. A player who switches loses if and only if the player happens to initially pick the door hiding the car.  For example, if the player initially picks door 1 and then swaps they will win the car if it is behind either door 2 or door 3 (2 chances out of 3) and loses only if the car is behind door 1 (1 chance out of 3).  Conversely, a player who does not swap wins only if the car is behind the initially chosen door (1 chance out of 3). Martin Hogbin (talk) 15:15, 14 April 2012 (UTC)


 * I found your shorter version to be somewhat more confusing than the longer version that preceded it. I do not agree that they are saying the same thing. Your version does not contain any mention of a preselected strategy, which is a useful way of thinking about the problem. And I find your "pointlessly confusing, irrelevant, and long winded" comment to be rather insulting. --Guy Macon (talk) 17:22, 14 April 2012 (UTC)


 * Insulting to whom? My comments were about the text, which I was asked comment on because it is being proposed for inclusion in the article.  I find the proposed text confusing because it talks about a different problem from the one actually posed.  It is irrelevant because it does not address the question that was actually asked and it is long winded because it is longer than is required to solve the actual problem.


 * Perhaps you could tell me how you were confused by my version. Martin Hogbin (talk) 21:13, 14 April 2012 (UTC)


 * Martin's text seems nearly the same, but without the contextual framing of this as the answer to a simpler problem. And correct is not the issue - ease of understanding is the issue.


 * We know vos Savant's simple solution was unconvincing (she continued to be inundated with letters even after her first followup column).


 * We know Eisenhauer's published opinion of vos Savant's solution : "what could and should have been a correct and enlightening answer to the problem was made unconvincing and misleading".


 * We know, per Krauss and Wang, that the standard version of the problem statement leads the reader to believe that the case of interest is where Door 3 has been opened. They say "Note that, once formed, this assumption prevents the problem solver from gaining access to the intuitive solution illustrated in Figure 1" (essentially identical to vos Savant's simple solution).


 * We know Grinstead & Snell (a standard reference work in the field) addresses the simpler problem before continuing with the standard problem and comparing with standard reference works is recommended by WP:TECHNICAL.


 * We know Carlton's published opinion is that "it helps" to present the simpler problem before presenting a solution to the standard problem.


 * Martin is perfectly entitled to his opinion that what I'm suggesting is pointlessly confusing, irrelevant, and long winded. However, we have empirical evidence that presenting a simple solution to the standard problem is not convincing (thousands of letters to vos Savant), a tertiary reliable source that explicitly says this (Eisenhauer), another reliable source that provides an explanation for why this is the case (K&W), a different approach used by a standard reference work (G&S), and another reliable source who explicitly says this approach helps (Carlton).


 * Should we believe Martin, or should we believe reliable sources that directly address the very issue we're talking about here? -- Rick Block (talk) 20:01, 14 April 2012 (UTC)


 * You have jumped from talking about how your text might be modified to sources. Of course sources are important and we can discuss them, but I suggest that we do so in a new section below.  Most of your quotes are irrelevant to the difference between the two versions.   It seems that you agree that what I have written is actually correct (if not, perhaps you could explain where the error lies).  Martin Hogbin (talk) 21:13, 14 April 2012 (UTC)


 * No. I've jumped from talking about the text based on personal opinions, to talking about the approach it exemplifies based on reliable sources.  Again, "correct" is not the issue.  Ease of understanding is the issue.  Choosing to present something more or less equivalent to vos Savant's solution directly in response to the standard problem implies we think we can do this in a more convincing fashion than Marilyn vos Savant, despite Eisenhauer's published opinion that doing this is unconvincing, and despite K&W's reasoning that this simple solution becomes inaccessible after being presented the standard problem.  In contrast, presenting a solution to the "strategy" problem (as done by G&S) is described by Carlton as helping.  These sources are exactly relevant, to the exact issue we're talking about.  -- Rick Block (talk) 22:28, 14 April 2012 (UTC)
 * Firstly can we just establish that my version is equally correct, you do not seem to want to admit that.


 * I responded to your question,' Is there any way we can modify it to address whatever concerns you might have?'. The only difference between my version and yours is that in your version the player is required to have a pre-planned strategy and in mine he is not. There are no sources which say that the player in the standard MHP has a pre-planned strategy. This approach is simply  a way in which the Morgan solutions can be justified.   If you would like to start a section below on what sources say about this subject, I would be happy to respond.  Martin Hogbin (talk) 23:20, 14 April 2012 (UTC)


 * I am not responding to your question about correctness because (IMO) this has absolutely nothing to do with the issue we're talking about, which is, and remains, ease of understanding. Or, are you trying to imply the text I'm suggesting is incorrect?  It's referenced - feel free to compare to the sources.  If you'd actually like to discuss correctness start a new thread (I'd suggest on the Arguments subpage - although since I'm pretty sure we each knows what the other thinks and I'm absolutely sure EVERYONE else is sick to death of this, I really don't see the point ).


 * I'm not claiming any sources say that the player in the standard MHP has a pre-planned strategy - are you thinking I'm claiming they do, or perhaps trying to imply the text I'm suggesting does not accurately reflect the sources I've cited?


 * I've said why I'm suggesting we include this new section and it has nothing whatsoever to do with the "Morgan solutions", or are you saying the sources I'm citing use the simpler problem to justify the "Morgan solutions"? If the latter, I'd be very interested in how you're concluding this.


 * I'm talking about what sources say about ease of understanding in this section, and I'll continue doing so. If you want to talk about what sources say about anything else in some other section, go for it.  -- Rick Block (talk) 05:35, 15 April 2012 (UTC)


 * Rick, you started by asking for my opinion on your proposed new section and I gave it. You then started talking about sources which you say support the addition of your new section.  I have no objection to discussing that subject but I think it would make it easier for others to follow if that discussion were in a new section of the talk page.  I have marked the two proposals [R] and [M]. If you believe that there are sources which prefer [R] to [M] as a means of explaining the basic problem and solution (that is to say the probability of winning by switching is 2/3 and it matters that the host always shows a goat) please give then, with supporting rational in a new section below.


 * I find it odd that you are not concerned whether either section is logically correct. Neither is close to the wording of a source they are both our own words.  We are, as editors, allowed to consider the truth of what we write, but it must be supported by sources.  We can argue that below.  Martin Hogbin (talk) 08:40, 15 April 2012 (UTC)


 * Yes, I started by asking your opinion on a proposed new section - and, BTW, my text is VERY close to the wording in both G&S and Carlton (without being quotes). Your response is clearly that you do not see the point of adding a new section explicitly introducing a related, simpler problem that has an intuitively obvious solution with the same numeric answer as the standard problem.  So, I responded with a source-based opinion supporting the addition of such a section.  Or are you suggesting adding the text labeled [M] above in a new section - if so, what would it be called?  And, what source(s) would the [M] section cite?  Since you've deleted the problem the rest of the [R] text refers to, sourcing this text to G&S or Carlton would be highly inappropriate.  I really can't say much about [M] (including whether it's "correct" - which in the context of this page I'll interpret to mean accurately represents its sources) unless you tell me where this text would go in the article and what its sources are.  -- Rick Block (talk) 16:33, 15 April 2012 (UTC)
 * I was not proposing my text for the article, just showing that yours added nothing new. Martin Hogbin (talk) 17:05, 15 April 2012 (UTC)
 * I'm sorry, maybe I missed something. Is the [M] text already in the article someplace?  I'm pretty sure a discussion of a related, simpler problem that is easier to understand than the standard problem is not in the article.  In what sense would this not be new?  -- Rick Block (talk) 01:29, 16 April 2012 (UTC)


 * Reliable sources are not limited to establishing correct / non-correct. They can also be used to establish confusing / non-confusing. --Guy Macon (talk) 23:02, 14 April 2012 (UTC)


 * I know of no WP policy which says that sources should tell us how to write the articles in WP. Sources tell us the facts and we as editors decide how to best present those facts, however,I am perfectly willing to discuss what sources say about which of the two versions above is easiest to understand.


 * You have yet to tell me what you find hard to understand about my version or who you think I was insulting. Martin Hogbin (talk) 23:20, 14 April 2012 (UTC)


 * Oppose addition. I don't see it helping the article. Glrx (talk) 04:44, 15 April 2012 (UTC)
 * So, you think Carlton is incorrect in his opinion that starting with this intuitive explanation (identical to what G&S call a "simpler, related question") does help? -- Rick Block (talk) 17:55, 15 April 2012 (UTC)

Misleading Article
This page is misleading, in that it states that switching doors leads to a higher probability of winning. It should state the the probability of having selected the correct door increases. These are two different probabilities because the probability of initially selecting the correct door does not change when the host opens the door, while the probability of the car being behind a given door does. Any argument can be refuted with the following simple scenario:

The contestant has a choice of door A, door B, or door C. Before the contestant chooses a door, the host opens door C to reveal a goat. The remaining doors (A & B) have equal probability of containing the car.

The only difference between this scenario and the 'Monty Hall problem' is that the contestant chooses a door before the host opens door C. Because the act of selecting a door cannot affect which door the car is behind, doors A & B will always have equal probability of containing the car, no matter what sort of door re-picking steps are taken (assuming door C contains a goat). — Preceding unsigned comment added by 50.47.134.57 (talk) 00:47, 27 April 2012 (UTC)
 * Every reliable source, including pretty much every introductory probability textbook, says your understanding is incorrect. If you'd like to discuss this, please start a thread on the /Arguments subpage.  -- Rick Block (talk) 05:48, 27 April 2012 (UTC)

Suggestion
Rather than argue about this ad infinitum, is there something productive we might be able to do? For example, how about changing the "Simple solution" section to a section that explicitly addresses what many authors present as a simpler problem, i.e. whether you should use a preselected switch or stay strategy changing the decision point from after the host opens a door to the very beginning of the game (much in the same way Grinstead&Snell treat it, and the same way Carlton suggests)? I think this has the advantage of progressing from simple to more complex (which, as I understand it, is Martin's goal), but it also alerts the reader that what follows may not totally address the problem they're currently thinking about (easing the shift to thinking about the "unconditional" problem). This is, more or less, the same structure the article had following its last successful FAR - but rather than (as the article did then) transition from a simple to more complex solution by saying the simple solution doesn't quite address the problem as stated, presents an explicitly different problem that stone-cold simple solutions definitely address (the choice is always stay or always switch, if you always stay you win 1/3 of the time so if you always switch [to whichever door the host opens] you must win 2/3 of the time). Maybe we'll argue about which simple solutions should be included in this section, but I think this could make the article much more to the point, significantly shorter, easier to understand, and far less contentious. -- Rick Block (talk) 06:06, 9 April 2012 (UTC)

Martin's comments

 * At first sight that might look like a new and positive suggestion, which I would welcome, but it is, in fact, just a restatement of your original position, that we should start with the simple solution provided they contain a 'health warning' that they do not answer the question as asked.


 * The only real consensus we have ever had here to start with the simple solutions, as given in reliable sources, and then to state the objections to those solutions that some sources give, and then to give the Morgan-style solutions given by some other sources. This does not push any POV but simply does what all good text books and encyclopedias do which is to start simple, then mention any complications, then discuss the more complicated aspects.


 * If you like, I can show you where there was a consensus to do this. Unfortunately you and a few other editors have consistently ignored this in favour of your own POVs. Martin Hogbin (talk) 08:28, 9 April 2012 (UTC)
 * Me however, I fully support Rick's suggestion. And I do not like to be accused of pushing forward my own POV, as such POV is found in reliable sources. Nijdam (talk) 09:30, 9 April 2012 (UTC)
 * Yes, of course, but some reliable sources say different without imposing the view that Morgan is correct and other sources are wrong. Martin Hogbin (talk) 10:04, 9 April 2012 (UTC)


 * Your claim about consensus is at best dubious (as I recall, it was after nearly all the editors who favor featuring a conditional approach had stopped commenting here). My suggestion does not just seem different, it actually is different.  The idea is fundamentally to present material that no source disputes.  No source disputes the simple 1/3 if you stay, 2/3 if you switch solution if you move the decision point to before the host opens a door (in fact, several sources such as Gillman that have issues with the simple solutions suggest this variation).  No source disputes that the conditional probability approach to the standard version is correct. We could even add a transition section (with content no source disputes) explaining that the "simplified" problem must have the same answer as the non-simplified problem as long as the problem is symmetrical (which is forced by the constraint on the host that he open an unchosen door randomly [uniform] if the player's pick happens to hide the car).  The structure would be:


 * Lead (more or less as is)
 * Problem description (more or less as is)
 * Related, simpler problem (new section, presenting the version and its stone cold simple solution where the decision is between the strategy of staying vs. the strategy of switching - referencing Grinstead&Snell and/or Carlton and/or Gillman and/or Gill etc.)
 * Solution
 * New section (explaining why the standard problem where the decision is after the host opens a door must have the same answer as the simpler problem if the problem is symmetrical - referencing Falk and/or Gill)
 * Other simple solutions based on the premise that the problem is symmetrical (if you want - although I suspect this is in all likelihood not even necessary)
 * Solution using conditional probability (another approach, showing the same result, presented in nearly all introductory probability textbooks, is ...)
 * Informal
 * Formal
 * Solution using game theory (yet another approach, where the problem is viewed as a 2-player game ...)
 * Variants
 * History


 * No health warnings. No POV pushing (in either direction).  -- Rick Block (talk) 20:10, 9 April 2012 (UTC)


 * Essentially you want to say, right at the start that the simple solutions do not solve the MHP in its normal form. Most sources say that it does. Martin Hogbin (talk) 21:10, 9 April 2012 (UTC)


 * No. I want to say right at the start that there is a related problem that inarguably permits a very simple solution - followed immediately with why it is that this solution also solves the MHP in its normal form (assuming symmetry).  Rather than this be a discussion between us, I'd be interested in comments from others as well.  Anyone else have any opinions about this?  -- Rick Block (talk) 01:13, 10 April 2012 (UTC)
 * I am not quite sure what you are suggesting then. The most well known statement of the problem is that by vosSavant/Whitaker.  If you are suggesting that we offer the simple solutions as solutions to this problem that is fine by me.  If, on the hand you are suggesting that we propose some different problem to which the simple solutions are an answer that this is just our original position restated.


 * Most sources offer simple solutions to the vosSavant/Whitaker formulation. Martin Hogbin (talk) 08:34, 10 April 2012 (UTC)


 * Is the outline above not clear? I'm suggesting adding a NEW section between the "Problem" description (where the standard problem is described) and the "Solution" section, and in this new section describing the related, simpler problem (posed and answered by several sources) where the question is changed to "what is the best pre-selected strategy to follow, always stay or always switch" which explicitly moves the decision point to before the host opens a door.  That the answer to this question is 1/3 chance of winning the car for the staying strategy, and 2/3 chance of initially selecting a goat (and, thus, winning the car) for the switching strategy seems to be intuitively obvious to most people.  I'm suggesting including in this section only this one, extremely intuitive, solution (the one provided by sources that pose this alternate problem - not saying anything in this section about any sources that present this same solution or any other solution to the standard problem).  Then, I'm suggesting starting the immediately following "Solution" section (which addresses the standard problem), with a section explaining why the answer to the standard problem must be the same as the answer to the simpler problem.


 * The point of this is to lead the reader to the insight that the intuitive 1/3:2/3 answer of the "always stay/always switch" argument actually applies to (and is correct for) the standard problem. Any other solutions we include would follow this new material.


 * If you'd like, I could write a draft. And, again, I'm still interested in other opinions about this.  -- Rick Block (talk) 15:46, 10 April 2012 (UTC)


 * Rick, you seem to think that saying exactly the same thing in different words makes it a new idea. You want to have a different question (that is to say not the standard MHP) that the simple solutions do answer.  That is exactly the same as saying that the simple solutions do not answer the standard MHP.  Most sources do not support this view.


 * There is a compromise which was accepted as a consensus, and that is to have the simple solutions first, without health warnings or qualifications or alternative questions, and then to have a description of why some sources consider these solutions to be incomplete, followed by the Morgan-style solutions. This is how the rest of the world does things. Martin Hogbin (talk) 16:53, 10 April 2012 (UTC)


 * Would you please read what I'm actually saying? You apparently either don't understand or are completely misinterpreting. -- Rick Block (talk) 19:04, 10 April 2012 (UTC)
 * You seem to make it fairly clear above. See my comments in italic below. Have I got it right? Martin Hogbin (talk) 22:22, 10 April 2012 (UTC)
 * No. I've marked up your text below (and see the explanation further below as well).  Is this more clear? -- Rick Block (talk) 06:35, 11 April 2012 (UTC)
 * Lead (more or less as is) <-- yes.
 * Problem description (more or less as is) - That is the standard vos Savant/Whitaker statement <-- not quite. It's the K&W explicit version (the "standard" version, per sources like Barbeau) including constraints on the host omitted from the vos Savant/Whitaker version, in particular including the constraint that the host must choose randomly between two "goat doors".
 * Related, simpler problem (new section, presenting the version and its stone cold simple solution where the decision is between the strategy of staying vs. the strategy of switching - referencing Grinstead&Snell and/or Carlton and/or Gillman and/or Gill etc.)
 * So that is a different problem? Not the standard one <-- Yes. But it is not one that anyone is going to say "simple solutions" generically solve.  It's one for which the sources that present it include a solution.  That solution, and only that solution, will be presented in this new section.  Perhaps you're confused by the fact that this solution is also presented (by other sources) as a solution to the standard problem.
 * Solution To the the non-standard problem. <-- no. The new section above would include the one and only solution provided by the sources presenting the non-standard (simpler) problem.  This section returns from the slight diversion (I'm imagining the new section is about two paragraphs - considerably less text than this section of the talk page - it would actually be easier to draft it than to keep discussing it like this) to the "standard" problem and presents its solutions.

So, Martin, do you understand now? Although you might not think so, I do want your comments as well. -- Rick Block (talk) 14:52, 11 April 2012 (UTC)


 * The related simpler problem is an insignificant and un-notable problem that is of no interest to anyone except as an introduction to the Morgan-style solutions. It will be utterly confusing to most readers to have the problem split into two before it has be solved. Martin Hogbin (talk) 19:28, 11 April 2012 (UTC)

Any other opinions about this suggestion?
I would really like comments about this idea from folks other than Martin (and I would appreciate it if Martin refrained from commenting in this section). -- Rick Block (talk) 19:04, 10 April 2012 (UTC)


 * The reliable sources supporting this approach (specifically, introducing and solving a related, simpler problem) include Grinstead&Snell, and Carlton (detailed references in the article).  And, yes, I think this would make the article more readable to the Wikipedia reader.  -- Rick Block (talk) 19:04, 10 April 2012 (UTC)
 * Hold on a minute Rick! Is this a section where I am asked to refrain from commenting but you can comment freely? Martin Hogbin (talk) 22:29, 10 April 2012 (UTC)
 * Exactly. I ask this because our discussions tend to get so voluminous that nobody else can get a word in edgewise.  It's not that I don't want anyone to hear what you have to say, but if you could say it elsewhere my hope is others will be more likely to add their comments here.  I've added a header for the section with your comments (just above).  Fair enough?  -- Rick Block (talk) 06:35, 11 April 2012 (UTC)
 * That seems rather an odd system. I can understand both of us keeping quiet for a while to let others have their say but not a section where everybody can edit except me.  However I am willing to try anything for an easy life so I will keep out of this section. Martin Hogbin (talk) 14:33, 11 April 2012 (UTC)
 * That would the "Carlton" who is reliably sourced with a simple solution in the current article? As well as the recently "recanted" Devlin, who is also reliably sourced with a simple solution in the current article? Seems counter-intuitive, no? Especially with three active editors already disagreeing. What is your Grinstead and Snell support sourcing? Is your approach consistant with Selvin's letters? 208.54.80.173 (talk) 19:21, 10 April 2012 (UTC)
 * Without question, this proposal is contrary to vos Savant's writings. Would you describe this as the "predminant" theory among the vast published reliable sources? 208.54.80.173 (talk) 19:32, 10 April 2012 (UTC)
 * [citation needed]No recantation has been entered into evidence. ~ Ningauble (talk) 23:28, 11 April 2012 (UTC)


 * My similar suggestion last July at Would it help to disambiguate "the" question?, to associate simple solutions with a simple interpretation of the problem, foundered on concerns about "health warnings" and on the problem of how to identify the "simple" interpretations and solutions in a neutral way. Unfortunately, the distinction I tried to use proved to be too abstract. Rick's suggestion of using a distinction between choosing a strategy and choosing a door is also an abstraction and, although it admits of a simple solution, it is an abstraction that complicates the question rather than simplifying it. I think Martin is entirely correct that characterizing the simple interpretation as a "related problem" distinct from some imagined "standard problem" is deprecatory, all the more when the simple approach is arbitrarily defined so as to be insufficient after the door is open. As Seymann says ("Comment on Let's make a deal: The player's dilemma," American Statistician 45: 287-288), "Without a clear understanding of the precise intent of the questioner, there can be no single correct solution to any problem." The most notable distinguishing characteristic of simple interpretations of the problem is the host's indifference in choosing a goat to reveal (i.e., symmetry) or, as Seymann puts it (ibid.), "the host is to be viewed as nothing more than an agent of chance." This is a basic, simple interpretation of the problem, not an extra stipulation for the purpose of making deficient solutions contingently viable. ~ Ningauble (talk) 19:58, 10 April 2012 (UTC)


 * For the nth time, I'm not suggesting associating any simple solutions with the simple problem except the one (consistent) solution the authors presenting the simple problem themselves provide. What I'm precisely talking about is the section in Grinstead and Snell  on page 137, which poses the simpler problem I'm talking about: "We begin by describing a simpler, related question. We say that a contestant is using the “stay” strategy if he picks a door, and, if offered a chance to switch to another door, declines to do so. ..."


 * Their solution (the next paragraph) is this: "Using the “stay” strategy, a contestant will win the car with probability 1/3, since 1/3 of the time the door he picks will have the car behind it. On the other hand, if a contestant plays the “switch” strategy, then he will win whenever the door he originally picked does not have the car behind it, which happens 2/3 of the time. "


 * Simiarly, Carlton (whose "intuitive explanation" is mis-characterized in the current article as a simple solution to the standard problem) says: "Before presenting a formal solution to the Monty Hall Problem to my students, I find that it helps to give an intuitive explanation for the 1/3 - 2/3 solution. Imagine you plan to play Let's Make a Deal and employ the 'switching strategy.' As long as you initially pick a goat prize, you can't lose: Monty Hall must reveal the location of the other goat, and you switch to the remaining door - the car. In fact, the only way you can lose is if you guessed the car's location correctly in the first place and then switched away. Hence, whether the strategy works just depends on whether you initially picked a goat (2 chances out of 3) or the car (1 chance out of 3). "


 * Note that this is exactly the same simpler problem presented by Grinstead & Snell, with exactly the same solution.


 * The suggestion is to include a NEW section, including this specific simpler problem with the specific solution presented by those sources that present the simpler problem (not all "simple solutions", just the ones offered by the sources presenting this problem).


 * Doing this says nothing whatsoever about any other solution. It's sourced to two impeccably reliable sources (a probability textbook and a peer reviewed academic paper written by a professor of probability).


 * The point of this NEW section is to begin to convince the reader that switching (in the standard version, too) doubles your chances of getting the car (it's fairly obvious in this version of the problem).


 * AFTER this section, we'll present solution(s) to the standard problem. I'm further suggesting we start the next section, which does address the standard problem, with an explanation of why the solution to the standard problem must be the same as the solution to the simpler problem (i.e. symmetry).  And only then, AFTER presenting and explaining the simpler problem, and explaining why the answer to the standard version must be the same as the answer to the simpler problem, proceed with other solutions (including simple solutions) to the standard problem.


 * Is there anybody here at all who understands what I'm talking about? -- Rick Block (talk) 00:14, 11 April 2012 (UTC)
 * Of course, I am. And I'm getting a little tired of well-meaning layman not understanding the problem. Also I like to state again that not any source on the problem is automatically reliable. For instance Devlin should not considered to be a reliable source. Richard Gill, an expert in the field, along with other sources, at least states that the simple solution is not to be considered a solution to the standard formulation, but to the simplified form. If it were up to me I'd make this clear from the start. But this is Wikipedia and there it is possible to just misinform people. Nijdam (talk) 09:35, 11 April 2012 (UTC)
 * The bald assertion that I do not understand the problem is unwelcome, but I would genuinely welcome an explanation of where I said something here that is factually or logically incorrect. It would also be helpful if you identified which you mean by "the simple solution," "the standard formulation," and "the simplified form." Since you appear to respect Richard Gill's expertise in this area, I will frame the remarks that follow in terms of his writings. It is my opinion that the stipulations in Gill's Proposition 3 constitute the best choice of "simplified form" to use for introducing the subject to laypersons. Rick is essentially talking about using Gill's Proposition 1 as the "simplified form", but it presupposes understanding what unconditional really means and sparks strenuous objections that it is rather far removed from the usual narrative in which the question is posed after the door is open. I appreciate the sense in which Proposition 1 is mathematically "simple" in that it involves fewer constraints, but I am looking at it from the perspective that the constraints impose a simplifying symmetry.  I confess that my opinion in that regard is unsubstantiated, as research into laypeople's understanding of the problem has largely focused on why they jump to the incorrect 50/50 answer rather than which maths they understand. I also acknowledge that in corresponding with Richard Gill at Wikipedia, he expressed concern that the symmetry imposed by the essential constraint that distinguishes his Proposition 3 from propositions 1 and 2 may be fuzzy for laypersons; however, he also remarked that solutions exploiting the symmetry are so obvious that it is not worth writing a research paper about it. (My opinion: It is a trivial exercise in applied combinatorics for undergraduates.)  Richard did agree with me (in the first exchange linked above) that the essential difference between the main interpretations of MHP is whether or not the host is impartial with respect to which goat is revealed. The difference between these problem versions is what gives rise to endless confusion and interminable disputation. I really believe this essential difference between problem versions is best understood by laypersons in terms of the concrete stipulation of Proposition 3 about whether the host is an impartial agent of chance, rather than the much more abstract distinction about a fixed strategy of always switching as expressed in Proposition 1. In a textbook treatment of conditional probability it makes sense to frame the issue in terms of unconditional or conditional probability, before or after the door is opened; but for a layperson trying to understand what they are up against, the distinction between an impartial host and a biased one is much more fundamental and concrete. I strongly believe that  the latter is the better way to introduce the difference between versions for a general audience. ~ Ningauble (talk) 23:16, 11 April 2012 (UTC)


 * I agree I'm suggesting introducing what amounts to Richard's proposition 1 (I assume you're talking about propositions 1,2 and 3 from ) - however, the way Grinstead&Snell and Carlton both phrase this avoids the unfamiliar "unconditional/conditional" jargon. Note that I'm not attempting to distinguish a "biased" host version from an "unbiased host" version (which I think makes sense only if you're already talking about an "after" situation).  I really don't think laypeople have trouble with before and after (which are, of course, one of the main uses of the unconditional/conditional formalism).  In particular, there isn't anyone who fails to understand that the probability the car is behind each of the doors before the host opens one is 1/3.  And (IMO) the only reason the simple "1/3 if you stay vs. 2/3 if you switch" argument fails to sway anyone is that it is typically posed as a solution to the "standard" version of the problem which, with or without an unbiased host constraint, creates a context where the host has already opened a door (and, since there are only two doors left and the location of the car is still unknown, most people assume it must be a 50/50 choice between the two remaining doors).  I think putting this same argument in the before context makes it accessible to nearly anyone (although anecdotal, Carlton says "it helps").  It would be lovely if there were more research about this, but I don't recollect any.


 * I think you're effectively suggesting distinguishing Richard's propositions 2 and 3, which I strongly suspect is fairly far beyond the grasp of most laypeople. Krauss&Wang talk about this where they say that most people assume the "standard" rules (embodied by proposition 3) whether they're explicitly mentioned or not.  Even stronger, Mueser&Granberg (, referenced in the article) report no statistical difference between how many subjects switch given 3 substantially different problem descriptions (including one they call "standard" which is the same as what we're calling "standard" here but without the symmetry constraint, and another that is essentially the same as the "host forgets" version where the probability of winning actually is 50/50) - the implication being that even if these sorts of host constraints are provided most people simply ignore them.


 * The sequence I believe makes most sense (would be most accessible to a general audience) is


 * 1) get them to understand the switch/stay probabilities are 1/3:2/3 in the "simplified" problem, which makes it clear we're talking about the before context


 * 2) get them to understand the switch/stay probabilities in the standard (symmetrical) version must be the same as the probabilities in the simplified version (I suspect this is harder than the first step, but not nearly as hard as doing this without first establishing the 1/3:2/3 solution to the simplified problem)


 * 3) whatever else we want to present (most people will probably lose interest)


 * Can you explain why you strongly think distinguishing biased vs. unbiased host would be better? You do, of course, know that host bias does not change the "always stay" vs. "always switch" probabilities (which remain 1/3:2/3 regardless of any host bias). -- Rick Block (talk) 03:38, 12 April 2012 (UTC)


 * (Yes, that is the paper by Gill to which I should have referred.) Your final remark is illuminating. Yes, I do, but the fact that you raise it suggests you don't really have so much confidence in laypeople understanding it. It has even been alleged very publicly that someone reported to have a record setting IQ simply doesn't get it. Talking about deciding in the before context clarifies little if readers don't appreciate how this relates to concepts of conditional and unconditional probability, or how it relates to the constraints of the standard formulation, or, much less, why one would choose to ignore available information. The problem with redefining the question it in terms of unconditional probability (or worse, obfuscating it by not saying so), although it is entirely appropriate in the context of a textbook motivating the subject of conditional probability, is that our readers are not going on to read the rest of the textbook, and most people won't get it. (Note that Grinstead&Snell, who set out to show (I use Gills' proposition numbers as shorthand) that a solution for P1 does not answer P3 but instead show that it fails for an instance of P2 which violates the constraints of P3, leave it for the reader to discern the essential difference between P1 and P3, presumably from reading the rest of the chapter(s).) The reason I believe emphasizing equidistribution in the standard problem is better than introducing a different problem is because, as noted above, the symmetry makes the result so blindingly obvious that few mathematicians would even mention it. I confess that my perspective on probability is distinctly Bayesian and information-theoretic and, as indicated in Gill's aforementioned paper, the situation may be less transparent for a frequentist. Stipulating that the unobserved frequency of left-handed Montys is equidistributed allows even the most stalwart frequentist to treat the problem as a simple, symmetrical one. ~ Ningauble (talk) 21:55, 27 April 2012 (UTC)


 * The point of calling it a different problem (although it is, of course, simply an explicitly unconditional version of the standard problem) is only that this seems to make it easier for people to grasp that its solution really is 1/3:2/3 - in a way that emphasizing the symmetry of the standard problem does not. Perhaps some later section of the article might delve into the differences between Gill's P1, P2, and P3 (which was similar to the distinction between three types of solutions in one of the many drafts I suggested, which even explicitly highlights symmetry) - but I suspect people won't continue reading if they find the initial sections of the article completely unbelievable.  And, for whatever reason, after being presented with anything remotely close to the standard version of the problem many people's initial reaction to the standard "simple" solutions is apparently disbelief - whether the problem is explicitly symmetrical or not.  -- Rick Block (talk) 05:46, 28 April 2012 (UTC)

It really does not matter who is right. Let us write the article for the benefit of our readers.
Let us forget which solutions are the 'right' ones. Some people think the simple ones are correct, some think the Morgan ones are correct. Both can be found in sources.

The MHP is one of the few simple problems that fools most of the people most of the time and many people do not get it even when it is carefully explained to them. We therefore need to make every effort to explain the problem and solutions as simply as possible before our readers give up. Even if I were an avid Morgan supporter I would still want to start by grabbing and holding our readers' attention by explaining the basic paradox and solution before going into the Morgan details. Here are two ways to do that:

The wrong way

 * This is the standard problem (vS/Whitaker + usual (K&W) assumptions)


 * Here is the solution, but it actually is wrong/incomplete/answers a different question

By this time most new readers will have lost interest


 * More stuff

The right way

 * This is the standard problem (vS/Whitaker + usual (K&W) assumptions)


 * Here is a simple solution.


 * This is why the answer is 2/3 and not 1/2


 * This is why it matters that the host knows where the car is.

''So far the article is simple and interesting. Those who wish can read further ''


 * Some sources say that the above solutions are wrong/incomplete/answer a different question


 * This is why they think that the above solution actually is wrong/incomplete/answers a different question


 * These are the solutions that they prefer


 * More stuff.

We make no judgement at all as to which solutions are 'right' but we give both in the way that helps our readers understand the subject. Martin Hogbin (talk) 15:18, 11 April 2012 (UTC)


 * My problem with this is it DOES make a judgment as to which solutions are 'right', in particular is says the "simple solution" is right (why else does the article lead with this solution?). Contrast this approach with the approach I'm suggesting above (which DOES NOT make any such judgment).


 * This is the standard problem (vS/Whitaker + usual (K&W) assumptions)


 * Here is a related problem and its easily graspable, intuitive, simple solution


 * Here is why the 1/3:2/3 answer to the simple problem must be the same as the answer to the standard problem (could include why the answer is not 1/2:1/2 as well in this section)


 * Here are a variety of solutions to the standard problem (simple, conditional, game theory)


 * More stuff (possibly including the controversy over the validity of the "simple solutions" to the standard problem in a much later section)


 * Unlike the structure you're suggesting, this structure (based on the approach used in a standard probability textbook and exactly echoed in another highly reliable source) accomplishes the same goal of helping the reader understand the subject without taking any stance whatsoever on whether any of the solutions to the standard problem are "more" correct than any others (because it presents them all equally). In the structure you're suggesting, the conditional and game theory approaches (you mean to include a game theory approach someplace, right?) are presented as if they are the ones that are controversial, where in reality the simple solutions are (somewhat) controversial and the conditional (and game theory) solutions are completely undisputed (indeed, both Selvin and Devlin follow up their own "simple solutions" with unassailably correct conditional solutions).  Your oft-repeated assertion that the structure you prefer makes no judgment as to which solutions are "right" is incorrect.  It definitely does.  And the judgment it makes is completely inverted from what the most reliable sources we can find actually say (some of which dispute the validity of the simple solutions while none dispute the validity of the conditional solutions).


 * Please read WP:STRUCTURE again (Segregation of text ... may also create an apparent hierarchy of fact where details in the main passage appear "true" and "undisputed", whereas other, segregated material is deemed "controversial", and therefore more likely to be false). Also (from WP:TECHNICAL) it may be helpful to compare with a standard reference work in the particular technical field to which the subject of the article belongs - like, say, Grinstead&Snell. -- Rick Block (talk) 20:14, 11 April 2012 (UTC)

@Martin: Some people think the earth is flat, and happily no-one consider them to be reliable sources. Of course is what you call "The wrong way" the wrong way. Neither me nor Rick would support this. But what you call "The right way" isn't the right way. Here is the right way:


 * This is the standard problem (vS/Whitaker + usual (K&W) assumptions)
 * Here is the (correct) solution.
 * This is why the answer is 2/3 and not 1/2
 * This is why it matters that the host knows where the car is.

"So far the article is simple and interesting. Those who wish can read further ''


 * Some sources present a simple solution, but this is incomplete, it is an answer to a different question
 * Most maybe all presented simulations make the same mistake.
 * More stuff.

We just quote reliable sources and do not make judgements of our own.

Got it? Nijdam (talk) 20:17, 11 April 2012 (UTC)


 * I agree with you about the simulations. Speaking from professional experience using stochastic modeling, I believe the main reason people find them so convincing is because they give no thought to the assumptions embodied in the models, apparently believing that "garbage in, garbage out" does not apply to model design. For MHP, I have always thought it absurd that people even consider stochastic models when the modeled problem space is small enough to completely enumerate on a 3x5 card. I assume it is because enumerating it forces you to examine the assumptions, about which they feel some doubts, but faith in the almighty computer allows them to ignore the existence of assumptions embodied in the model. Do you have citable sources for asserting the bogosity of the models or, more charitably, for clarifying which interpretations of MHP they actually model? ~ Ningauble (talk) 00:10, 12 April 2012 (UTC)
 * Right, but there are of course simulations which you perform yourself (in fact you play the game), whereby one experiences the necessary assumptions. About the enumerations, I'm interested how you would enumerate the possibilities of the MHP. Nijdam (talk) 06:54, 12 April 2012 (UTC)
 * How I would enumerate the model space would (a) depend on how the question is framed, and (b) be original research. ~ Ningauble (talk) 22:03, 27 April 2012 (UTC)


 * Regarding citable sources asserting the bogosity of the models, the (much maligned on this page and particularly detested by Martin) Morgan et al. (cited in the article) has this to say:


 * This analysis is one of 6 of what they call the most appealing "false solutions". -- Rick Block (talk) 15:44, 12 April 2012 (UTC)


 * In the same journal, however, was Seymann's criticism of this paper which Ninguable has quoted above.


 * Later, Morgan et al, say, '...had we adopted conditions implicit in the problem, the answer is 2/3, period.'  They also say that they should also have pursued the effect of the player having observed previous plays of the game.  In other words the exact mathematical model of the game is a matter of choice not fact.


 * But none of this matters. Although I believe that the Morgan solutions refer only to an academic extension of the problem I am proposing a genuine compromise in which both solutions are fully and openly presented. Martin Hogbin (talk) 16:27, 12 April 2012 (UTC)

Structure
Rick quotes from WP:Structure, Segregation of text ... may also create an apparent hierarchy of fact where details in the main passage appear "true" and "undisputed", whereas other, segregated material is deemed "controversial", and therefore more likely to be false.

The point is that we clearly state the criticism of the simple solutions by certain sources, giving the reasons for this criticism, thus the order of the two sections is irrelevant to the argument about which one is 'right'. On the other hand it is very relevant to the probability that our readers will understand what we are talking about.

Most physics text books and all practically all mathematics books start by ignoring some of the complexities of the subject; they have to otherwise they would be impossible to understand. Books on electromagnetism often start with Coulomb's law for example, but this is in fact wrong, it only applies to the non-exstent idealised case where nothing ever moves. I could give hundreds of examples of this approach. Subject have to introduced slowly bit-by-bit or they are incomprehensible. I defy anyone to find me a maths book or encyclopedia article that does not gloss over some subtlety at the start in order to get started.

Even if I were an avid Morgan supporter I would want to do things in this way. It gives us a chance to discuss the issues raised by Morgan et al in a proper and scholarly way without making the article inaccessible to the general reader.

The only alternative that I can see is endless argument that 'Morgan are right', 'Morgan are wrong', while the article languishes in limbo.Martin Hogbin (talk) 16:51, 12 April 2012 (UTC)


 * You say this, but yet you're apparently opposed to the structure I'm suggesting that does the very thing you're talking about (starts with solving a simpler problem). I'm confused. -- Rick Block (talk) 19:50, 12 April 2012 (UTC)
 * The simpler problem is an artificial construction whose main purpose is to explain the Morgan solution. That is why it is used in G&S for example, which is aimed at people wishing to learn more seriously about probability.  Many of our readers will not be interested in doing that.  I have no objection to mentioning the simpler problem in the introduction to the Morgan solutions. Morgan themselves make exactly the same point, because it helps to clarify their reasoning.


 * We need to start with the MHP, as it is generally understood, and then explain why the answer is 2/3 and why the host's knowledge matters because those are the two points which puzzle most people. If you are in any doubt about that just look through the talk page history.  We get a regular stream of newcomers claiming that the answer is 1/2 and that it cannot possible matter what the host knows.  Surely that gives us a clue as to what most of our readership want from this article.


 * Just to clarify again, my compromise is to give the Morgan solutions a level of prominence equal to that of the simple solutions (even though I think there are actually an academic backwater) but not to give them at the start of the article. Martin Hogbin (talk) 22:52, 12 April 2012 (UTC)


 * You keep saying you're compromising when what you're actually doing is simply insisting that you get your way. The point of my suggestion is NOT to explain the "Morgan solution", but to explicitly put the reader in a frame of mind where your beloved "simple solution" actually becomes intuitive to most people.  The regular stream of newcomers claiming the answer is 1/2 has not stopped, even though the structure of the article has been what you are arguing for, for approximately a year now.  Surely that gives us a clue that this structure (the one you are insisting on) does not provide the benefit you claim you want (making the answer easily understood).  My suggestion is a NEW idea - fundamentally unrelated to whatever disagreements we might have had in the past about the validity of this or that solution.  Frankly, I'm past that.  It would appear to me that you're not. -- Rick Block (talk) 00:14, 13 April 2012 (UTC)
 * The current structure is not what I am suggesting it is now a complete mess. Martin Hogbin (talk) 08:37, 13 April 2012 (UTC)


 * I have a challenge for you. Pose the Monty Hall problem (using whatever version you'd like) to anyone you'd like who's over the age of 12.  Before they answer, ask a few more questions (in this order).  1) What is the probability the car is behind door 1 BEFORE the host opens a door?  2) So if you decide you're going to stick with your original choice no matter what the host does, what is the probability you'll win the car?  3) What is the probability there's a goat behind door 1 BEFORE the host opens a door?  4) So if you decide you're going to switch to whichever door the host opens (i.e. no matter what the host does), what is the probability you'll win the car? My hunch is nearly everyone gets all of these right.  Establishing the right context is the key. -- Rick Block (talk) 00:26, 13 April 2012 (UTC)


 * Lest you think I'm kidding, tonight I actually did what I'm suggesting you do. 3 out of 3 (fairly random adults - and I understand this is not a statistically significant sample size) got the first 3 questions correct.  They had more trouble with the 4th, but with (minimal) more explanation were able to grasp that the answer to the 4th (however unintuitive it may be) must be the same as the answer to the 3rd (the probabilistic complement of the answer to the 2nd).  Just FYI.  -- Rick Block (talk) 06:42, 13 April 2012 (UTC)


 * We all know that it is the fact that the host opens a door to reveal a goat that makes the problem unintuitive. We all know that the player makes their choice after they see a door opened. The disputed point is whether it matters which door the host opens. Your challenge is an irrelevance as is your own OR on the subject. Martin Hogbin (talk) 08:37, 13 April 2012 (UTC)


 * Who is disputing whether it matters which door the host opens? I'm not.  I don't think Nijdam is either.  If you actually think this is what the dispute is about I think you're completely mistaken.


 * My challenge to you has nothing to do with OR. You're simply asserting that the presentation you prefer makes the 1/3:2/3 solution easier for most people to understand.  I'm suggesting an alternative, following an example from a standard reference work (Grinstead&Snell) as suggested by WP:TECHNICAL, backed by a different reliable source (Carlton) that says "it helps".  The challenge is merely an idea of a way for you to see for yourself whether this structure might make any difference - since you apparently don't believe Carlton (who, BTW, apparently teaches this).  I'm not suggesting adding "m out of n random people understand the MHP better with this presentation" to the article. Ergo, not OR. -- Rick Block (talk) 15:11, 13 April 2012 (UTC)
 * The long running dispute has always been about whether a solution needs to explicitly show that fact that the host could have opened two doors but, in fact opened a specific one. Martin Hogbin (talk) 18:42, 13 April 2012 (UTC)


 * Like you, I really would appreciate some input from others, although you are welcome to comment too. Does anyone else see the point of starting simple?  Martin Hogbin (talk) 08:37, 13 April 2012 (UTC)

Glkanter sockpuppetry deleted
(Comments by proxy 192.35.79.113 deleted. See below.)


 * I will respond to either Martin or Rick as I have been asked by Rick to do a very specific task here involving Martin and him. Sunray (talk) 16:07, 26 April 2012 (UTC)


 * Hmmm. an IP that pops up sounding exactly like yet another sockpuppet of Glkanter‎. At first I thought it might not be, based upon the edit history, but the Geolocate button shows "Confirmed proxy server". The "findings of Ownership by the ArbCom" statement makes it clear. There is only one person on earth who is still obsessed with this one small part of an ArbCom case from over a year ago. I am invoking WP:DUCK and deleting it. --Guy Macon (talk) 15:07, 27 April 2012 (UTC)

Cutting to the chase
OK. We've been discussing this (for years?) without a resolution. Should the article

1) present the standard version of the MHP, which encourages the reader to think the question pertains to the specific situation where the player has picked Door 1 and the host has opened Door 3 with two and only two alternatives being stay with Door 1 or switch to Door 2, immediately followed by a collection of solutions that actually address staying with Door 1 or switching to either of Door 2 or Door 3

or

2) do something, virtually anything, else

Martin argues #1 is the one and only thing that we can ever consider. I've suggested a number of alternatives for #2 (including, just recently, introducing a source-based section introducing a simpler, related problem precisely corresponding to the "always stay" vs. "always switch" intuitive solution).

I'm really tired of this.

I carried this article through the initial WP:FAC process and two subsequent WP:FARs, which I think demonstrates an ability to bend to the wishes of the community at large. Martin (and various others) have instigated a variety of challenges culminating in an arbcom action, the article has lost its featured status, and it is currently "a mess" ( per Martin!).

Please indicate below who you'd rather keep this page on their watch list, me or Martin. I certainly can't speak for Martin, but if the 50% + 1 consensus here is Martin, I'll delete this page from my watch list and never post here again.

Thanks for all the fish. -- Rick Block (talk) 04:56, 17 April 2012 (UTC)
 * Rick, I am sorry that you feel the need for one of us to go. I have put another suggestion below which I think is preferable and which I hope will be acceptable to you and which is more in line with WP as a cooperative encyclopedia. If the idea is acceptable to you, I will suggest a few more details for brief discussion before we go ahead.Martin Hogbin (talk) 09:51, 17 April 2012 (UTC)


 * As you know, in the past I have tried to shepherd this content dispute through Content Dispute Resolution, staying as unbiased as I could possibly be. That effort failed because I could not get agreement by all parties as to exactly what question to ask uninvolved editors. This isn't just a dispute over article content; there is a unresolvable dispute over what the content dispute is! Like so many before me, I walked away, but have continued monitoring the talk page. I just went back and skimmed the last hundred or so posts again, and I am now going to give my personal opinion about who's vision of the MHP page I would choose. Before I give that opinion, let me say that I do not believe that either vision is bad or wrong or stupid or bad for the encyclopedia, but that I do believe that not agreeing for year after year is bad for the encyclopedia, or at least bad for the article.  I would also note that, because of the unresolvable dispute over what the content dispute is, Martin will no doubt disagree with Rick's description of the dispute above, and may even write his own description, which of course Rick will disagree with. Because of that I am ignoring the wording of the question asked above, and am simply saying who's vision of the MHP page I would choose, based upon the totality of the arguments posted over the last year. With those caveats, my personal preference is Rick Block's vision for the MHP page. --Guy Macon (talk) 06:44, 17 April 2012 (UTC)


 * Thanks for revealing your personal position, I think that is a good thing. Rather than cast a vote, which is just one of many that have been cast over the years of this dispute, I think you should explain and discuss your reasoning now, as you have refrained from doing this in the past in order to take a neutral stance.


 * If we are to try to reach a final decision we should have another RfC. I suggest that Rick and I present our respective side's arguments at the start (in an agreed number of words if desired and possibly on a separate page) and then all involved editors (say anyone who has edited here in the last year) keep quiet and we leave the decision entirely to uninvolved editors.


 * Win or lose I would accept the result and I think others should too. Martin Hogbin (talk) 09:40, 17 April 2012 (UTC)


 * I just made up my mind on this this week, and it pretty much comes down to completely subjective personal preference with me. I really cannot point to any flaw in your vision, and I think that it would also lead to a fine article.  In fact, if I see a tie, I will almost certainly change my vote to give you a two "vote" majority.


 * If the two of you decide to do an RfC, I have thought about that a lot. I suggest doing it like this:


 * First, pick a hard size limit, with an optional link to something on your your userpages with no limits. 500 words, perhaps?


 * Second, you each write an argument without seeing what the other is doing. You could agree to email them to me at exactly 12:00 and then post them after that, and I would confirm no changes other than typo fixes.


 * Third, you each write a limited-size rebuttal, again without seeing each others work.


 * Fourth, you each get another X number of words total to address any and all questions, rebut bad arguments, etc. Use them up and you are done commenting. --Guy Macon (talk) 10:33, 17 April 2012 (UTC)


 * That plan is rather more complex than I was envisaging but I would be happy to go along with it. Regarding the final decision, I would suggest that whoever closes does so on the basis of the number of votes unless there is a very strong and compelling argument based strictly on WP policy and reliable sources not to do so.  There have been some real experts in this subject who have contributed here along with many others who have discussed the problem for years.  To expect a uninvolved person to close the RfC by 'weighing up the strength of the arguments' is in those circumstances an absurdity. I am fully aware of WP:PNSD but in this case no one can claim that we have not had a full discussion.  Rick and I and others want to move on and I am happy to abide by the result of a vote of uninvolved editors.  That does not mean that I will leave the page completely but I will refrain from pressing my opinion on the disputed topic. I suggest that we set a time for the RfC to be closed in advance. Martin Hogbin (talk) 12:44, 17 April 2012 (UTC)

Uninvolved editor here; Rick can I ask, are you married to the wording " The answer, as shown below by numerous solutions to the standard problem, is that they're not! ", it doesn't seem completely encyclopedic in tone to me.Number36 (talk) 11:31, 17 April 2012 (UTC)


 * No, not married to any particular wording. It's a wiki.  Anyone can change anything.  -- Rick Block (talk) 15:01, 17 April 2012 (UTC)


 * Rick, are you happy to go with my/Guy's proposal?


 * Guy, are you happy with my suggestions and willing to administer the process? Martin Hogbin (talk)


 * Am I happy? No, not particularly.  Mostly because as far as I can tell you've never been willing to describe the conflict in a way that even remotely hints what the real issue is or (IMO) says what you actually want.  I suspect this would make an RFC confusing and probably pointless.  Perhaps we could find a third party (one of the mediators perhaps?) who could write up a mutually agreeable RFC statement.  If we could find someone would you be willing to try this? -- Rick Block (talk) 20:15, 17 April 2012 (UTC)


 * I have no objection to using a third party to write the RfC statement but I think this will not be necessary. The statement could just say something like,'Editors are asked to chose between two different proposals for the article format'.  I think we should do all we can to attract as many people as possible to the RfC, so perhaps we could point out that detailed discussion will not be required as the issues in dispute have been under intensive discussion for years. Would you agree to accept the decision if the vote goes against you?Martin Hogbin (talk) 21:45, 17 April 2012 (UTC)

In my opinion, "as far as I can tell you've never been willing to describe the conflict in a way that even remotely hints what the real issue is" is just another way of saying what I said earlier: there is a unresolvable dispute over what the content dispute is. I think that both sides can honestly say that the other side won't describe what the real issue is, and that both sides can honestly respond be saying that they cannot imagine that being true.

I also strongly suspect that this is recursive; there is a unresolvable dispute over what the unresolvable dispute over what the content dispute is is, there is a unresolvable dispute over what the unresolvable dispute over what the unresolvable dispute over what the content dispute is is is, and so forth (smile).

Because of this, I don't think any 3rd party can describe what the content dispute is, thus my proposal for each party to make a single limited-length statement and rebuttal, and then to have a limited amount of words in the discussion that ensues. This has four advantages; first, it bypasses the recursive problem. Second, it does not allow the RfC to grow without bounds. Third, knowing that your statement of the dispute will have a rebuttal that you cannot respond to will result in a lack of easily rebutted points in the statements, and fourth, any new argument put forth only in the rebuttal will be naturally be viewed with skepticism.

Meanwhile, a little voice in the back of my mind keeps asking me how my proposal for resolving the dispute will crash and burn this time... --Guy Macon (talk) 22:38, 17 April 2012 (UTC)
 * I agree with you, we should make our statements and leave it to others to decide, if a bunch of crazy editors who think I am wrong turn up and vote against me that is too bad. I will accept the result. The RfC should just say chose between the two stated views.


 * I hope you agree that we must accept a simple vote, the subject has been discussed to death and to expect someone new to come along and make a decision on the basis of the strength arguments is crazy. Rather than one person come along and say which argument is stronger better for many people to say which argument they think is stronger and then go with the majority.  There is an argument for having someone close the RfC who has no interest in or understanding of the subject but who is an experienced wikipedian. Do you have any suggestions?


 * This is not an argument between myself and Rick, it is an argument between two groups of editors and I would not exclude anyone from the discussion. Although Rick and I may act as spokesmen for the two sides I assume that we do not object to discussion between the two sides on the best way to present their case.


 * If work to an agreed set of rules then this resolution will not crash and burn due to me. I will abide by a vote. Martin Hogbin (talk) 08:36, 18 April 2012 (UTC)


 * Re: having someone close the RfC, I believe I know where I can find that person, once we all agree. Re: voting, I agree, but someone is likely to bring up Polling is not a substitute for discussion. That guideline only applies if someone is trying to force someone else to obey the result of a vote, not to editors who agree beforehand to voluntarily abide by the result of a vote. Heck, you can agree to flip a coin or hire an astrologer when deciding whether to continue working on a page. --Guy Macon (talk) 10:51, 18 April 2012 (UTC)
 * I agree, but is everyone else willing to abide by a vote? Martin Hogbin (talk) 13:59, 18 April 2012 (UTC)


 * I do not like this section. It starts with a simple issue -- addressing the quantified v literal interpretation of opening door 3. But it cannot stop there. Instead of staying on that simple point, it is compelled to broaden the debate to include a nebulous and contentious option of "something, virtually anything, else". Then it goes off the rails with the crazy notion of a duel at dawn that will leave one of two editor-protagonists dead in the dusty street. Instead of a simple debate on a well defined issue, we have a death match.
 * To add to the story, another editor finally chooses a side and promotes this final solution. Strangely, he's willing to immediately surrender his recent opinion, ignore all argument, and let the flip of a coin or the alignment of the planets decide the issue.
 * The duel theme is not new. Instead of finding ways to improve the article, there's been this continual desire to take some grand either-or visions to a third party for a final decision. A binary decision in a multi-dimensional space. The tone has not been let's identify some issues to work out. It's been let two protagonists write position papers for an external judge. That's a screwy form of abdication.
 * The subtext of this section is that the duel is for the WP:OWNERSHIP of this article. None of us should own this article. None of us should control it. Both protagonists happen to have influence simply by engaging in the endless and unrestricted debate upon these pages. I don't think they like it, but that's what happens. Other WP articles have had factious, violent, and seemingly irreconcilable disagreement, yet they've been able to isolate issues, decide how to handle them, and produce a reasonable compromise.
 * That the duel would be voting one of the protagonists off the island also illustrates the failure to divide the issues into smaller ones. The sensible approach is to divide issues and look at them separately. Instead of a binary decision, a sensible evaluation would have four possible results -- it would include the options that both protagonists might stay or both might get voted off. I'm not proposing that we have such a vote -- I'm just using it to illustrate the absurd monolithic option that invests this talk page.
 * I like some of what Rick Block says. He has a particular clarity on some issues, but Rick also goes too far with it. The same goes with Martin. I like some of it, but I'm unhappy with other parts. Consequently, I'm not happy with either's vision, and I detest the continual take option A or option B because there are no other options mindset.
 * No editor should be voting to turn the keys over to either Martin or Rick.
 * I also don't like the notion of trying to banish well intentioned editors. In any event, an article talk page is also not the appropriate venue for that process; that jury should be unbiased and disinterested. However, I am mindful that a staggering amount of text is generated on this talk page for little or no impact on the article. Those walls of text are intimidating to other editors.
 * The poorly focused debates need to stop. If someone wants to open a debate on a distinct issue such as door quantification and its impact on the article (or even WP:BRD it), then go ahead. Identify something that's wrong with the article and fix it. Just don't let it snowball into an intractable contest for world domination.
 * For example, here's an issue. The statement "If there are two doors left, then why isn't each door 1/2?" is in the Monty Hall problem section about 1/4 the way down the article. It is the mindset that makes the puzzle, and a newspaper editor might comment that it buries the lead. Should a similar statement be made earlier in the article to explain the thinking of many contestants?
 * Glrx (talk) 21:29, 18 April 2012 (UTC)
 * I have only been arguing for just one clearly defined thing and that is to have the simplest solutions present at the start with no disclaimers or 'health warnings' followed by a clear explanation of the two things that nearly every newcomer wants to know: why the answer is 2/3 and not 1/2, and why it matters that the host know where the car is. Do you disagree with that idea? Martin Hogbin (talk) 22:01, 18 April 2012 (UTC)


 * Glrx, you have made some allegations about me which I do not think are justified. I have no desire to 'hold the keys' to this article or claim ownership to it.  I came here long ago because of an RfC and because of an allegation of ownership, which has been dealt with by Arbcom.


 * As you know there has been a long and protracted debate, involving many editors, about a single issue, which is the relative importance of what for simplicity I will call the 'simple' solutions and the 'Morgan' solutions. Some editors believe that the 'simple' solutions are what the MHP is all about whilst others believe that the 'Morgan' solutions are the only correct ones.  There has been endless debate on this subject with no progress being made.  Rick and I have made suggestions for solutions to this specific problem which involve mainly changes in the way the article is structured.  I have proposed, and Guy has supported putting this matter up for a community consensus by means of an RfC.  No one else has made any proposal to resolve this debate.


 * Once that decision is made and the article has been changed in accordance with it, other editors will be free to improve the article without fear of stepping into a hornet's nest. I would expect that the chosen structure would not be changed without clear consensus but apart from that the article could be improved in any way that people think fit.


 * On the above basis described above are you willing to allow this process to continue? Martin Hogbin (talk) 12:59, 20 April 2012 (UTC)


 * I agree with many things Glrx says here. There are actually several issues and there is no way a single RfC will resolve them all.  On the other hand, I agree with Martin there is one main dispute between us, which I think boils down to a POV issue.  It must.  There's absolutely no way a dispute can last this long and not be a POV dispute.  I've said this before, but IMO the only way to resolve this is to treat it as a POV dispute.  As far as I can tell, Martin's stance is that it's NOT a POV dispute but insists that what's needed is a binary vote on an entirely reasonable sounding, somewhat nebulous article "structure" where discussions of simple solutions come before more complex ones (but with a POV-laden tag-along constraint that the so-called "simple solutions" must be presented as if they are universally understood to be the complete and correct answer for the standard version of the problem - with no mention, whatsoever, that any other approaches even exist until later sections of the article intended to be of interest only to experts).  In contrast, I've been pushing the stance that it's simply a POV dispute and should be resolved as one (not that the article must reflect my POV), and have made a series of specific, detailed, source-based suggestions for text that might accomplish this.  I think the asymmetry in these stances is at least one thing that makes "the argument" so difficult to resolve.  Rather than repeat (yet again) things I've already said, I'll refer the interested reader to the archives for the following:


 * Drafts I've suggested for what is intended to be a completely neutral, NPOV (combined) solution section (each and every one rejected essentially out of hand by Martin):


 * A completely different approach, suggested recently (likely still above, also rejected out of hand by Martin)


 * Comments (by me and others) about what I consider to be the real issue


 * One area in which I disagree with Glrx is the notion that I promote an either/or mindset. I think the links above demonstrate a willingness to not only consider but  propose actual text for many different alternatives.  The only thing I'm arguing against is the notion that the article should present the so-called "simple solutions" as if they are universally understood to be the complete and correct answer for the standard version of the problem while everything else is treated as an academic diversion of interest only to experts.  This apparently directly conflicts with the only thing Martin wants.  Furthermore, he apparently won't engage in any serious attempts at editing until the article's structure is "fixed".  If anyone can think of any way to help resolve this it would be most appreciated.  -- Rick Block (talk) 20:23, 20 April 2012 (UTC)
 * I am happy to engage in editing but I suspect that it will simply become an edit war. If you want me to improve the article now, just say so. Martin Hogbin (talk) 09:18, 21 April 2012 (UTC)
 * If person A makes a change that person B reverts, this is not an edit war. It becomes an edit war if person A then reverts the revert (followed by person B reverting the revert of the revert, etc.).  Are you saying you expect any changes you make to be reverted and that you will then revert any reverts of any changes you make?


 * Regarding the proposed RfC (section below), I still don't see how we can expect this to succeed if we don't have a focused description of the conflict on which we agree. I suggested above trying to find a 3rd party willing to help create a statement.  I suggest (again) that we proceed with this. -- Rick Block (talk) 03:58, 23 April 2012 (UTC)


 * I do not believe that this is possible. I tried and failed - there was no description of the conflict which the two of you were willing to agree upon. That is why I structured the RfC below so as to work without requiring such agreement. Perhaps someone else can write such a description, but I think that the task is impossible. I am going to withdraw again and go back to silently monitoring this page (other than reverting and reporting vandalism and sockpuppetry) and mark my proposal as failing from lack of agreement. I wish you luck in finding that third party, and I really do hope that I am wrong. --Guy Macon (talk) 11:05, 23 April 2012 (UTC)
 * I agree with Guy, there is no need for a third party to write the RfC statement because it will simply say something along the lines of 'Editors are requested to chose between the two proposals shown'. I am sure Guy could write this without any problem. We are both free to write our proposals in any way that we wish.  What is wrong with that, Rick?


 * Guy, I would ask you to persist a little longer with this approach. I cannot see what Rick objects to.


 * I would also be happy to work with a neutral third party, if we can find one, to come up with an RfC statement. I will ask around. It might be best to get someone with little interest in the subject itself but who is experienced in dispute resolution.  Martin Hogbin (talk) 13:19, 23 April 2012 (UTC)


 * The problem with proceeding with two individual statements and not an agreed upon statement is that Martin's statement will likely say something relatively inarguable like "should the article's structure proceed from simpler to more complex descriptions consistent with WP:TECHINCAL" while mine will also likely say something relatively inarguable like "should the article adhere to WP:NPOV". These sound like two completely different issues, the answer to both of which is clearly yes - how can they possibly be in conflict?  Making a judgment on these requires completely different skills and backgrounds - judging POV requires familiarity with WP policies and the bulk of the sources, while judging ease of understanding actually requires cognitive psychology skills (although I suspect most respondents would simply express their own personal preference). I think this is one of the main points Glrx is making above.  Without boiling these down to discrete issues, people who comment will have no actual idea what we're asking them to comment about.


 * For example, I think one of the foundations of Martin's stance is that he believes the Morgan et al. paper (and anything else that says anything similar) should be considered as representing a "fringe" POV. I'm not sure we even agree what POV this paper presents, but this seems like a specific, resolvable topic - and one that picking Martin's approach to the article implicitly decides.  -- Rick Block (talk) 14:41, 23 April 2012 (UTC)


 * You are quite correct about my personal stance regarding Morgan but my proposal will be the compromise position that we deal with the simple solutions first (without disclaimers of any kind) then we discuss the central issues, then we give the Morgan criticism and subsequent solutions full and completely and discussed in a scholarly manner. I cannot see that is too hard to understand. Martin Hogbin (talk) 15:02, 23 April 2012 (UTC)

Rick, it's a given that you will think that Martin's argument will sound like you are talking about two completely different issues. That's why you have a rebuttal, so you can point that out to the reader. Likewise it is a given that Martin will find flaws in your argument, which is why he gets a rebuttal.

And if that's not enough, both of you get to post comments in the ensuing discussion hammering home the flaws you see in the others arguments. You can even make a strategic decision such as keeping your argument / rebuttal very short and thus having more words in the ensuing discussion, or using more words in the argument or the rebuttal in an attempt to make it relatively bulletproof against counterarguments.

The way I see it you have two choices; argue until one of you dies of old age or the Copyright Lobby finally kills the Internet, or simply explain in your own words why you think your vision should prevail and why Martin's is flawed, meanwhile allowing him to do the same thing. You are never going to be able to stop him from making what you think are the wrong arguments / problem description / characterization of your arguments and he sees as him making the best arguments and rebuttals he can. Likewise, you are never going to allow him to stop you from making the best arguments and rebuttals you can. And of course all of the above is just as true when addressed to Martin.

I say make your best argument, do your best in your rebuttal, and trust the readers to evaluate both arguments and both rebuttals. If the other fellow writes something that is misleading or wrong, point it out and trust the reader to understand why it is flawed. --Guy Macon (talk) 15:50, 23 April 2012 (UTC)


 * @ Martin. I'm not not accusing you of wanting unreasonable things. I think both you and Rick are interested in improving the article. The problem is rather one of the failure to converge. Glrx (talk) 23:57, 27 April 2012 (UTC)


 * @ Rick. Both you and Martin are willing to compromise on some things. The either-or view arises because everything gets glued together. Even a little discussion snowballs to be either view 1 or view 2. Glrx (talk) 23:57, 27 April 2012 (UTC)

The health warning
Where are we on a health warning?

Say somebody could write a simple, VS-style description of the MHP with arbitrary doors and simple solutions to start the article off.

Say somebody else could require a health warning in that section. The health warning should be a small part of the description - maybe 5 but no more than 10 percent of the section.

What would the health warning have to say?

To keep things from snowballing, do not debate other editors' answers (i.e., not indented comments). We'll save that part for later. Remember the 5 to 10 percent limitation, so a long response would defeat the notion of a warning. Start your response with an asterisk to give a bullet. If the health warning is unacceptable, then say * Oppose and give a short reason why. If no health warning is needed, then say * No warning and give a short reason. Otherwise, do * Warning and give your warnings and/or point to other editors' choices (possibly adding or subtracting).

Glrx (talk) 00:37, 28 April 2012 (UTC)


 * Confused (I'm not sure I understand) - It sounds like you're suggesting somebody write an original description of the MHP, followed by published solutions claiming to solve the "standard" MHP. If so, I think this would be WP:OR and the health warning would have to be cn.  The new section I suggested above possibly has the same intent (simple problem followed by simple solution), but avoids the OR issue since the description and solution in this section are from the same sources.  If we stay strictly within what sources say (which I believe we should), and the article is broadly NPOV (which I believe it has to be) no warning of any kind should be necessary.  -- Rick Block (talk) 07:00, 28 April 2012 (UTC)
 * Already planned. That is exactly what this debate is all about, 'To what degree should we mention claimed deficiencies of the simple solutions and where?'. — Preceding unsigned comment added by Martin Hogbin (talk • contribs) 09:43, 28 April 2012
 * Ambiguous. I think there may be a general consensus against health warnings per se, but there appears to be disagreement about what constitutes a health warning: one person's health warning is another person's aid to understanding. ~ Ningauble (talk) 14:51, 28 April 2012 (UTC)

Whac-A-Mole
As some of you may have noticed, banned user Glkanter has been attempting to evade his ban by under various IP addresses. His latest trick is to use Proxy Servers.

We have addressed the immediate problem by semi-protecting this page so that only autoconfirmed users can edit it, but we don't want to do that permanently -- other IP editors sometimes contribute.

After some discussion with the clerks and admins, the advice they gave me was to play a game of Whac-A-Mole as a countermeasure. If you see any edit by a Glkanter sockpuppet, delete it on sight. If it turns out to be a false positive we can easily undo the deletion. Whatever you do, do not reply.

So how do you identify a Glkanter sockpuppet? Please use the "" method. The first clue is that he usually refers to arbcom sanctions that have expired. This is almost always inappropriate; sanctions are not meant to be a "badge of shame" after the sanctions have expired, and the sanctioned editors are allowed and encouraged to make a fresh start. This of course does not apply to lifetime bans, such as the ban given to Glkanter.

Again don't be overly concerned about false positives. The vast majority of legitimate IP users talk about the Monty Hall problem, not about year-old disputes with other editors. I will be watching this page and will evaluate every deletion. I will check the geolocation, and whether it is a proxy, and if needed I will get help from a clerk. If you accidentally delete something from an IP that is not a Glkanter sockpuppet, I will restore it.

In case anyone is curious about whether Glkanter can ever come back, yes, he can. First he has to stop vandalizing Wikipedia with sockpuppets. Next, he has to convince an uninvolved admin that he understands what got him blocked and that he will will not repeat his misbehavior. This would get the indefinite block removed, but he would still be subject to his lifetime MHP topic-ban. To get that removed he will have to spend at least six months editing other articles without the misbehavior that got him banned, then request an unbanning. If he convinces arbcom to lift the ban, he will then be watched closely and instantly blocked and banned if he goes back to his old ways.

There are a lot of antivandalism tools that we have not been applied to this case because they have undesirable side effects on other IP editors, in some cases making them register in order to edit this page. Glkanter cannot win this battle and he will only hurt other innocent IP editors if he tries.

Remember, do not respond, delete on sight. Good luck with your Mole Whacking! --Guy Macon (talk) 22:58, 28 April 2012 (UTC)

Cutting DIRECTLY to the chase, minus the usual debates
Once again we are going down the rabbit hole. Debating whether there is a single issue or multiple issues. Debating whether the issue is a POV issue. Debating about who is being flexible and who is being rigid. We could spend another unhappy year debating just what has been brought up (again) in the "cut to the chase" section above. Here is my proposal for bypassing all debates and arriving at a solution.

First I am going to state some findings of facts / unsupported assumptions (take your pick):

There exist two individuals who we shall refer to as "Martin" and "Rick".

They disagree about something - something that we will not attempt to define.

They have been talking about this for a very, very long time, with no end in sight.

They are both tired of doing this.

They are free to resolve this any way they choose. Other editors are free to opine on the proposed method of resolution, but only Rick and Martin have to agree on it.

No other editor is bound in any way by any of this.

Proposed solution:

An RfC is created.

Rick and Martin are to be given 1000 words total. They may link to new material in their user space that has no size limit, and they may link to previous arguments. Links count toward the word total. The count will be done with the word count feature of UltraEdit and the running total appended to each post (the running totals are not themselves counted as words).

The 1000 words are to be split as they wish among the following:

Each writes a statement that they wish the voters to vote on / agree with. Neither gets to see the others statement before it is published.

Each reads the statements and writes a rebuttal, again without seeing each others work.

In the discussion / voting, each is allowed to address questions, make comments, rebut bad arguments, etc., up to 1000 words total. Any words past that will be deleted on sight. Going back and removing words posted earlier does not increase the total allowed.

Voting will be for two four weeks.

At closing, an independent and unbiased editor who has never edited any MHP page (including talk) will be asked to count the votes, throwing out obvious bad votes from blocked users, sockpuppets, etc. The discard decisions may be debated, but she/he has the final decision.

If (and only if) the result is a 50:50 tie, voting will be extended one two additional weeks.

Whoever loses the vote will graciously step aside and let the winner have his way. Again, nobody else is agreeing to pay any attention to this, so the winner of the vote may very well face opposition from other editors - but not from the loser of the vote.

Editors are politely requested to only comment on the nuts and bolts of this proposal, and to place any other arguments (including should we do this) in the section above this one. --Guy Macon (talk) 08:23, 21 April 2012 (UTC)


 * Sounds good to me. I think the decision made must be accepted as a consensus, but only regarding the specific issue voted on, otherwise the whole thing will start all over again with a new editor of the losing side taking the place of the original loser. Martin Hogbin (talk) 09:23, 21 April 2012 (UTC)


 * I suggest voting is for four weeks. The more outsiders the better. Martin Hogbin (talk) 09:23, 21 April 2012 (UTC)


 * Done. --Guy Macon (talk) 10:25, 21 April 2012 (UTC)


 * While we are debating "should we do this" in the section above, does anyone have any criticisms / suggestions about the nuts and bolts of the plan above? --Guy Macon (talk) 15:54, 23 April 2012 (UTC)

Preparing an RfC statement
Rick contacted me ask for assistance in working with Martin and him to prepare an RfC statement. I note in the above discussion that it has been suggested each of them prepare a statement. That is certainly a possibility, and the other approach would be to work out neutral language that both Martin and Rick agree to. I'm willing to help with that. How should we proceed? Sunray (talk) 01:36, 26 April 2012 (UTC)
 * Working out neutral language that both Martin and Rick agree to would be my first choice. I was unable to accomplish that, which is why I proposed the two-statement scheme above. It may very well be that you have no problem getting agreement; I have mild Asperger's syndrome, and sometimes I miss nuances. If you are not able to get agreement on what the disagreement is about, the above scheme is, I believe, a good alternative that doesn't require agreement. With that said, I am going to watch this page without commenting for a while. --Guy Macon (talk) 02:34, 26 April 2012 (UTC)


 * First I think we need to clarify a bit what the result would be, and then make sure Martin is on board. As I've mentioned, I think the two independent statement approach suggested above is likely to end up with jarringly different statements that will tend to make it difficult for folks to comment.   Instead of two separate statements, the suggestion is Martin and I work out a single, focused statement - with Sunray's help - and not have anything else (not two statements with two rebuttals, just the one statement).  Perhaps this might start with two separately written statements that we then merge into one, but if we agree this is the goal then I'm pretty sure we can work out the procedural details.  Martin - are you OK with trying this?  I'd further suggest if the answer is yes, that Sunray set up a subpage with his thoughts on how to proceed.  For Sunray's reference, previous efforts at coming up with RfC statements are in the archives (I'd suggest starting reading about here). -- Rick Block (talk) 03:06, 26 April 2012 (UTC)


 * Sunray, once we go to RfC, how would you suggest that a final decision is made?


 * Have you had any involvement in this subject before? Your name seems familiar. Martin Hogbin (talk) 08:34, 26 April 2012 (UTC)


 * https://en.wikipedia.org/w/index.php?title=Special:Search&limit=100&offset=0&redirs=1&profile=all&search=Sunray+Monty+Hall Requests for mediation/Monty Hall problem:


 * "This case was initiated in January 2010, and underwent extensive formal mediation from then until the end of 2010. During that period of mediation, multiple mediators were in turn assigned to the case: first, User:Andrevan; second, User:Sunray; and third, User:Sunray and User:AGK. The case was then referred to the Arbitration Committee at the start of 2011, with a request for arbitration being filed by one of the parties, User:Rick Block, on 9 February 2011. This request resulted in the arbitration case at Wikipedia:Arbitration/Requests/Case/Monty Hall problem and the resulting decision. During arbitration, this mediation case was placed on hold; after arbitration (which ended on 25 March 2011), mediation remained on hold, and was then closed on 28 April 2011." --Guy Macon (talk) 13:08, 26 April 2012 (UTC)


 * Martin - I thought asking one of the mediators previously involved would be a reasonable choice (able to hit the ground running). My first thought was AGK, but his talk page suggests he's otherwise occupied IRL.  I can't imagine why you would, but if you object in any way to having one of the previously involved mediators help us work out a consolidated statement I'd suggest we ask someone else on the mediation committee, or possibly one of the arbitrators.  I haven't talked about this with him, but I expect Sunray would not be in the least offended if you'd prefer we try to find someone else.  -- Rick Block (talk) 15:20, 26 April 2012 (UTC)


 * @Martin: Yes, as Guy has pointed out, I was one of the mediators during the long history of this dispute. As I said, I would be happy to assist in developing an RfC statement. In addition to being a mediator, I have real world editing skills, which may be useful; also some familiarity with the topic. And yes, I am open to you choosing someone else. Sunray (talk) 16:10, 26 April 2012 (UTC)


 * Sunray, what about my other question? Once we go to RfC, how would you suggest that a final decision is made? Martin Hogbin (talk) 17:05, 26 April 2012 (UTC)


 * One option would be to agree on the decision rules we would apply. This is often done in consensus environments were a clear decision is needed. Usually that means defining a supermajority. Wikipedia examples where a supermajority is the decision rule applied include RfAs and AfDs. Sunray (talk) 17:53, 26 April 2012 (UTC)


 * As there are just two proposals why would we not just use a simple majority? Martin Hogbin (talk) 21:35, 26 April 2012 (UTC)


 * The problem I see with that is that a simple majority is not consensus (according to the commonly accepted definitions of consensus). Editorial decisions are to be made by consensus. That doesn't mean unanimity, but to avoid any misunderstanding, it is best to define what constitutes consensus. A supermajority qualifies as consensus. Sunray (talk) 01:29, 27 April 2012 (UTC)


 * So what will happen if, say, Rick's proposal gets 51% of the votes? Martin Hogbin (talk) 09:54, 27 April 2012 (UTC)

There are two separate things you can do with a RfC, both of which are perfectly legitimate.

First, there is achieving consensus. Let's say that 100 people vote. 80 or 90 voting one way would be a clear consensus. 51 votes would be no consensus. Consensus is very desirable because everybody, including Glrx, Ningauble, Nijdam, Me, and even Glkanter‎ (should he convince the admins that he is willing to behave and gets unbanned) is required to follow consensus when making editing decisions.

Second, two editors can agree to let a vote - even a 51% vote - decide a content dispute. That's because individual editors are free to make or not make any edit that is within policy for any reason or for no reason at all. There have been occasions when I was in favor of something, someone else ran an RfC and got 75% for it, and I decided to unwatch the page and let someone else make the changes. I certainly had a right to do that - nobody is forcing me to edit Wikipedia. So the two of you could agree to abide by a 51% vote, or you could each decide that you need at least 60%, all the way up to the point where there is a consensus.

Deciding by agreement has several disadvantages. First, anyone can go back on the agreement with no repercussions other than a few folks going Tsk Tsk. Second, no gentleman's agreement by Martin and Rick is binding on anyone else, so Glrx, Ningauble, Nijdam, etc. may very well have a content dispute with the "winner". Making such an agreement does have the advantage of being decisive, and it will identify a consensus if there is one.

Rick Block indicated his willingness to accept 51% when he started this thread, saying "I certainly can't speak for Martin, but if the 50% + 1 consensus here is Martin, I'll delete this page from my watch list and never post here again." Of course he has every right to do that. (He can, of course, take it back - nowhere did he agree to details about the RfC that were developed later.)

So, it appears that we have all the components of a solution except one (both parties agreeing to it). We have a proposed criteria that is 100% sure of achieving a decision (see my previous comments to see why 50%/50% cannot happen). We have an uninvolved third party (Sunray) willing to help you to make a single statement that you both agree on. We have my proposal as a second choice in case he cannot achieve that goal. We have other choices in the wings in case anyone objects to Sunray or me being involved with this. We have a proposal that limits word count so that the discussion cannot grow without bounds.

Nonetheless, a little voice in the back of my mind says that this too will crash and burn. Please show me that I am wrong... --Guy Macon (talk) 14:43, 27 April 2012 (UTC)

Proposed steps
I appreciate your clarity, Guy. I think that whichever course of action is taken, the steps should be: I agree to assist in developing the RfC statement and decision rule and will tally the results of the RfC. Guy, if Rick and Martin agree would you be willing to assist with the process? Sunray (talk) 16:56, 27 April 2012 (UTC)
 * 1) Martin and Rick agree on an RfC statement, or statements.
 * 2) They then agree on a decision rule
 * 3) Input is requested from other regular editors on the proposed wording and parameters of the RfC
 * 4) The RfC is then posted and opened to comment from all autoreviewed editors


 * Would Rick and Martin please comment on the above? Other editors may wish to comment on point #3. I'm unsure whether that step is necessary. Sunray (talk) 17:02, 27 April 2012 (UTC)


 * I am happy to help in any way I can. I am also happy to step aside if that would be better. --Guy Macon (talk) 17:09, 27 April 2012 (UTC)


 * Sounds fine to me. I'm mot sure if step #3 is necessary - but wouldn't object to this if anyone thinks it is. -- Rick Block (talk) 19:54, 27 April 2012 (UTC)


 * As a "third party" who has followed this article for years, I think point #3 is worth keeping. I suspect that last year's attempt to arrive at a formal RFC failed in large part due to there being too many issues in play for a single answer, and in no small part due to the fact that "Les querelles ne dureraient pas longtemps, si le tort n'était que d'un côté." ("Quarrels would not last long if the fault were only on one side." – François de La Rochefoucauld, Reflections; or Sentences and Moral Maxims, Maxim 496. ) I was prepared last year to !vote "no" on what was becoming framed as an "either/or" choice between unsatisfactory alternatives. If this starts heading down a wrong track again, or if (a very big "if") anybody sees an agreeable way to re-factor the question(s) in a way that might actually settle them, then input from other editors should be encouraged. I do not fault Guy for the way it fell apart last time because it is very, very difficult to disentangle the issues. I will be very pleasantly surprised if the parties succeed on their own. ~ Ningauble (talk) 22:12, 27 April 2012 (UTC)


 * I agree with Rick that point 3 is unnecessary and would go further and say that it is unhelpful. My point, or in fact the point made by the clear majority of editors who have visited this page is that the Morgan solutions complicate an already difficult problem and make it harder for new readers  to understand the central paradox.  Therefore we should not mention anything about the Morgan solutions or claimed deficiencies in the simple solutions until the central paradox has been full explained and discussed.  I cannot see that point being too hard to understand and it is just that principle that I would like a vote on.


 * Another round of discussion with proposed variations and compromises would complicate the central issue which I have given above.


 * Also, have we dropped that idea that Rick and I (as representatives of the two camps) would have a limited number of words to explain what we want and why? I cannot see how new editors could be expected to come here and make a decision based on a short statement.  Martin Hogbin (talk) 09:41, 28 April 2012 (UTC)


 * It was my understanding that Rick and Martin were to be representatives. As neither of them are in favour of point 3, I've struck that. I invite them to begin this process here. Sunray (talk) 17:50, 28 April 2012 (UTC)


 * I am not sure of the basis for saying that these parties represent anyone but themselves. If, as Guy suggested at "Cutting DIRECTLY to the chase, minus the usual debates" above, the bottom-line question is to be whether one or the other should be recused from this article, then I am ok with framing it as a private dispute between them, and will wait for the question to be brought before the community for decision. ~ Ningauble (talk) 20:35, 28 April 2012 (UTC)


 * Martin and Rick represent themselves only. Other editors may partially agree or disagree with Rick or Martin (and thus are "represented" in a very limited sense) and of course are free to make whatever decisions (within policy) they choose after seeing the results of the RfC, but nobody is asking them to do so.


 * Also, while this is indeed designed to settle a long-running content dispute between Martin and Rick, it may in addition identify a clear consensus on some points, and of course we are all required to follow consensus once we know what it is. That being said, it is perfectly reasonable to conclude that the RfC was about a different question. To me, this implies that any consensus that we identify will be useful for editors who in good faith want to follow the consensus and particularly ill-suited for telling another editor that "consensus is against you" --Guy Macon (talk) 21:33, 28 April 2012 (UTC)


 * I do object to this being classified as a dispute between myself and Rick. I have listed all the editors who, prior to the Arbcom case, expressed an opinion about the status of the Morgan solutions .  Please have a look at them and read what they have to say.  You will see that 7 editors expressed opinions in favour of giving prominence to the Morgan solutions and 22 expressed opinions against this.


 * As a result of the Arbcom case, two editors were sanctioned for showing a degree of page ownership. Since then new editors have continued to express strong opinions against giving the Morgan solutions undue prominence.


 * After the Arbcom case we had a consensus to accept a compromise solution that I proposed but for some reason that was ignored.


 * I do wonder what needs to happen for the clearly expressed supermajority opinion to be accepted as a consensus here. Martin Hogbin (talk) 09:08, 29 April 2012 (UTC)


 * Are you backing out of the proposed RfC process and suggesting, instead, considering the results of a straw poll to be taken as consensus? -- Rick Block (talk) 16:58, 29 April 2012 (UTC)
 * I am not backing out of anything but I note that you refrained from taking part in Guy's proposal above. Martin Hogbin (talk) 20:48, 29 April 2012 (UTC)
 * If you'll recall, I had responded with the suggestion to find a 3rd party to help draft the RfC statement (which you agreed to, further saying you would "ask around"). I thought we were each trying to find such a 3rd party, and as you can plainly see I have found one.  Your insinuation that I'm not being cooperative here is quite offensive.  -- Rick Block (talk) 21:24, 29 April 2012 (UTC)
 * So let us leave the personal comments here and get back to discussing the subjectMartin Hogbin (talk) 08:31, 30 April 2012 (UTC)

Here's my current understanding: Martin and Rick do not represent particular factions; they have been delegated to prepare the opening statement(s) for an RfC. I will assist them in developing the opening statement(s) and in determining what will constitute consensus (i.e., a decision rule). The RfC will make clear how consensus will be determined. On that basis, shall we proceed? Sunray (talk) 17:21, 29 April 2012 (UTC)
 * I agree that I have no rights to speak for anybody else but I see no reason why I should draw attention to what other editors have said on the subject. I am happy to work with you to come up with a statement that can be put to RfC and to agree some decision rules.


 * Can I suggest that, when we go to RfC, Rick and I (and any other regulars here who so wish) should be given a limited number of words to state their case and respond to comments. Martin Hogbin (talk) 20:44, 29 April 2012 (UTC)


 * Yes, I think that leanness of expression is usually beneficial. How about limiting posts to 200 words? Sunray (talk) 21:07, 29 April 2012 (UTC)


 * What Sunray suggests sounds fine to me (including a 200 word limit). I have already suggested a format (with partial content) for an RfC statement on the page he linked above.  -- Rick Block (talk) 21:24, 29 April 2012 (UTC)


 * Sunray, do you mean 200 words per post or a total of 200 words for the whole RfC process? Martin Hogbin (talk) 08:31, 30 April 2012 (UTC)


 * I would urge caution based upon past experience; discussion on this topic tends to grow without bounds. If we end up going with my (not as good, second choice) proposal with separate statements I think the word limits I specified are good ones. If we end up going with Sunray's (better, first choice) single statement that both parties agree upon, there should be a limit of the size of the initial statement (200 sounds right, but I am open to arguments that another number is better) and a separate limit on total words-per-participant used in the subsequent discussion (I like 1000, but again this is open for discussion). If Sunray gets both parties to agree on a statement, I support whatever ground rules Sunray prefers, but I advise some sort of limit - these discussion really do tend to grow and grow forever. --Guy Macon (talk) 17:12, 30 April 2012 (UTC)


 * I meant 200 words per post, but I like Guy's proposal that there be a limit on total words-per-participant. Somewhere between 500 and 1,000. What do you think Martin? Sunray (talk) 23:55, 30 April 2012 (UTC)
 * I think a total limit is a good idea, but I am not quite sure exactly what is being proposed here. I liked Guy's proposal above where Rick and I (and anyone else who so wishes) made clear proposals for the future of the article and these proposals were voted on in an RfC. In that case, all the proposers should have a word limit to stop them from filibustering other editors into submission.  Is that what you have in mind? Martin Hogbin (talk) 09:24, 1 May 2012 (UTC)
 * Re. the parenthetical "anyone else who so wishes": You have already said that input from other editors in framing the RFC would be unnecessary and unhelpful, and I thought the whole point of Guy's "Cutting DIRECTLY to the chase" framework was to identify and settle whatever it is that the two of you have been filibustering about all these years. Having found it futile to offer suggestions in last year's discussions, I appreciate the wisdom of Guy's approach as a last resort to break the deadlock. ~ Ningauble (talk) 13:14, 1 May 2012 (UTC)
 * I cannot quit understand the point you are making. I fully supported Guy's suggestion and still do. He at least made a clear and positive suggestion for a way forward and I would be perfectly happy to go with it.
 * My only concern here is to get a clear decision on the issue which concerns myself and 22 other editors, which is the necessary complication of the article by the Morgan solutions. Once that issue is finally resolved we will be able to get back to cooperative improvement of the article. If you have a better suggestion for a way forward let us hear it. Martin Hogbin (talk) 15:10, 1 May 2012 (UTC)
 * Been there (attempted simplified statement of dispute, ignored), done that (suggested compromise for contextualizing "simple" treatments in a non-judgmental way, talked to death), tried as far back as 2008 before I even registered an account. My point is that I am expressly barred from doing so at this time, on your say-so (Rick was ambivalent), so inviting suggestions from me or "anyone else who so wishes" is not in order for the stipulated process. I do not wish to be chided for failing to offer constructive suggestions when I am actually barred from trying to help.
 * (I am trying to avoid commenting on what the process ought to be, or on the substance of the RfC; but I recognize that I am skirting close to violating the decision to limit these matters to the parties and the facilitator. If the facilitator feels I am crossing a line in pointing out what the decision means, I would not object to having my remarks stricken.) ~ Ningauble (talk) 17:33, 1 May 2012 (UTC)
 * In what way are you 'barred from trying to help'? Martin Hogbin (talk) 17:40, 1 May 2012 (UTC)
 * Ninguable, I am sorry if my comments sounded like I did not want contributions form anyone but myself and Rick. Let me explai what I am getting at.  If you are going for a vote you need to present the two (or more) sides of the debate clearly, with each side having the freedom to make clear exactly what it is advocating.  There is absolutely no point in presenting a compromise position to an RfC; if we can reach a compromise acceptable to all then the job is done.  Unfortunately, we have never been able to do this.  All I object to is someone diluting my proposal (now supported by 25 others). Martin Hogbin (talk) 18:38, 1 May 2012 (UTC)
 * I don't see where you posted the specific proposal to which you refer, nor was I aware that voting had begun. I was looking for it here. Would you please provide a link to the appropriate location? (I have replied at greater length regarding process on Martin's user talk page.) ~ Ningauble (talk) 20:29, 1 May 2012 (UTC)
 * As Guy says below, nothing has started yet. My proposal, for the purposes of this discussion, has not been written yet but you know what it is going to say, it will be the same as it ever was, which that there should be no health warnings until the main points have been covered fully. Martin Hogbin (talk) 12:23, 2 May 2012 (UTC)

The only specific proposal so far has been mine. It is of course open to discussion/change. Sunray is working on a single statement of what the conflict is that Rick and Martin agree with to replace my two-statement scheme. If he succeeds I will put forth an edited proposal. Either way, the purpose is to settle by agreement a conflict between Martin and Rick, so I am not asking Ningauble to agree with or be bound by any part of my proposal.

I think the statement "All I object to is someone diluting my proposal (now supported by 25 others)" is partially accurate (under my proposal Martin alone chooses his statement and rebuttal, with nobody diluting it) and partly inaccurate (his statement has not been written yet, and any previous statement - and the number of editors who support it - is irrelevant.)

Now of course Ningauble is welcome to make any suggestions he wishes, and Rick / Martin are well-advised to consider them, but under my proposal the decision of what is in their statement is theirs alone. I assume that Sunray will do something similar; allow anyone to make suggestions but only Martin and Rick need to agree. The key here is that we are also asking Rick and Martin to agree to certain actions based upon the results, while we are not expecting Ningauble to be bound by it in any way.

I previously mentioned that we may identify an overwhelming consensus, but what are the chances that Ningauble or anyone else will say "that's the exact question I would have asked, worded the way I would have worded it!!!"? Unless that unlikely event happens, I see zero reason why Ningauble should consider a consensus on something that he considers to be unrelated and poorly worded to be binding on him.

I would also emphasize that I have not gotten both Martin and Rick to agree with my proposal, and that I don't expect them to until it becomes clear that Sunray has succeeded or failed. So right now it is just a proposal. --Guy Macon (talk) 21:18, 1 May 2012 (UTC)

Some remarks at MHP, the publications, and the discussions
The reason why the MHP got so famous is that the suggested 2/3-solution for the formulated problem was wrong.

The publicists of the problem and its supposed solution did not realize that it is a critical assumption for the 2/3-solution that the host is forced by the rules of the game to open an unchosen door with a goat after the contestant has "chosen" a door. (Another necessary rule which may be an obvious implicit assumption is that the host opens each of the two possible doors with probability 1/2 if he has the choice. But variants without this assumption may also be considered ...)

Without the assumption that the host is "forced" by the rules of the game the MHP together with the suggestion of a 2/3-solution is a joke.

And it is this (wrong) combination of problem and solution which led to widespread protests of people who insisted rightly on a 1/2-solution.

The height of fall between the formulated problem and the suggested "counterintuitive" 2/3-solution only occured because this solution was wrong. (And this indeed is a real reason for the "protests", not that the MHP is a miracle.)

And up to today we can read new publications containing this error. A popular argument indicating this wrong understanding is the statement that the probability 1/3 for the chosen door does not change if the host opens another door with a goat. The "advantage" of this argument is that it is applicable to the correctly formulated problem which has a 2/3-solution as well as to the widespread problem which has not.

One of the uncountable publications which demonstrate this error are the articles of Keith Devlin.

It is worth to look exactly at his phrasing of the problem, especially at the addition in 2005:

Keith Devlin 2003:

''Now comes the twist. Instead of simply opening the chosen door to reveal what lay behind, Monty would open one of the two doors the contestant had not chosen, revealing that it did not hide the prize. (Since Monty knew where the prize was, he could always do this.) He then offered the contestant the opportunity of either sticking with their original choice of door, or else switching it for the other unopened door.''

Addition in paranthesis in Keith Devlin 2005:

(As the game was actually played, some weeks Monty would simply let the contestant open their chosen door. The hypothetical version of the game described here, where Monty always opens the door and makes the "switch or stick" offer, is the one typically analyzed in statistics classes.)

What's this? The MHP with the 2/3-solution suddenly is only a hypothetical version analyzed in statistics classes? Not the one which had been published in Parade in 1990, and in the german weekly newspaper DIE ZEIT in 1991, and in many other publications including Wikipedia(s)?

But does Keith Devlin himself understand the difference?

In both articles his argumentation for the 2/3-solution is:

[1] The probability that the prize is behind door B or C (i.e., not behind door A) is 2/3. [2] The prize is not behind door C.

Combining these two pieces of information, you conclude that the probability that the prize is behind door B is 2/3.

[1] holds before the host opens another door with a goat. But [1] is false after he has opened the door if he sometimes simply let the contestant open their chosen door. In Ein Auto und zwei Ziegen we can read at this point:

''In my opinion many people within the "2/3 fraction" up to today don't see the difference between the blank fact that the host opens a not chosen door with a goat after the first choice, and the enforcement by the rule of the game which leads to this action. And this enforcement is crucial for the 2/3-solution, especially too for reenactments and "computer proofs".''

[1] holds in the version with the formulated enforcement not because the probability for the chosen door "stays" 1/3 (or, equivalently, 2/3 for B or C), but because (1/3 * 1/2)/(1/3 * 1/2 + 1/3 * 1) = (1/6)/(1/6 + 1/3) = 1/3. This is an important step in the proof that "1/3 stays". Those who replace this step by the blank claim "1/3 stays" cannot explain why this proposition does not hold in the case where there is no enforcement, as is the case in the articles of Keith Devlin and many others. (And they cannot explain why the MHP should be any problem at all.)

Those who really understand the problem are not "shocked" about the 1/2-solution in the variant, where one of the remaining doors is opened by accident revealing a goat. For they know that the essential condition leading to a 2/3-solution is the enforcement by the rules.

It is true that at the beginning p = 1/3 for the chosen door. And it is true that the contestant (provided correct rules) will win the price by switching exactly if it is behind one of the two not chosen doors. But there have to be additional arguments such as symmetry considerations to establish the probability of 2/3 for the second choice; as an alternative to the computation above.

For the article to be serious it should strictly seperate the original problem stated by Whitaker/vos Savant in Parade which had no 2/3-solution from later formulations which often are the result of strange adjustments to hide the fact that their original solution had been wrong.

Every problem formulation which is not equivalent whith the following phrase in the critical passage does not have a 2/3-solution:

The contestant now determines two doors, from which the host has to open one door with a goat.

A complete correct formulation of the problem you can find in Ein Auto und zwei Ziegen:

''In a television show you as a contestant are faced with three closed doors. Behind one of the doors which had been determined by accident there is the price, a car. Behind the other two doors there are goats as blanks. The host knows the door with the car. Now you have to determine two doors, and the host has to open one of them with a goat. If the host has a choice, he determines the door by accident. Then you may chose one of the two remaining doors. Name the probability to win the car for both doors.''

''Example: You ask the host to open door 2 or 3 with a goat, and he opens door 3. Name the chances for door 1 and door 2.''

(Note: Of course "by accident" means a probability of 1/3 in the first, 1/2 in the second case.)

Here is an analogous problem:

''Soon the final game in the European football Champions League between FC Bayern München and FC Chelsea will take place. Suppose the probability to win is 1/2 for both teams. Your neighbor is a supporter of the team of Chelsea who will act as follows:''

After the game he will hoist the colors of Chelsea upon his house if his team has won the game.

''If Chelsea has lost the game, he will flip a coin and hoist the colors of Chelsea if heads occur. If tails occur, he doesn't hoist anything.''

In the next morning you (not knowing the result of the game) see the banner of Chelsea on the supporter's house.

What is the probability that Chelsea is the new champion?

As with the MHP we can modify the problem as follows:

The supporter does not flip a coin if Chelsea lost the game, but draws one ball from a box which contains 1000 balls numbered from 1 to 1000. If the number of his ball is 777 he will hoist the colors.

Where are the heavy protests? Where is the paradox? Where are the endless discussions? Where is the spook? Where is the miracle?

--Scharzwald (talk) 13:05, 9 May 2012 (UTC)
 * A lot of remarks, but what is your point? Nijdam (talk) 13:28, 9 May 2012 (UTC)
 * You are quite right about the 2/3 answer being dependent on the host being forced to choose a goat but this is the assumption that is generally made. Whitaker's statement says that the host knows what is behind the doors.  It is usually assumed that this means that he must reveal a goat. Martin Hogbin (talk) 15:20, 9 May 2012 (UTC)


 * The assumption about the host being forced to choose a goat has not at all been generally made. And the host not only knows the position of the car; he also knows whether the contestant has chosen the car or not. I think we agree that a consequence of this rule is that the host does not have any freedom in the game. But unfortunately most publicists since 1990 try to embed the problem in a "story" which shows their wrong understanding. For example Gero von Randow (and many "followers" including Krauss, S. & Atmaca, S. (2004). Wie man Schülern Einsicht in schwierige stochastische Probleme vermitteln kann. Eine Fallstudie über das "Drei-Türen-Problem". Unterrichtswissenschaft, 1, 38-57) who wrote the most popular articles (and a book) in Germany about "Das Ziegenproblem" lets the host say "Now I'll show you something", before he opens the door with a goat. This remark is not compatible with the enforcement by the rule. (And in his book there is a chapter "My error" in which he writes that first he had thought that the game with the 2/3 solution would work without any enforcement. But he didn't change the problem formulation in his following publications ...) You know another example: Keith Devlin. His articles mentioned above would look completely differently if he had clearly assumed this rule. And guessing a "shock" for those who hear that missing this rule will lead to a 1/2-solution shows his bad understanding of the problem. For from the beginning it is exactly this rule which leads to the 2/3-solution. You will find a great many of other examples. A guideline may be: If someone gives any freedom to the behaviour of the host, he is wrong. Another guideline: Look whether an author makes clear from the beginning (and not in additional assumptions, footnotes, appendixes and so on) what problem he is dealing with. And: Check whether the assumptions of the problem appear in the "proofs", and, of course, if the assumptions made during the argumentation appear in the problem set.


 * --Scharzwald (talk) 17:24, 9 May 2012 (UTC)


 * One of the "proofs" of the fact that the assumption of "the forced host" is not generally made, is the film 21. At the critical moment the teacher says, that his offer to switch may be a trick of the host to disabuse the contestant of the winning door. What lets the 2/3-solution collapse like a house of cards ...


 * But instead of studying curious sources including so-called "reputable" ones we can make it easier:


 * Set a friend the task in the version of Marilyn vos Savant. Explain the 2/3-solution saying: "Now the host is forced to open another door with a goat." And if your friend now asks "Why is he forced?" you are on your wit's end.


 * --Scharzwald (talk) 08:54, 11 May 2012 (UTC)

Another remark/point concerning the article:

1. What is the probability of Bryan winning the car if he will abandon his initial pick? The answer is:

If the host opens the right of the not chosen doors, Bryan wins (with the left door). If he opens the left door, Bryan has a chance of 1/2 (with the right door).

There is no essential difference between 1. and 2.

And The answer to the first question is not 2/3.

Also the following statement at the beginning of the paragraph is wrong:

The popular solutions correctly show that the probability of winning for a player who always switches is 2/3.

The probability of winning for a player who always switches is exactly the same as for any other "switcher". (And it will not happen often that a contestant has the opportunity to switch more than once ...)

True is: Providing the correct rules, at the beginning of the game the probability of winning by switching is 2/3 - not only for those who always switch. And the probabilities just before the second choice have to be proven, maybe by symmetry considerations.

Especially the paragraph Criticism of the simple solutions seems to be part of the widespread confusion about the MHP.

--Scharzwald (talk) 16:13, 9 May 2012 (UTC)

(editing)
Who is Bryan? Nijdam (talk) 08:30, 11 May 2012 (UTC)


 * Bryan is Bryan of the paragraph Criticism of the simple solutions. --Scharzwald (talk) 12:03, 11 May 2012 (UTC)
 * Okay, then to answer "What is the probability of Bryan winning the car if he will abandon his initial pick?", I want to know: Is Bryan's initial pick known? Nijdam (talk) 19:17, 11 May 2012 (UTC)


 * Bryan's initial pick is not known; but there is a clear and correct answer which I gave above:
 * Does Bryan suffer from Alzheimer? Nijdam (talk) 11:08, 13 May 2012 (UTC)


 * Other users may continue this highly sophisticated discussion.--Scharzwald (talk) 12:16, 14 May 2012 (UTC)
 * You're apparently not that sophisticated, otherwise explain to me how Bryan made his choice and yet doesn't know what his choice was. Nijdam (talk) 22:21, 15 May 2012 (UTC)


 * If the host opens the right of the not chosen doors, Bryan wins (with the left door). If he opens the left door, Bryan has a chance of 1/2 (with the right door).


 * The problem set contains all informations to give exact probabilities for the situation just before the second choice.


 * And with this general solution you can exactly solve every special case, for example:


 * 2. What is the probability of the car being behind the door 2 if Bryan first picked door 1 and Monty revealed a goat behind door 3?


 * We apply the general solution of 1. to solve 2.:


 * Door 3 ist the rightmost of the two possible (unchosen) doors; therefore p = 1 for (the leftmost) door 2.


 * In my view this is normal elementary mathematics.


 * Maybe you meant the following question:


 * What is the probability of Bryan at the beginning of the game winning the car if he will abandon his initial pick?


 * The answer is 2/3.
 * --Scharzwald (talk) 12:23, 12 May 2012 (UTC)

Remarks concerning the whole article:

In its actual version the article is strongly characterized by the basic orientation that the solution of Marilyn vos Savant's problem published in Parade in 1990 is 2/3 for "switching"; and that the lot of people who protested against this solution insisting on "1/2 : 1/2" were wrong because of a typical misleading intuition.

But the solution "1/2" is the best answer to this problem. And if a teacher would set his pupils this task and regarding their 1/2-solution as an error because the 2/3-solution is correct providing that one makes seven highly plausible assumptions (Donald Granberg), the pupils probably would think that the teacher has gone mad.

The most interesting aspect in Granberg's list is the fact that the most important point (that the host is committed to showing an incorrect, unchosen alternative deliberately after the initial guess) is hidden among other really obvious points (The host is truthful), intending the reader to think: "These points are really pedantic".

And in my opinion there is also a great difference between the rule that the host is committed to open an unchosen door with a goat, and the rule that the host makes his choice randomly with equal probability if he has a choice. For the second rule is a very obvious assumption. As in other problems of this kind, where there is no reason for a player to prefer one of two possibilities, p = 1/2 is the only reasonable assumption. In this argumentation I agree with Marilyn vos Savant, though there were heavy protests since Selvin's solution against this opinion. Curiously many publications place special emphasis on this minor aspect but overlook the critical rule to the present. Surely you may set the minor rule in the problem, too; but if not there will be no serious man who will protest if you make clear in your argumentation that it is your assumption that the host opens each of the possible doors with probability 1/2 if he has a choice.

In America as in Germany there had been a few letters at the beginning of the debate which pointed out that the problem in its published form had no 2/3-solution. But this letters were not published (for example see Ein Auto und zwei Ziegen). If they had been published - or if the following publishers had taken account of Martin Gardner's and Monty Hall's falsification in Tierney's article in the New York Times in 1991 - we would not have the "Monty Hall Problem" at all.

So we have to turn a huge tanker in a new direction. I suggest to do this by creating a Wikipedia article which is like a small speedboat which has a clear direction:

Start with Marilyn vos Savant's/Whitaker's problem set and her suggestion of a 2/3-solution.

Mention the plenty of protests against this solution.

Highlight the fact that the only reason for a 2/3-solution is that the host is commited by the rules of the game to open an unchosen door with a goat; and that the task set of Marilyn and of plenty of her "followers" lack this rule.

Don't overload the article with considerations and speculations about the case if the host has a choice. And don't suggest that there is an "unconditional" and a "conditional" interpretation of the problem. For everyone knows that the question is whether you should switch in the situation after the host has opened the door with a goat.

The essence of the correct argumentation is the fact that the probability that the host opens "his" door is twice as big if the car is behind the not chosen remaining door as if it is behind the chosen door. Of course, this assumes that the host makes his choice randomly with equal probability if he has a choice. You may set this assumption in the problem, or make clear that it is an assumption in your argumentation. But surely, you also may present a problem set and solution where these probabilities are arbitrary values p and q with p + q = 1 instead of p = 1/2 and q = 1/2. But this has nothing to do with the widespread debate "1/2 or 2/3" and should be clearly seperated from the main argumentation.

(See also my remarks above.)

--Scharzwald (talk) 12:03, 11 May 2012 (UTC)


 * I have been suggesting this structure for a long time but some editors consider explaining about conditional probabilty is more important. The two interesting points about the MHP are:
 * That the answer is 2/3 not 1/2
 * That it matters that the host must reveal a goat.  Martin Hogbin (talk) 17:46, 11 May 2012 (UTC)


 * I think it is better to formulate it in one point: Only if the host must reveal a (not chosen) goat, the answer is 2/3.


 * For the versions of Marylin vos Savant, Gero von Randow and their "followers" a simple correct answer is:
 * There are two possibilities: Either the host tries to disabuse the contestant of the winning door, or he offers a second chance. Probabilities: fifty-fifty.''
 * This also is essentially what Martin Gardner and Monty Hall himself said within the second part of the New York Times article in 1991, when John Tierney wished to finish the debate by asking the four experts which he thought are best capable of doing this. The other two persons were Persi Diaconis, who said that the formulated problem cannot be solved by mathematical reasoning alone; and Marilyn vos Savant, who acknowledged that the ambiguity did exist in her original statement.
 * But it seems that most people did not read this second part of the article.
 * --Scharzwald (talk) 18:14, 11 May 2012 (UTC)
 * The important fact is that with Bertrand's box problem and the Three prisoners problem, most people will give the answer 1/2, and are very reluctant to accept the correct answer. Now the MHP in its standard form is equivalent to these problems, and most of the commotion comes just from the idea of most people of equal odds for the two remaining doors. Most of the discussion you are mentioning has nothing to do with this "paradox". Nijdam (talk) 19:28, 11 May 2012 (UTC)


 * This is a very strange argumentation. I'll try to understand: Most people will give the answer 1/2 with Bertrand's box problem and Three prisoners problem (10000 letters against the solution 2/3?). "MHP in its standard form" is equivalent to these problems. And it doesn't matter whether the odds for the two remaining doors are in fact equal for the formulated ("non-standard") problem. For we only have to assign the protests against the solution of the non-standard problem, where they are justified, to the standard problem where they are not. And we have an additional surprising result: Because the standard MHP is equivalent to Betrand's box problem and Three prisoners problem there has been a huge protest storm against the 2/3-solution of these two problems. Very strange ...
 * And in my opinion all what I have written above has to do with what you call a "paradox" including a clear unambigous formulation and a simple mathematical solution; and similar problems for the better understanding of what you call a "paradox". Please don't reply my arguments by ignoring them.


 * --Scharzwald (talk) 20:27, 11 May 2012 (UTC)

Scharzwald, we agree at least on the two most important things about this puzzle but, regarding the interpretation that the host must reveal a goat, there are good reasons to believe that this is how the the question should be interpreted:


 * 1) The Whitaker/vos Savant questions says (my bold), '...the host, who knows what's behind the doors '. What possible reason could there be for saying that the host knows what is behind the doors other than to imply that he always reveals a goat?
 * 2) Years earlier Selvin had made clear that the host knew where the car keys were with the obvious intention of meaning that the host must not reveal the prize.
 * 3) It is hard to see how a game show would proceed if the host revealed the car. What would happen then, would they just say 'tough luck, you lose, because I spoiled things for you'?
 * 4) Kruass and Wang say in their interpretation of how the problem is generally understood '...the door [the host] opens must have a goat behind it'.

I think, therefore, that it is fair to say that, in the problem intended by both Selvin and Whitaker/vos Savant, the host always reveals a (unchosen) goat and that most people interpret the problem this way. On the other hand, I agree that the fact that the host cannot reveal the prize is a critical part of the problem and one that is surprising and of interest to most readers. Martin Hogbin (talk) 16:38, 12 May 2012 (UTC)


 * ad 1: the host, who knows what's behind the doors means for the contestant (and "us") that the host knows exactly what he will do now: For the other important fact is that he now knows which door the contestant has chosen. Now he is free to show him immediately whether he has won or lost the prize - by opening the door the contestant has chosen or by opening the door with the car - or to continue the game. In the case of Whitaker's setting he continues the game by opening a not chosen door with a goat and offering a switch; which means that he disabuses the contestant of the winning door, or that he gives to him a new chance. The contestant cannot know the motivation of the host. Therefore it is a simple and correct answer that the chances for staying and switching are equal.


 * The publishers of the problem now claimed that the behaviour of the host results in a 2/3-solution; and in the publications which resulted in a "huge protest storm" there was never highlighted that the host is commited by the rules of the game to open an unchosen door with a goat - and they couldn't truely do this, because it was exactly this condition, which leads to a 2/3 solution instead of a 1/2-solution, but which was missing in the problem set.


 * That the critical rule - which means that the first act of the contestant is not a real choice but that he determines two doors of which the host has to open one with a goat - is no "obvious assumption", is proven by a plenty of publications. Above I mentioned some and a simple way to find many others ... A pretty one found in the web I add here: A woman confronted with the problem said: "I don't think that the host will cheat me; therefore I should switch."


 * More specifically: Many of the "2/3 fraction" have thought (and think down to the present day) that for the 2/3-solution to prove the blank fact suffices that the host opens one of the not chosen doors with a goat (knowing the door with the car).


 * And my opinion is that the key mistake (up to today) was the following: Every "reenaction", computer proofs (and assuming probabilities in more mathematical approaches) seem to "simulate" the MHP just like it is formulated originally in Parade and "followers"; that is: without the critical rule. The result is a 2/3-solution. But this is only because the "simulation" automatically includes an assumption which is not set - and the "simulators" overlooked this error.


 * (The presentation of the MHP without the critical rule combined with the 2/3-solution is a joke - and to present it in 2004 or 2012 in the original form suggesting the 2/3-solution, adding assumptions in further remarks, footnotes, appendixes, other publications and so on is more than a joke ... this holds if the publications are "reputable" or not.)


 * ad 2., 3., 4.: See ad 1. and my other comments.


 * --Scharzwald (talk) 10:43, 13 May 2012 (UTC)
 * You are entitled to your opinion but most literature and the vast majority of editors here, on both sides of the ongoing argument, accept that the original poser of the question that vos Savant answered intended the rule that the host must reveal a goat to be part of the problem, even though this is not clearly stated. This, generally accepted, interpretation of the problem is given early in the article in the 'Extended problem description'. Once that point is made clear, the answer is 2/3.  Martin Hogbin (talk) 15:08, 13 May 2012 (UTC)


 * Thank you for your answers and to Wikipedia who admitted me to demonstrate my arguments, even against the vast majority. Surely my suggestions for a better article persist. Maybe sometime the majority changes; but it should be a majority of arguments, not of opinions. And they will see that the starting position for a well-documented Wikipedia article is excellent. For there is plenty of reputable nonsense.


 * Suppose you're on a game show ...... Is it to your advantage to switch your choice of doors?


 * --Scharzwald (talk) 18:13, 13 May 2012 (UTC)

Marilyn vos Savant
Why does this article seem to revolve around Marilyn vos Savant? She did not originate the MHP but merely wrote a 1990 article about it in Parade magazine which contains, IMO, a flawed truth table to demonstrate what she thinks is the solution. I have always been dubious of Ms. vos Savant's self-promotional claims. — Preceding unsigned comment added by Chris319 (talk • contribs) 21:37, 4 May 2012 (UTC)
 * There will be an RFC of some kind in the not too distant future about this very topic - requesting community input on a suggestion this article be changed to revolve considerably MORE around Ms vos Savant's solution (and other similar solutions). I would suggest anyone with opinions about this stay tuned. -- Rick Block (talk) 04:24, 5 May 2012 (UTC)
 * Welcome Chris319, what do you see as the flaw in vos Savant's solution? Martin Hogbin (talk)
 * No use in answering your question if a perfectly valid demonstration of the solution is unilaterally deleted from the article. Care to explain that move, Ningauble? Gotta love Wikipedia. — Preceding unsigned comment added by Chris319 (talk • contribs) 19:12, 7 May 2012 (UTC)
 * The short answer to Chris319's question about vos Savant is that, right, wrong, or indifferent, her article popularized the problem and is the most widely cited source. Whether they love it or hate it, virtually everyone who writes about the problem refers to it. ~ Ningauble (talk) 15:22, 5 May 2012 (UTC)


 * Chris319 - are you talking about this edit, which asks for a citation for the solution you added? As far as I can Ningauble didn't delete anything, although if you can't find a reliable source for what you've added it certainly might be deleted. -- Rick Block (talk) 19:32, 7 May 2012 (UTC)
 * On reflection, I am not able to give a satisfactory explanation for that action: I don't know why I didn't just go ahead and delete it on the spot. It appears to be unsubstantiated original research employing a novel interpretation of what a truth table is. ~ Ningauble (talk) 13:46, 8 May 2012 (UTC)
 * Having received no rebuttal for over two weeks, I am removing this unsourced addition now. ~ Ningauble (talk) 16:47, 24 May 2012 (UTC)

What is shown in the solution table?
Regardless how I try, I can't relate the information present in the table to the solution given afterwards and the problem description. The description says that the host opens door #3, but the table displays #3 in the last row as having the car (i.e. the host cannot open it). My question thus: what is being shown in the table? It either needs to state what were the player choices (given the goat-car combinations) or deal with only two cases car-goat-goat and goat-car-goat because third one, goat-goat-car is impossible. — Preceding unsigned comment added by 79.181.31.243 (talk) 10:14, 21 May 2012 (UTC)
 * The table shows all possibilities for the car location, and whether a player who has picked Door 1 wins or loses by switching. Some sources are critical of this approach since it at least seems to change the problem.  It's perhaps more clear if you explicitly think of this as answering whether a strategy of switching is better than a strategy of staying with the original choice.  Another way to think of this solution is that it moves the decision point to before the host opens a door (you pick Door 1, and decide whether you're going to switch or stay knowing the host is going to open one of the other doors and show you a goat).  Given either of these, does it then seem obvious that switching wins 2/3 of the time and staying with your original choice wins only 1/3 of the time?


 * From there, the question becomes what changes when the host opens a door. If the player picks Door 1 there are only two cases - the host opens Door 2 or the host opens Door 3.  If the overall odds of winning by switching are 2/3 and there's nothing special about either of these cases, then in each case it must be that the odds of winning by switching are 2/3.  It's maybe a little indirect, but this is all definitely correct.


 * Vos Savant didn't show it this way, but perhaps this table showing the results of 300 "shows" might help:
 * {| class="wikitable" style="margin:auto; text-align: center;"

! colspan=4 | Situation BEFORE the host opens a door !colspan=4 | Situation AFTER the host opens a door ! Door 1 || Door 2 || Door 3 || total cases !colspan=2 | host opens Door 2 !colspan=2 | host opens Door 3 ! || || || || cases || result if switching || cases || result if switching
 * Car || Goat || Goat || 100 || 50 || Goat || 50 || Goat
 * Goat || Car || Goat || 100 || 0 || N/A || 100 || Car
 * Goat || Goat || Car || 100 || 100 || Car || 0 || N/A
 * }
 * Goat || Goat || Car || 100 || 100 || Car || 0 || N/A
 * }
 * }


 * A direct solution is shown in the Decision tree section of the article. -- Rick Block (talk) 13:44, 21 May 2012 (UTC)


 * I'm in total agreement with the solution, it's just that table shows something it claims to be a truth table but it's really not. The last column of the table would need to have the truth values that "solve" the table, for example. It, as it is, may lead one to assume that some order of the doors and cars may be winning or that there are 3 possible ways for the situation to develop (while there are only two, i.e. choose from 3, choose from 2). IMO it would be best if the table followed the precise description of the problem and, indeed drew a probabilistic truth table. $$X \land_{1/3} Y \land_{1/2} Z$$ (hope I'm doing it right, I'm not very good at probabilistic logic...) I.e. there is a way to represent the problem using rigorous mathematical notation, and I think it would be better to have one that does it in a formal way, and yet another one that does it informally (if someone thinks it is helpful - why not). — Preceding unsigned comment added by 79.181.31.243 (talk) 06:23, 22 May 2012 (UTC)


 * My problem is that I can see the logic in both answers: both 1/2 and 2/3 if switching. But why is the logic that leads to the latter answer better than that which leads to the former? After all, if door 3 shows a goat, then the last row of the table above is not applicable. We can only use the first two rows. Therefore, we're left with 1/2. So, the article must explain why, when both logical concepts seem right, the 1/2 one is wrong while the other is right. So far, the article singularly fails to explain the 'paradox' here. Which, surely, is the point of the article... Malick78 (talk) 12:56, 28 May 2012 (UTC)

The left column "host opens Door 2" shows that, by swapping to door 3 in 150 cases, you will get the Goat in 50 cases and will get the Car in 100 cases. And the right column "host opens Door 3" shows that, by swapping to door 2 in 150 cases, you again will get the Goat in 50 cases and again will get the Car in 100 cases. So, in a total of 300 cases, by swapping you will get 100 Goats and 200 Cars. And by staying you will get 200 Goats and 100 Cars. Regards, Gerhardvalentin (talk) 17:34, 28 May 2012 (UTC)
 * Malick78, the table above shows in total 300 cases. Just read the columns "Situation AFTER the host opens a door" on the right half of the above table:
 * You missed my point. I understand the table above and agree that, using it, we get 2/3 for switching. But, my point was actually the one (I assume) that all people who think there's a 1/2 chance think. The state after opening door 3 is basically: there are two doors, one has a car, the other a goat. Choose one. You have a 50% chance of being right. "Choosing" is via switching, sure. Yet, how can this choice between 2 not be 50%? For me, both approaches to the "paradox" are logical and should be right, yet are mutually exclusive. Why does the use of the table above beat the seemingly simple choice between two doors? Nothing has explained that part of the problem to me yet. May I add... I'm not thick ;) I'm just awaiting a better explanation. Malick78 (talk) 18:04, 28 May 2012 (UTC)


 * This is the author of the original question (sorry, I can't remember my Wiki password). I really don't know what is the most accepted form for displaying decision trees, but the article calls that table "a truth table" - which is something I know how it looks like, and the table is certainly not a truth table (truth table consists of the columns for each variable, then (optional) columns for intermediate results displaying what happens after application of each logical operation, and the last (now mandatory) column with the result). Formal logic has only so many as T and F values, but probabilistic logic has $$T_{1/2..1}$$ and $$F_{1/2..1}$$, $$T_{1/3} = F_{2/3}$$ and so on. As this problem is about probabilities, I would imagine that having an actual probabilistic truth table would be beneficial for the "formal" solution. Unfortunately, I'm not sufficiently proficient in the probabilistic logic (that's why I went to read the article in the first place!) so I'm afraid that if I drew one such table, I've a good chance to mess things up. But would really like to see one here. I think that the table would be built upon 2 truth variables: A = you guess the first time, B = you switch. Operations that it would show should yield something that fits into these rules: if you guess the first time and you switch, then the result is 0, if you don't guess the first time and switch, the result is 1/2, if you guess the first time and don't switch, the result is 1, if you don't guess and don't switch, the result is 0. I.e. now you have total chances of winning the car whatever you do 1+1/2 out of 4, of those when you switch: 1 of 2, of those you don't switch 1/2 of 2. Does it make sense? Ah, and then, of course it should be noted that you chance to guess at first is 1/3. — Preceding unsigned comment added by 79.181.31.243 (talk) 10:10, 29 May 2012 (UTC)
 * Is this more or less what you're asking for (this shows all outcomes given the player initially picks Door 1):
 * {| class="wikitable" style="margin:auto; text-align: center;"

! Car behind Door 1 || Car behind Door 2 || Car behind Door 3 || Host opens Door 2 || Host opens Door 3 || Probability
 * yes || no || no || yes || no || 1/6
 * yes || no || no || no || yes || 1/6
 * no || yes || no || no || yes || 1/3
 * no || no || yes || yes || no || 1/3
 * }
 * This is essentially equivalent to the tree that is shown in the Decision tree] section of the article (which is also equivalent to the large figure in this section). If you pick Door 1 and don't switch, the first two lines apply so your probability of winning is 1/3.  If you pick Door 1 and do switch, the last two lines apply so your probability of winning is 2/3.  This also shows the outcomes if you pick Door 1 and the host opens Door 3.  Only the 2nd and 3rd lines apply (the host opens Door 2 in the other cases), so if you don't switch you win with probability 1/6 and if you do switch you win with probability 1/3.  It's customary to express these as conditional probabilities, which you do by dividing them each by their sum (1/2) so given you've picked Door 1 and the host opens Door 3 the probabilities of winning are 1/3 if you don't switch and 2/3 if you do.
 * no || no || yes || yes || no || 1/3
 * }
 * This is essentially equivalent to the tree that is shown in the Decision tree] section of the article (which is also equivalent to the large figure in this section). If you pick Door 1 and don't switch, the first two lines apply so your probability of winning is 1/3.  If you pick Door 1 and do switch, the last two lines apply so your probability of winning is 2/3.  This also shows the outcomes if you pick Door 1 and the host opens Door 3.  Only the 2nd and 3rd lines apply (the host opens Door 2 in the other cases), so if you don't switch you win with probability 1/6 and if you do switch you win with probability 1/3.  It's customary to express these as conditional probabilities, which you do by dividing them each by their sum (1/2) so given you've picked Door 1 and the host opens Door 3 the probabilities of winning are 1/3 if you don't switch and 2/3 if you do.
 * This is essentially equivalent to the tree that is shown in the Decision tree] section of the article (which is also equivalent to the large figure in this section). If you pick Door 1 and don't switch, the first two lines apply so your probability of winning is 1/3.  If you pick Door 1 and do switch, the last two lines apply so your probability of winning is 2/3.  This also shows the outcomes if you pick Door 1 and the host opens Door 3.  Only the 2nd and 3rd lines apply (the host opens Door 2 in the other cases), so if you don't switch you win with probability 1/6 and if you do switch you win with probability 1/3.  It's customary to express these as conditional probabilities, which you do by dividing them each by their sum (1/2) so given you've picked Door 1 and the host opens Door 3 the probabilities of winning are 1/3 if you don't switch and 2/3 if you do.


 * To contrast the situation where the host knows what's behind the doors (the usual problem) and where the host opens an unchosen door randomly, it's perhaps useful to expand this showing the "impossible" cases:
 * {| class="wikitable" style="margin:auto; text-align: center;"

! Car behind Door 1 || Car behind Door 2 || Car behind Door 3 || Host opens Door 2 || Host opens Door 3 || Probability
 * yes || no || no || yes || no || 1/6
 * yes || no || no || no || yes || 1/6
 * no || yes || no || yes || no || 0
 * no || yes || no || no || yes || 1/3
 * no || no || yes || yes || no || 1/3
 * no || no || yes || no || yes || 0
 * }
 * which can be directly compared with a similar table where the host randomly opens one of the unpicked doors:
 * {| class="wikitable" style="margin:auto; text-align: center;"
 * no || no || yes || yes || no || 1/3
 * no || no || yes || no || yes || 0
 * }
 * which can be directly compared with a similar table where the host randomly opens one of the unpicked doors:
 * {| class="wikitable" style="margin:auto; text-align: center;"
 * {| class="wikitable" style="margin:auto; text-align: center;"

! Car behind Door 1 || Car behind Door 2 || Car behind Door 3 || Host opens Door 2 || Host opens Door 3 || Probability
 * yes || no || no || yes || no || 1/6
 * yes || no || no || no || yes || 1/6
 * no || yes || no || yes || no || 1/6
 * no || yes || no || no || yes || 1/6
 * no || no || yes || yes || no || 1/6
 * no || no || yes || no || yes || 1/6
 * }
 * From this table you can see the outcome if you pick Door 1 and the host opens Door 3 (without revealing a goat) from lines 2 and 4 - the probabilities are 1/6 and 1/6, which expressed as conditional probabilities are 1/2 and 1/2. -- Rick Block (talk) 15:51, 29 May 2012 (UTC)
 * no || no || yes || yes || no || 1/6
 * no || no || yes || no || yes || 1/6
 * }
 * From this table you can see the outcome if you pick Door 1 and the host opens Door 3 (without revealing a goat) from lines 2 and 4 - the probabilities are 1/6 and 1/6, which expressed as conditional probabilities are 1/2 and 1/2. -- Rick Block (talk) 15:51, 29 May 2012 (UTC)
 * }
 * From this table you can see the outcome if you pick Door 1 and the host opens Door 3 (without revealing a goat) from lines 2 and 4 - the probabilities are 1/6 and 1/6, which expressed as conditional probabilities are 1/2 and 1/2. -- Rick Block (talk) 15:51, 29 May 2012 (UTC)


 * Yup, the first table is very good IMO, one thing, I'm not sure how to do, which is a technical thing about formatting. What you say in a sentence before it about which rows are the rows when the player switches, and which rows are the rows when she doesn't, if those could be added on either side of the table (I think at the start) - that would be perfect.  — Preceding unsigned comment added by 79.181.31.243 (talk) 10:50, 31 May 2012 (UTC)

Another table:
 * Or could that table help to show all possible outcomes?
 * Gerhardvalentin (talk) 16:14, 29 May 2012 (UTC)



Why is this caption incorrect? Please, please don't say: "Because" :) Malick78 (talk) 18:08, 28 May 2012 (UTC)
 * One way of understanding what is wrong with the caption (quoting myself from four years ago about the "charm of the problem") is just this:
 * "We are to understand that the car is initially placed behind one of three indistinguishable doors with equal probability; and we are to understand that, in the event he needs to choose, Monty does not distinguish between goats but reveals one of the two with equal probability; but it is naive to conclude in the end that the car will be found behind one of the two closed doors with equal probability, for these doors are now distinguishable."
 * The caption treats doors #1 and #2 as equivalent, but they are not: Monty selectively reveals a goat from doors #2 and #3, and door #1 is special because he is not allowed to open it. ~ Ningauble (talk) 19:56, 28 May 2012 (UTC)
 * Thanks for the attempt :) But while part of my brain understands the factual accuracy of your reasoning, the other part doesn't see why opening door 3 doesn't reset the probability. After all, if the problem started from scratch with the pic to the right, wouldn't it be 50%? We'll forever have a problem (for newbies) with this page till someone explains why 'resetting' doesn't occur. Malick78 (talk) 20:07, 28 May 2012 (UTC)
 * That is the beauty of this problem. Even when logic and calculation tell you the correct answer there is something in your brain that tells you it is wrong.


 * You can only take probability to be uniformly distributed between the possibilities when you know nothing about the situation. In the MHP that is not the case because, as Ninguable says, you know the host could open door 2 or 3 but not door 1. Now it might not be obvious why that should result in an answer of 2/3 but it should be sufficient to explain why you cannot assume that the probability of hiding the car is not uniformly distributed between doors 1 and 2.


 * The MHP is an interesting exercise in symmetry and false symmetry. There is a true symmetry between the doors that the host might open, or the goats that he might reveal but the symmetry that you perceive between the two doors left closed is a false one and assuming it to be a true symmetry leads to a false conclusion.


 * To take a completely different track Malick78, do you find the 'Combining doors' solution convincing? Suppose that the host says, before he opens a door, you can have the prize if it is behind your originally chosen door or you can have both the other doors and you win if the car is behind either of them.  Would you stick to your one door then? Martin Hogbin (talk) 20:43, 28 May 2012 (UTC)


 * I like (and feel slightly humiliated by) the simplicity of your last suggestion ;)
 * Having reread the "Confusion" section, I find this sentence quite important: "However, if a player believes that sticking and switching are equally successful and therefore randomizes their strategy, they should, in fact, win 50% of the time, reinforcing their original belief." Many people, myself included, perhaps intuitively consider the previous events (choosing 1 door and opening another) to have been a distraction, and then reset the probability naturally in the head. The status quo bias perhaps then convinces them that since nothing is to be gained from switching, they won't bother switching (which might risk looking weak/indecisive. We all have our pride. (Shouldn't their be more on the psychology of it? I have a feeling it's even deeper than the Confusion section makes out)). Anyway, I'm almost at peace with this, many thanks. Though I expect to have a relapse in a day or two... ;) Malick78 (talk) 20:55, 28 May 2012 (UTC)
 * We're all saying the same thing, but here's another explanation. If you randomly pick between the two remaining doors you have a 50/50 chance, so in this sense your caption is correct.  However, doing this ignores the process by which the situation arose.  The host cannot open door 1, and must open one of the other doors.  If the car is behind Door 2 (which it is 1/3 of the time), the host must therefore open Door 3.  If the car is behind Door 1 (which it is 1/3 of the time), the host opens either Door 2 or Door 3.  Assuming this is a random choice, this has probability (1/3)*(1/2) = 1/6.  These are the only possibilities where Door 3 has been opened.  The host opened Door 3 either because the car is behind Door 2 (with probability 1/3), or because he randomly picked between Door 2 and Door 3 with the car behind Door 1 (probability 1/6).  These add up to 1/2 - the other 1/2 being the (similar) cases where the host opens Door 2. -- Rick Block (talk) 21:57, 28 May 2012 (UTC)
 * Thanks Malick78. The problem we have here is that, once you get your head round the correct answer, it is hard to put yourself back into the position of someone who has not seen the problem before. Martin Hogbin (talk) 22:29, 28 May 2012 (UTC)