Nowhere does it'say that he *always* opens a door with a goat as you claim!

This is a common mistake in the statement of the Monty Hall problem. Usually an outcome is given: Monty opens a door which shows a goat, rather than the strategy, Monty always opens a door which shows a goat. When only the outcome is given, the probability of switching could be anything from 0 to 1, depending on what Monty's strategy is.

It is extremely rare for the problem to be stated correctly. Without saying "Monty always opens a door with a goat" rather than "Monty opens a door which has a goat" you cannot condition on the probability.

Many sites with an incorrect statement of Monty Hall problem:
https://mathworld.wolfram.com/MontyHallProblem.html
https://math.ucsd.edu/~crypto/Monty/monty.html
https://www.cut-the-knot.org/hall.shtml
https://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html
https://mathforum.org/dr.math/faq/faq.monty.hall.html
https://www.comedia.com/hot/monty.html
https://people.hofstra.edu/staff/steven_r_costenoble/MontyHall/MontyHall.html
https://www.grand-illusions.com/monty.htm

One sites with correct statement of Monty Hall problem (as of Oct 28, 2006)
https://en.wikipedia.org/wiki/Monty_Hall_problem
(note wiki states both the incorrect and correct versions and gives a detailed explanation of why the correct is true.)

Quite recently a geometry text has been translated from Russian. In a proof of the SSS, the original considers three case. The translator, a Berkeley math professor, omitted one in translation as not interesting. (Curiously, in a faulty proof of SSA, one can easily overlook an analogous case and then wonder how is SSA possible.)

Parade magazine is not even a mathematical text; ambiguity is creative (as the hoopla around an omission of one word "always" has demonstrated.) While the https://en.wikipedia.org/wiki/Monty_Hall_problem article is great, its very existence is due to an instance of such a creative ambiguity which I think was a fortunate incident if not the intended goal.

However, I still would argue that the *always* is not simply the requirement of a nitpicker, but is important in a very fundamental way. I'll give two reasons here.

1) Conditional probability. There is the question of what you can condition on in probability. You always condition on events (or at a measure theoretic level on sigma-algebras representing information, but I'll keep it at the undergrad level for now.) Most of the solutions that I've seen to the Monty Hall problem instead condition on an *outcome*, that Monty opened a door with a goat.

The outcome only becomes an event when the full joint distribution of Monty and the Players behavior is specified. Up until that point, it is merely an outcome, and you just can't condition on specific outcomes. From a pedagogical point of view, it is an ill posed problem that leads students into thinking that they should be making extra assumptions to solve any conditional probability problem, when that is the last thing that they should be doing.

2) Preciseness. What is it about this puzzle that makes people think it is OK for it to be stated imprecisely? No one would state the Tower of Hanoi problem without saying that larger rings cannot be placed on smaller rings. Yet that is of course the only assumption that makes the problem "interesting".

Or who would state Fermat's Last Theorem without including "for integer x, y, and z". But that is exactly what happens with the Monty Hall problem! Is it'somehow OK to leave out an essential part of the problem just because one statement of the problem that received widespread attention in a magazine contained an error.

I guess what I am getting at, is that just because Craig wrote Marilyn one time with an incorrect statement of the problem, why do countless websites feel the need to replicate this error, and virtually no one feels the need to fix it? Why quote Craig's statement of the problem at all, when clearly Craig did not realize that he was leaving off an essential piece of the puzzle?

Mark "befuddled at the psychology of the Monty Hall problem" Huber

We also agree on the rest. The difference is in that I perceive a problem coming from a Parade column as a puzzle and try to understand and if necessary make up for inaccuracies in formulation just to make the problem interseting. I would criticize such a formulation in a text book. (Mine is a quote.)

>2) Preciseness. What is it about this puzzle that makes
>people think it is OK for it to be stated imprecisely?

Thousands of puzzles come in less than precise formulations. Theorems come with faulty proofs. Why such an overhyped indignity about this one problem?

>No
>one would state the Tower of Hanoi problem without saying
>that larger rings cannot be placed on smaller rings. Yet
>that is of course the only assumption that makes the problem
>"interesting".

I never read Marylin's column or Parade magazine, for that matter. I know nothing about her, except that she is married to Dr. Jarvik and features an entry in the Guinness book of records. However, I am sure that had she described the Tower of Hanoi problem in the manner you suggest and there was a correction sent, she would admit to mistake. I am pretty sure that if asked she would agree with everything that was written about the Monty Hall problem, because it'simply can't be denied. Why the whole hoopla? I grasped the meaning and enjoyed the problem. Many people out there were simply aroused by the opportunity to stick it to Marylin. I believe she did not find it necessary to set the original inquiry right because of the same attitude: she intuited the meaning that made the problem interesting and responded to that in good faith. That's all.

>Or who would state Fermat's Last Theorem without including
>"for integer x, y, and z". But that is exactly what happens
>with the Monty Hall problem!

No, this is not what happened with the Monty Hall problem. With the Monty Hall problem, a non-professional sent in an inquiry to which Marylin had replied. The inquiry admitted a second interpretation which Marylin chose to disregard because she judged it non-interesting and unworthy of being querried after.

After somebody noticed that the inquiry admited a second interpretation, the question had to be posed to the proposer as to which of the interpretations he had in mind. Instead ... you know what happened.

>Is it somehow OK to leave out
>an essential part of the problem just because one statement
>of the problem that received widespread attention in a
>magazine contained an error.

Depends on where you leave it.

>I guess what I am getting at, is that just because Craig
>wrote Marilyn one time with an incorrect statement of the
>problem,

A statement by a confused non-professional can't be correct or not. At best, it may be non-sensical or ambiguous. I saw an announcement on the outer wall of a grocery store: The fourth year in a row the winner of the 2005 prize. Was it incorrect? How could it be (except grammatically) if every one understood the intended meaning? This is what happened with the Monty Hall problem. Marylin chose not to dig into the language but to answer the problem the way she understood it.

>why do countless websites feel the need to
>replicate this error,

Because it's a reference and a story. Personally, up there I placed a quote and subsequently added letters from the visitors who opined differently.

>and virtually no one feels the need to
>fix it?

Can't talk for every one else. A fix would destroy the story unless you tell the whole of it like in wikipedia. I did not want to tell the whole story and instead chose the "laissez faire" attitude; let the story unfold.

>Why quote Craig's statement of the problem at all,
>when clearly Craig did not realize that he was leaving off
>an essential piece of the puzzle?

Because this is how a question/answer column works. They print the questions that they are getting asked.

Any puzzle that I ran across with a less than precise formulation I would be equally interested in changing the statement to be precise. Being as I teach probability and so am asked about this problem all the time, I will admit to having spent more time thinking about this particular puzzle than many others. I check in on Cut-the-knot occasionally (with weeks or months between visits) and what do I find but yet another new thread about this problem, with yet another individual (Silas) who misread the problem and automatically added an assumption not in the problem statement. I'd like people to follow the story as well, but simply put, they don't! Silas did not even read the entire Wiki article that he quoted! If it is overhyped, it is simply because the incorrect formulation is overhyped as being true.

And of course is goes without saying that a theorem with a faulty proof needs to be exposed.

>I never read Marylin's column or Parade magazine, for that
>matter. I know nothing about her, except that she is married
>to Dr. Jarvik and features an entry in the Guinness book of
>records. However, I am sure that had she described the Tower
>of Hanoi problem in the manner you suggest and there was a
>correction sent, she would admit to mistake. I am pretty
>sure that if asked she would agree with everything that was
>written about the Monty Hall problem, because it'simply
>can't be denied. Why the whole hoopla? I grasped the meaning
>and enjoyed the problem. Many people out there were simply
>aroused by the opportunity to stick it to Marylin. I believe
>she did not find it necessary to set the original inquiry
>right because of the same attitude: she intuited the meaning
>that made the problem interesting and responded to that in
>good faith. That's all.

I was certainly unclear earlier. I do not mean to say that Marylin was wrong to publish Craig's letter or that it was necessarily a bad thing that she made the assumption the way she did in her answer (although it is too bad that she did not state it explicitly). I do think that it is bad that those who quote this statement of the problem on their own websites continue to ignore the fact that they are making an assumption.

You write below the problem "Two controversial solutions are given after the puzzle. Which is the right one?" You imply here that one of these solutions is correct, when in fact either could be correct or both could be wrong based on what assumptions you make about Monty's behavior, assumptions that you are not making explicit on the page.

>>Or who would state Fermat's Last Theorem without including
>>"for integer x, y, and z". But that is exactly what happens
>>with the Monty Hall problem!
>
>No, this is not what happened with the Monty Hall problem.
>With the Monty Hall problem, a non-professional sent in an
>inquiry to which Marylin had replied. The inquiry admitted a
>second interpretation which Marylin chose to disregard
>because she judged it non-interesting and unworthy of being
>querried after.
>
>After somebody noticed that the inquiry admited a second
>interpretation, the question had to be posed to the proposer
>as to which of the interpretations he had in mind. Instead
>... you know what happened.

Again, I am not really talking about what happened with Marilyn's answer or the Parade column. What bothers me is the multiple websites that repeat the Craig formulation of the problem and then proceed to solve it without making their assumptions explicit.

>>Is it somehow OK to leave out
>>an essential part of the problem just because one statement
>>of the problem that received widespread attention in a
>>magazine contained an error.
>
>Depends on where you leave it.
>
>>I guess what I am getting at, is that just because Craig
>>wrote Marilyn one time with an incorrect statement of the
>>problem,
>
>A statement by a confused non-professional can't be correct
>or not. At best, it may be non-sensical or ambiguous. I saw
>an announcement on the outer wall of a grocery store: The
>fourth year in a row the winner of the 2005 prize. Was it
>incorrect? How could it be (except grammatically) if every
>one understood the intended meaning? This is what happened
>with the Monty Hall problem. Marylin chose not to dig into
>the language but to answer the problem the way she
>understood it.

The definition of puzzle that I was using here is "a particularly baffling problem that is said to have a correct solution". In this sense, Craig's formulation is incorrect because without the assumption the puzzle does not have a correct solution. But this is just semantics on my part.

What isn't semantics is that you write on your webpage "At long last, the truth was established and accepted." I have to disagree with this statement. I feel that "the truth" is neither widely established nor widely accepted given the multiplicity of websites that still do not understand the fundamental issues at play with this problem. And while I do feel you personally do understand the issues involved, I also see visitors to your site that after reading your page do not.

>>why do countless websites feel the need to
>>replicate this error,
>
>Because it's a reference and a story. Personally, up there I
>placed a quote and subsequently added letters from the
>visitors who opined differently.

You are telling the story, but that is not all that is on your page. You wrote a Java application for the page that implicitly includes the assumption that never appeared in the question to Marylin or her answer or subsequent columns. That means you are adding to the story, and your addition is misleading in that it implies that no other assumptions are necesssary by burying the assumption in the code.

>>and virtually no one feels the need to
>>fix it?
>
>Can't talk for every one else. A fix would destroy the story
>unless you tell the whole of it like in wikipedia. I did not
>want to tell the whole story and instead chose the "laissez
>faire" attitude; let the story unfold.

And as long as there are people like Silas who are just not interested in reading the whole story, there will be people like me who try to uncover it for them in the CTK exchange. Hopefully they will read the followups to their own letters even if they cannot be bothered to read whole web pages before quoting from them.

>>Why quote Craig's statement of the problem at all,
>>when clearly Craig did not realize that he was leaving off
>>an essential piece of the puzzle?
>
>Because this is how a question/answer column works. They
>print the questions that they are getting asked.

Here I was not being explicit enough. I'm not talking about Marilyn's column in the above paragraph, I'm talking about the multiple websites that have repeated the incorrect statement of the problem and given their own solutions without making their own assumptions explicit. Unfotunately, despite the overall excellence of the site, I believe cut-the-knot falls into this category on this particular problem.

Mark "expecting to receive a list of imprecisely formulated puzzles so he'll go bother them instead" Huber

The fact is you can't know in which circumstances more people learn more. Do you remember the cat statitstics from Marylin's book. If the memory serves, veterinarians could claim that judging by the condition of the cats brought for treatment, cats tend to land on their feet. The problem with this reasoning was that the cats that did not land on their feet, were not brought for treatment.

Yes, students and others keep asking this question and misreading the problem. Naturally, those that did not misread the problem do not ask the question. Irritating as it may be to return to the same question over and over again, still are you 100% sure that removing all the question marks will serve a noble purpose.

I feel like mentioning the reason for the controversy and making a reference to wikipedia, but do not have it in me to compete with the wikipedia. You know, if only writing everything right, dotting the i's and crossing the t's could assure that students learn the thing, we all would be great mathematicians.

You know, everyone can modify wikipedia articles (this is what "wiki" generically stands for.) I think you'll do public service by inserting this remark.