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Yet the naive definition, even when inappropriate, has a powerful hold on people's intuitions. When Marilyn vos Savant presented a correct solution to the Monty Hall problem in her column for Parade magazine in 1990, she received thousands upon thousands of letters from readers (even mathematicians) insisting that she was wrong.

Blitzstein. Introduction to Probability (2019 2 ed). pp 70, 71.

The Time Everyone “Corrected” the World’s Smartest Woman

Though her answer was correct, a vast swath of academics responded with outrage. In the proceeding months, vos Savant received more than 10,000 letters -- including a pair from the Deputy Director of the Center for Defense Information, and a Research Mathematical Statistician from the National Institutes of Health -- all of which contended that she was entirely incompetent:

The Monty Hall Problem

I put my solution of the problem on the bulletin board in the physics department office at the Naval Academy, following it with a declaration that you were right. All morning I took a lot of criticism and abuse from my colleagues, but by late in the afternoon most of them came around. I even won a free dinner from one overconfident professor. – Eugene Mosca, Ph.D., U.S. Naval Academy, Annapolis, Maryland

According to Google Answers, solely Robert Sachs "was one of the few with the grace to concede his mistake."

But Aaron Brown SB Applied Mathematics (Harvard 1974-1978), MBA Finance (Booth School of Business, Univ. Chicago 1980-1984) wrote a different side of the story on Quora. Who's correct?

You have to have lived through this ancient travesty to care about it. It’s probably best forgotten.

Marilyn Vos Savant published an incorrect answer to the Monte Hall problem. She got the correct answer from a lot of readers, including me. She then published a second incorrect answer claiming that all the letters she got—including from math PhDs were wrong. Finally she got the answer right—the one she undoubtedly got from any math PhD or intelligent person who wrote her—but continued to claim (a) that her previous answers were correct and (b) that all the letters were wrong.

As a result, the public is mostly misinformed about this simple problem.

The set-up is simple. There are three doors, one of which has a valuable prize behind it, and two of which have worthless prizes. You select one door. The host opens one of the other two doors to show you a worthless prize, and offers you the opportunity to switch your choice.

The key to this problem, which Marilyn finally saw in her third solution, is the knowledge and intentions of the host. If the host doesn’t know which door has the prize, there’s no reason to switch or not switch, it’s the same either way. If the host knows the door and is trying to hurt you, don’t switch. If the host knows the door and is trying to help you, switch.

Marilyn’s first answer was to always switch, regardless of the host’s intentions. This is also the answer that most people now believe. It’s a common error that people make all the time. That’s why people wrote in to correct her.

In order to justify her incorrect first answer, she claimed it was obvious that the host used the strategy of always opening a non-prize door and offering you the chance to switch. This does indeed justify switching. But that assumption was missing from her first and second answers, along with any discussion of the host’s knowledge and intentions mattering.

Her justification that the assumptions were too obvious to need stating was the actual television program Let’s Make a Deal, hosted by Monte Hall, which eventually gave the name to the problem. But anyone who watched the show knew that was not Monte’s strategy. Sometimes he opened another door and gave a chance to switch, sometimes he didn’t. He was well aware of the problem if he followed Marilyn’s assumed strategy, and was scrupulous about not giving any advantage to the contestant. You can tell this from statistics on the show—on average it didn’t pay to switch—and his published statements.

Clearly there are much bigger problems in the world, and it doesn’t pay to get upset about misinformation put out by popular vain people. If was very frustrating at the time, but it seems quaint that such things upset us in the 90s when there is so much worse information put out today by even more popular and more vain people.

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    $\begingroup$ What follows isn't intended as an answer to your question (hence my using a comment), but the events are sufficiently recent that online discussions from the time in question exist. For instance, see these sci.math threads that took place during December 1990 and July-August 1991. $\endgroup$ Aug 21 at 17:19
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    $\begingroup$ Who is "correct" is not a history question. Wikipedia has a long nuanced article on this now, with references which mentions that "most statements of the problem, notably the one in Parade Magazine, do not match the rules of the actual game show and do not fully specify the host's behavior or that the car's location is randomly selected" and that "chance was anything between 1/2 and 1 depending on the host's decision process". There are various opinions on whether this mattered or people made "standard assumptions" and still got it wrong. We won't do better here beyond adding more opinions. $\endgroup$
    – Conifold
    Aug 21 at 21:55
  • $\begingroup$ The problem statement as quoted here from A. Brown is incorrect. The correct problem statement has to read "... the host, who knows where the prize is opens one of ..." $\endgroup$ Aug 23 at 13:21
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To add just a bit to the great documentation in the question, and to Dave Renfro's and Conifolds accurate/helpful comments:

First, as in the question, the actual conduct of Monty Hall in the TV show was different from the iconic statement of the problem. (I remember seeing it, live, long ago...) That is, as in the question, he did not always open a door... thus, could not be counted on to impart information... I didn't think about it at the time (though I was a mathematically inclined kid), but, in hindsight, he was either deliberately or subliminally trying to prevent "gaming" his game.

It is not clear to me whether peoples' reading of the "Monty Hall Problem" accounts for the twist that Monty always opens a door.

But now let's I would not be at all surprised if tens of thousands of fairly serious math people either misread or jump to a conclusion... I do remember being caught off guard when I first heard of this "Problem", perhaps in the 1990s. My first reaction was, indeed, that "the probabilities changed"... sorta subconsciously based on the "principle" that Laplace already spoke against 200 years ago: ignorance is not a proof of equal probabilities. :)

In my intro to cryptology classes in the late 1990s and early 2000's, many, many students could not be persuaded that the probabilities did not change, despite a variety of explanations, and despite extensive simulations. :)

Apart from the misreading or the (over-) simplification of the game from the reality, I think this story is powerful evidence for an innate tendency of human beings to believe in that false principle already denounced by Laplace (and probably understood by earlier people who wanted to gamble relatively successfully...)

But, yeah, in my direct observation, lotta otherwise-serious math people "jump" at the naive/wrong conclusion. :)

(Now that I think further, I do suspect that a certain number of math people have taken offence at the "presumption" of an outsider to have public opinions on math things... Even apart from gender bias and such.)

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    $\begingroup$ In teaching aspects of probability theory or combinatorics, I used the "vos Savant incident" as an example of why it is important to state a problem as clearly as manageable to (hopefully) avoid misinterpretations. I think many of the "outraged" academics did not actually understand correctly which conditional probability was being asked about. (I won't dispute, though, the gender-bias issue or the "opportunity" to (attempt to) "take down" someone billed as having an extremely high IQ.) $\endgroup$
    – boojum
    Aug 22 at 8:02
  • $\begingroup$ @boojum Bertrand paradox had some recent afterlife that can help with stressing the point of how deep the (mis)interpretations can run. Although it can't compete with Monty Hall in juiciness of the public spectacle :) $\endgroup$
    – Conifold
    Aug 23 at 7:38

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