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.
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:
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.