This post was one of a series of three I wrote about the Monty Hall problem, and how Bayes’ Theorem can help to solve it. I wrote them back in 2020 as part of an internship with Tom Crawford, and so this post originally appeared on his site in 2021.
If you don’t know what the Monty Hall problem is, why not go back to part I?
Monty Hall Origins
Three people, and three magazines, are principally responsible for the Monty Hall problem. The first is Martin Gardner (1914–2010), long-time writer of the “Mathematical Games” column in Scientific American; in 1959, he presented it as the mathematically-equivalent “three prisoners” problem, in which two of three prisoners are due to be executed, and prisoner A asks the guard to name one of the other two who is due to face the electric chair.
In the less-grisly form involving two goats and a car, the first-known instance (which came with a correct solution) was in a letter in American Statistician by Steve Selvin in 1975. But the puzzle really took off when a reader wrote in with it to Marilyn vos Savant, for her “Ask Marilyn” column in Parade magazine, in 1990. Savant, once recognised by Guinness World Records as having the world’s highest IQ, also gave the correct answer, that it was better to switch.
Ten thousand people wrote in to Parade to tell Savant she was wrong, including around a thousand people with PhDs. She defended her answer in subsequent articles, and still the complaints came in, with the whole shebang eventually making the front page of the New York Times. Complicating the issue slightly was that Savant slightly mis-stated the problem, not making it clear that Monty will always reveal a goat—which, as we’ve seen, makes a huge difference. But given that ambiguity, while the “it doesn’t matter” respondents might have been right, Savant still wasn’t wrong.
Savant eventually suggested that schools carry out experiments on the problem, and count the proportion of times that the car was won, to get a real-life estimate of the probability: such experiments, naturally, suggested that switching made you twice as likely to win as sticking. In the end, most people in academia accepted the correct result, and some of Savant’s critics did later write to her to retract their letters; one later described the fact that he’d initially written to her (with his suggestion of “confessing [her] error and, in the future, being more careful”) as an “intense professional embarrassment”.
One person who was never fooled by the problem was Monty Hall himself (1921–2017), the game show host after whom the problem was named. The puzzle was inspired by the show Let’s Make a Deal, where contestants chose boxes with prizes in them, and where Hall would often show an empty box (or one containing a valueless prize) to the contestant after they’d made their pick. He knew that opening a dud box didn’t affect the proportion of the time the contestant won: a viewpoint which Hall confirmed both in a letter to Selvin after the latter’s column, and in an interview with the New York Times piece.
But, as he noted in the Selvin letter: on the real show, “once the first box is seen to be empty, the contestant cannot exchange his box.” That is, even if you know the switching strategy, in real life the house always wins. Well, two times out of three, at any rate.
References
Gardner, Martin. 1959. Mathematical Games. Scientific American. October 1959, pp. 180–182.
Garsia, Adriano. 2014. Now Playing Let’s Make a Deal. Mathematics 187: Introduction to Cryptography; University of California, San Diego. [Online] 2014. [Cited: February 17, 2021.]
Let’s Make A Deal (A Joint Venture). 1999–2012. The Monty Hall Problem. The Official Let’s Make A Deal Website. [Online] 1999–2012. [Cited: February 17, 2021.]
Selvin, Steve. 1975. A Problem in Probability (Letter to the Editor). The American Statistician. 1975, Vol. 29, 1, p. 67. [Cited: February 17, 2021.]
—. 1975. On the Monty Hall Problem (Letters to the Editor). The American Statistician. 1975, Vol. 29, 3, p. 134. [Cited: February 17, 2021.]
Tierney, John. 1991. Behind Monty Hall’s Doors: Puzzle, Debate and Answer. The New York Times. July 21, 1991, § 1, p. 1. [Cited: February 17, 2021.]
Vos Savant, Marilyn. 1990. Ask Marilyn. Parade. September 9, 1990, p. 16.
—. 2006–2012. Game Show Problem. marilynvossavant.com. [Online] 2006–2012. [Cited: February 17, 2021.]
Wikipedia contributors. 2021. Monty Hall problem. Wikipedia, The Free Encyclopedia. [Online] January 28, 2021. [Cited: February 17, 2021.]
Leave a Reply