Monty Hall Problem - History

History

The earliest of several probability puzzles related to the Monty Hall problem is Bertrand's box paradox, posed by Joseph Bertrand in 1889 in his Calcul des probabilités (Barbeau 1993). In this puzzle there are three boxes: a box containing two gold coins, a box with two silver coins, and a box with one of each. After choosing a box at random and withdrawing one coin at random that happens to be a gold coin, the question is what is the probability that the other coin is gold. As in the Monty Hall problem the intuitive answer is 1/2, but the probability is actually 2/3.

The Three Prisoners problem, published in Martin Gardner's Mathematical Games column in Scientific American in 1959 (1959a, 1959b), is equivalent to the Monty Hall problem. This problem involves three condemned prisoners, a random one of whom has been secretly chosen to be pardoned. One of the prisoners begs the warden to tell him the name of one of the others who will be executed, arguing that this reveals no information about his own fate but increases his chances of being pardoned from 1/3 to 1/2. The warden obliges, (secretly) flipping a coin to decide which name to provide if the prisoner who is asking is the one being pardoned. The question is whether knowing the warden's answer changes the prisoner's chances of being pardoned. This problem is equivalent to the Monty Hall problem; the prisoner asking the question still has a 1/3 chance of being pardoned but his unnamed cohort has a 2/3 chance.

Steve Selvin posed the Monty Hall problem in a pair of letters to the American Statistician in 1975. (Selvin (1975a), Selvin (1975b)) The first letter presented the problem in a version close to its presentation in Parade 15 years later. The second appears to be the first use of the term "Monty Hall problem". The problem is actually an extrapolation from the game show. Monty Hall did open a wrong door to build excitement, but offered a known lesser prize—such as $100 cash—rather than a choice to switch doors. As Monty Hall wrote to Selvin:

And if you ever get on my show, the rules hold fast for you—no trading boxes after the selection. —Hall 1975

A version of the problem very similar to the one that appeared three years later in Parade was published in 1987 in the Puzzles section of The Journal of Economic Perspectives (Nalebuff 1987). Nalebuff, as later writers in mathematical economics, sees the problem as a simple and amusing exercise in game theory.

Phillip Martin's article in a 1989 issue of Bridge Today magazine titled "The Monty Hall Trap" (Martin 1989) presented Selvin's problem as an example of what Martin calls the probability trap of treating non-random information as if it were random, and relates this to concepts in the game of bridge.

A restated version of Selvin's problem appeared in Marilyn vos Savant's Ask Marilyn question-and-answer column of Parade in September 1990 (vos Savant 1990a). Though vos Savant gave the correct answer that switching would win two-thirds of the time, she estimates the magazine received 10,000 letters including close to 1,000 signed by PhDs, many on letterheads of mathematics and science departments, declaring that her solution was wrong. (Tierney 1991) Due to the overwhelming response, Parade published an unprecedented four columns on the problem (vos Savant 1996:xv). As a result of the publicity the problem earned the alternative name Marilyn and the Goats.

In November 1990, an equally contentious discussion of vos Savant's article took place in Cecil Adams's column The Straight Dope (Adams 1990). Adams initially answered, incorrectly, that the chances for the two remaining doors must each be one in two. After a reader wrote in to correct the mathematics of Adams' analysis, Adams agreed that mathematically, he had been wrong, but said that the Parade version left critical constraints unstated, and without those constraints, the chances of winning by switching were not necessarily 2/3. Numerous readers, however, wrote in to claim that Adams had been "right the first time" and that the correct chances were one in two.

The Parade column and its response received considerable attention in the press, including a front page story in the New York Times in which Monty Hall himself was interviewed. (Tierney 1991) Hall appeared to understand the problem, giving the reporter a demonstration with car keys and explaining how actual game play on Let's Make a Deal differed from the rules of the puzzle.

Over 75 papers have been published about this problem in academic journals and the popular press. Barbeau 2000 contains a survey of the academic literature pertaining to the Monty Hall problem and other closely related problems as of the year 2000, and contains citations to 40 publications on the problem. At present the book Rosenhouse 2009 has the most recent comprehensive academic survey, and refers to at least 25 publications on the topic which appeared subsequently to Barbeau's book. Since then another 10 or so publications have come out. Nowadays, Wikipedia is frequently cited as a source for Monty Hall problem (it is cited, for instance, by Ruma Falk, Rosenhouse 2009, Gill 2012, Gnedin 2012).

The problem continues to appear in many venues:

• The syndicated NPR program Car Talk featured it as one of their weekly "Puzzlers", and the answer they featured was quite clearly explained as the correct one (Magliozzi and Magliozzi, 1998).
• Accounts of the Hungarian mathematician Paul Erdős's first encounter of the problem can be found in The Man Who Loved Only Numbers and Vazsonyi 1999; like so many others, Erdős initially got it wrong.
• The problem is presented in fictional form in the first chapter of the novel Mr Mee (2000) by Andrew Crumey.
• The problem is discussed, from the perspective of a boy with Asperger syndrome, in The Curious Incident of the Dog in the Night-Time, a 2003 novel by Mark Haddon.
• The problem is also addressed in a lecture by the character Charlie Eppes in an episode of the CBS drama NUMB3RS (Episode 1.13).
• Derren Brown explains the Monty Hall problem in his stage show Svengali. After asking a member of the audience to choose the location of his shoe from three boxes, he reveals an empty box from one of the ones not chosen. He then asks if they would like to change their mind and recommends that they do so, as it will increase their chances of winning. He explains this further by demonstrating on a large screen the same puzzle but with one hundred boxes. The member of the audience decides to stick with their decision and loses. The problem is also addressed in his 2006 book Tricks Of The Mind.
• The problem is featured in Ian McEwan's 2012 novel Sweet Tooth.
• Penn Jillette explained the Monty Hall Problem on the "Luck" episode of Bob Dylan's Theme Time Radio Hour radio series.
• The Monty Hall problem appears in the film 21 (Bloch 2008).
• Economist M. Keith Chen identified a potential flaw in hundreds of experiments related to cognitive dissonance that use an analysis with issues similar to those involved in the Monty Hall problem. (Tierney 2008)
• In 2009 a book-length discussion of the problem, its history, methods of solution, and variations, was published by Oxford University Press (Rosenhouse 2009).
• The problem is presented, discussed, and tested in the television show MythBusters on 23 November 2011. This paradox was not only tested to see if there was an advantage to switching vs. sticking (which, in a repeated sample of 49 "tests", showed a significant advantage to switching), but they also tested the behavior of "contestants" presented with the same situation. All 20 of the common "contestants" tested chose to stay with their original choice.