Edited by Charles B. Cross (University of Georgia)
|Summary||The Monty Hall Problem is a puzzle derived from the game show Let's Make a Deal, which first aired on American television during the 1960's and was for many years hosted by Monty Hall. Unlike most other philosophically interesting decision problems, the Monty Hall Problem has an uncontroversially correct solution, but this solution is easy to miss. The game show contestant is shown a series of closed doors and told that s/he may have what is behind exactly one of Door #1, Door #2, or Door #3. S/he is told that there is a new car behind exactly one of the three doors and nothing of much value behind either of the other two. The contestant selects exactly one of the three doors (say, Door #1), but this door is not opened. At this point the host opens one of the other doors (say, Door #2) revealing that the car is not behind that door. The host now asks whether the contestant would like to keep what is behind Door #1 or switch and take what is behind the unopened Door #3. Assuming that (1) the host must open a door that is not the contestant's initial choice and that does not have a car behind it, and (2) if the host has two non-car-concealing doors to choose from, the host will choose at random which of them to open, and (3) each door has an initial probability of 1/3 of concealing the car, it turns out that when the host opens Door #2, the probability of finding the car behind Door #3 increases from 1/3 to 2/3, which makes switching to Door #3 the correct decision. This result is surprising and counterintuitive, since, until the assumptions of the problem are taken into account, it may seem that the host has simply eliminated Door #2 and thereby given each of the remaining two doors a probability of 1/2 of concealing the car.|
|Key works||A version of the Monty Hall Problem appears in Steve Selvin, A Problem in Probability (Letter to the Editor), The American Statistician 29, No. 1 (Feb., 1975), p. 67. The problem was popularized in Marilyn vos Savant's column, Ask Marilyn, in Parade Magazine (9 September 1990, p. 16; 2 December 1990, p. 25; 17 February 1991, p. 12; 7 July 1991, p. 26). Since there is no issue concerning which solution to the Monty Hall Problem is correct, discussion of the problem in the philosophical literature has focused on how the problem is related to other issues, such as single case probabilities (Baumann 2005, Baumann 2008, Sprenger 2010), self-locating beliefs (Bradley 2007), and other decision problems including Judy Benjamin, Sleeping Beauty, and Doomsday (Ferreira 2010, Bradley & Fitelson 2003).|
|Introductions||For a presentation of the Bayes Theorem calculations that yield the correct solution, see Cross 2000.|
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