4 years ago by in Mathematics, Mathematics

The Monty Hall problem is a probability puzzle, loosely based on the American television game show Let’s Make a Deal and named after the show’s original host, Monty Hall. The problem was originally posed in a letter by Steve Selvin to the American Statistician in 1975 (Selvin 1975a), (Selvin 1975b). It became famous in the following form, as a question from a reader’s letter quoted in Marilyn vos Savant’s “Ask Marilyn” column in Parade magazine in 1990 (vos Savant 1990a):

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

Vos Savant’s response was that the contestant should switch to the other door. (vos Savant 1990a)

The argument depends on the assumptions, made explicit in more extended solution descriptions given by Selvin (1975a) and by vos Savant (1991a), that the host always opens a different door from the door chosen by the player and always reveals a goat by this action – something he can always do because he knows where the car is hidden. Leonard Mlodinow says: The Monty Hall problem is hard to grasp, because unless you think about it carefully, the role of the host goes unappreciated.(Mlodinow 2008)

Contestants who switch have a 2/3 chance of winning the car, while contestants who stick have only a 1/3 chance. One way to see this is to notice that there is a 2/3 chance that the initial choice of the player is a door hiding a goat. When that is the case, the host is forced to open the other goat door, and the remaining closed door hides the car. “Switching” only fails to give the car when the player had initially picked the door hiding the car, which only happens one third of the time.

Many readers of vos Savant’s column refused to believe that switching is beneficial, despite her explanation of her answer. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming that vos Savant was wrong (Tierney 1991). Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy. (vos Savant 1991a) Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation confirming the predicted result (Vazsonyi 1999).

The Monty Hall problem has attracted academic interest because the result is surprising and the problem is simple to formulate. Furthermore, variations of the Monty Hall problem can easily be made by changing the implied assumptions, and the variations can have drastically different consequences. For example, if Monty only offers the contestant a chance to switch when the contestant has initially chosen the car, then the contestant should never switch. If Monty opens another door at random and only happens to reveal a goat, then it makes no difference (Rosenthal, 2005a), (Rosenthal, 2005b).

The problem is a paradox of the veridical type, because the correct result (you should switch doors) is at first sight ludicrous, but is nevertheless demonstrably true. It is mathematically closely related to the earlier Three Prisoners problem, and both problems bear some similarity to the much older Bertrand’s box paradox.


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