3 Variants

With several minutes remaining in the game, Monty Hall chose two contestants for the "Big Deal". Behind one of three doors was the grand prize. Each contestant was allowed to choose a door (not the same one).

In this scenario, a variant of Selvin's problem can be stated. Monty eliminates a player with a goat behind their door (if both players had a goat, one is eliminated at random, without letting the players know about it), opens the door and then offers the remaining player a chance to switch. Should the remaining player switch?

The answer is no. The reason: a switcher in this game will lose if and only if either of two initial choices of the two contestants was correct. How likely is that? Two-thirds. A sticker will win in those 2/3 of the cases. So stickers will win twice as often as switchers.

Altenatively by enumerating possibilities (again you play 1 and the other player plays 2)

Ejected Player Probability Switch Strategy Stick Strategy 1 2 3 C G G 2 1/3 Lose G C G 1 1/3 Lose G G C 1 1/6 Win Lose 2 1/6 Lose Lose

Player 1 wins 1/3 of the time with the stick strategy, or 1/6 of the time with the switching strategy. Half the time he is eliminated. Given that he is not eliminated there is 2/3 probability of winning with the sticking strategy.

There is a generalization of the original problem to n doors: in the first step, you choose a door. Monty then opens some other door that's a loser. If you want, you may then switch your allegiance to another door. Monty will then open an as yet unopened losing door, different from your current preference. Then you may switch again, and so on. This continues until there are only two unopened doors left: your current choice and another one. How many times should you switch, and when, if at all?

The best strategy is: stick with your first choice all the way through but then switch at the very end. With this strategy, the probability of winning is (n-1)/n. This was proved by Bapeswara Rao and Rao.

4 Origins

The game used in the Monty Hall problem is similar to three card monte, a gambling game in which the player has to find a single winning card among three face-down cards. As in the Monty Hall problem, the dealer knows where the winning card is, but here the dealer always tries to trick the player into picking the wrong card. As the card is often a Queen court card, it is also known as Find the Lady.

An older puzzle in probability theory involves three prisoners, one of whom (already chosen at random but unknown to the prisoners) is to be executed in the morning. The first prisoner begs the guard to tell him which of the other two will go free, arguing that this reveals no information about whether he will be the victim; the guard responds by claiming that if the prisoner knows that a specific one of the other two prisoners will go free it will raise the first prisoner's subjective chance of being executed from 1/3 to 1/2. The question is whether the analysis of the prisoner or the guard is correct. In the version given by Martin Gardner, the guard then performs a particular randomizing procedure for selecting which name to give the prisoner; this gives the equivalent of the Monty Hall problem without the usual ambiguities in its presentation.

5 Anecdotes

After this problem's solution was discussed in Marilyn vos Savant's "Ask Marilyn" question-and-answer column of Parade magazine in 1990, many readers including several math professors wrote in to declare that her solution was wrong. An equally contentious discussion of Marilyn's discussion took place in Cecil AdamsCecil Adams is the pen name of the author of The Straight Dope since 1973, a popular question and answer column published in The Chicago Reader and syndicated in thirty newspapers in the United States and Canada, and available online. Billed as the "World's column The Straight Dope.

The Monty Hall problem is elegantly discussed, from the perspective of a boy with Asperger's syndromeAsperger's disorder or Asperger's syndrome AS is a pervasive developmental disorder most closely related to autism, and commonly referred to as a form of "high-functioning" autism. The term "Asperger's syndrome" was coined by Lorna Wing in a 1981 medical, in The Curious Incident of the Dog in the Night-timeThe Curious Incident of the Dog in the Night-time is a novel by Mark Haddon. The title is a quotation of a remark made by the fictional detective Sherlock Holmes in Arthur Conan Doyle's Silver Blaze''. Written as the first-person narrative of Christopher, a 2003 novel by Mark HaddonMark Haddon is a novelist, who was educated at Merton College, Oxford. While there, one of his major achievements was to "clock" the Gravitar video game, a fiendishly difficult task which proved beyond the college's most obsessive video nerds. He won the.

6 References

• Bapeswara Rao, V. V. and Rao, M. Bhaskara (1992). "A three-door game show and some of its variants". The Mathematical Scientist 17, no. 2, pp. 89–94
• Bohl, Alan H.; Liberatore, Matthew J.; and Nudick, Robert L. (1995). "A Tale of Two Goats ... and a Car, or The Importance of Assumptions in Problem Solutions". Journal of Recreational Mathematics 1995, pp. 1–9.
• Gardner, Martin (1959). "Mathematical Games" column, Scientific American, October 1959, pp. 180–182.
• Selvin, Steve (1975a). "A problem in probability" (letter to the editor). American Statistician 29(1):67 (February 1975).
• Selvin, Steve (1975b). "On the Monty Hall problem" (letter to the editor). American Statistician 29(3):134 (August 1975).
• Tierney, John (1991). "Behind Monty Hall's Doors: Puzzle, Debate and Answer?", The New York Times July 21, 1991, Sunday, Section 1; Part 1; Page 1; Column 5
• vos Savant, Marilyn (1990). "Ask Marilyn" column, Parade Magazine p. 12 (Feb. 17, 1990). [cited in Bohl et al., 1995]

7 External links

• THE MONTY HALL PROBLEM (at letsmakeadeal.com; quotes Monty's letter to Steve Selvin in full)
• Monty Hall Paradox (let's make a deal) (lengthy bibliography)
• Monty Hall Dilemma (Monty Hall simulation, discussions, and generalization)

Mathematical recreations and puzzles Paradoxes

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