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Problem of the Week 1104

Monty Hall Revisited

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Background -- The infamous Monty Hall problem: There are three doors, with a car behind one door and a goat behind the others. I pick, Monty opens a door showing me a goat and ask me if I want to switch. I should switch, as that changes my probability of getting the car from 1 in 3 to 2 in 3.

Suppose there are four doors. After my first choice Monty opens a door, shows me a goat, and offers me a chance to switch with one of the two unopened doors. After I make my decision, Monty opens a door, shows me a goat, and offers me a chance to switch to the remaining closed door. What should my two decisions be?

Note: I know in advance that Monty will follow this protocol, showing me two goats, regardless of choices I make, and making his choice randomly.

Source: M. Bhaskara Rao, on a game-show problem of Marilyn Vos Savant and its extensions, American Statistician 46 1992 241-242. Also in the book Impossible, by J. Havil.

© Copyright 2008 Stan Wagon. Reproduced with permission.

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22 September 2008