The Monty Hall Problem

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 [but the door is not opened], 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?


The solution presented by Vos Savant shows the three possible arrangements of one car and two goats behind three doors and the result of staying or switching after initially picking Door 1 in each case:

behind Door 1 behind Door 2 behind Door 3 result if staying at door #1      result if switching to the door offered
Car Goat Goat Car     Goat
Goat Car Goat Goat     Car
Goat Goat Car Goat     Car

A player who stays with the initial choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three. The probability of winning by staying with the initial choice is therefore 1/3, while the probability of winning by switching is 2/3.

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