One hundred doors

Question

A game show has 100 doors, with a prize behind exactly one. You choose a single door and leave it shut. The host, who knows where the prize is, then opens 98 of the other 99 doors, every one revealing nothing. One other door stays closed alongside yours.

What is your probability of winning if you switch to that remaining closed door?

Accepted Answer

Your original guess is right with probability 	frac{1}{100} and wrong with probability 	frac{99}{100}. The host never opens the prize door, only empties. So: - If your first door was the prize (probability 	frac{1}{100}), the other closed door is empty and switching loses. - If your first door was wrong (probability 	frac{99}{100}), the prize sits among the other 99 doors. The host opens 98 empties from those, and the only door he can leave closed is the prize itself. Switching wins. Therefore P(	ext{win by switching}) = \frac{99}{100}. The many doors make the effect vivid: almost certai…

One hundred doors

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