The Monty Hall problem

Question

On a game show, a car sits behind one of three doors and goats behind the other two. You pick a door, say Door 1. The host, who knows where the car is and will always open a goat door you didn't pick, opens Door 3, revealing a goat. He offers to let you switch to Door 2.

Should you switch, and what is your probability of winning if you do?

Accepted Answer

Your initial pick is correct with probability 	frac{1}{3}. The key is that the host's behavior is not random: he always opens a goat door from the two you didn't pick. - With probability 	frac{1}{3}, your door has the car. The host opens either goat door; switching loses. - With probability 	frac{2}{3}, your door has a goat. The remaining goat is the only door the host can open, so the other unopened door must hold the car; switching wins. So switching wins with probability P(	ext{win} \mid 	ext{switch}) = \frac{2}{3}, versus 	frac{1}{3} for staying. Formally, by Bayes with you on Door …

The Monty Hall problem

Your answer

More Probability questions