Monty Hall with five doors

Question

A car sits behind one of five doors, goats behind the other four. You pick Door 1. The host, who knows where the car is, opens three goat doors from among the four you did not pick, leaving exactly one other door closed. He offers to let you switch to it. Should you switch, and what is your probability of winning if you do?

Accepted Answer

Your initial pick is correct with probability 	frac15. As always with Monty Hall, the host's action is forced and informed, which is what concentrates probability. - With probability 	frac15, your door has the car. Then the other four are all goats; the host opens three of them, and the last closed door is a goat. Switching loses. - With probability 	frac45, your door has a goat. Then the other four doors hold the car and three goats; the host opens exactly those three goats, so the single door he leaves closed must be the car. Switching wins. So P(	ext{win} \mid 	ext{switch}) = \frac{4}…

Monty Hall with five doors

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