The Three Doors Problem, Without the Shouting

A prize sits behind one of three doors. You pick one. The host, who knows where the prize is, opens a different door that has nothing behind it, then offers you a switch. Should you?

Yes. Switching wins two thirds of the time. This answer has started more arguments per word than almost anything in mathematics, so let us go slowly.

The short version

Your first pick is right one time in three. That never changes, because the host can always open an empty door whatever you picked. So the other two doors together held the prize two times in three, and the host has just shown you which of those two to take.

Count it out

Say you always pick door 1, and always switch.

• Prize behind 1: you switch away and lose.
• Prize behind 2: the host opens 3, you switch to 2 and win.
• Prize behind 3: the host opens 2, you switch to 3 and win.

Two wins out of three. We also ran it 200,000 times in a short simulation and switching won 66.7 percent of them.

The fine print that matters

All of this depends on the host knowing where the prize is and always opening an empty door. If the host opened a door at random and happened to reveal nothing, switching would make no difference. How the information reached you is part of the information.