Three doors
A prize is behind one of three doors, each equally likely. You pick a door. The host, who knows where the prize is, then always opens one of the other two doors that has no prize behind it, and always offers you the chance to switch to the remaining closed door.
What is your probability of winning if you switch?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
When does switching lose?
Your first pick is right with probability , and the host's action never changes that.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Switching loses only when your first pick was right. The host always opens an empty door no matter what, so his action does not change the chance that your first pick was right.
Step 1. Your first pick hides the prize with probability .
Step 2. If your pick was wrong (probability ), the prize is behind one of the other two doors, and the host must open the other one. The remaining closed door then hides the prize, so switching wins.
Step 3.
Why the rules matter. The answer relies on the host always opening an empty door and always offering the switch. If the host opened a door at random and it happened to be empty, switching would win only half the time. With doors and the host opening empty ones, switching wins with probability .