This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
You're about to play on a game show. There are three doors; behind one is
a valuable prize, behind the other two, junk. You'll get to choose a
door, and then Monty Hall (who knows where the prize is) will open one of
the other doors, showing you junk. At that point, you'll have a chance to
"switch" your choice to the remaining unopened door. After that, you'll
win whatever is behind the door you have chosen. Should you switch?
It is advantageous to switch: your probability of winning is 2/3 if you do
so. The probability that your first guess is wrong is 2/3, and switching
doors will gain you the prize if and only if your first guess was wrong.