The Monty Hall problem
Behind one door is a car, behind the other two are goats. Pick a door. The host, who knows where the car is, opens a different door to show you a goat. Then you choose: stay with your door or switch to the other closed one. Play a few rounds, then let the computer play ten thousand.
Play
Simulate
What's going on
Your first pick is right one time in three. That does not change: a host who knows where the car is can always open a goat door, so seeing him do it tells you nothing about your own door. So two times in three the car is behind one of the other two doors, and the host has just told you which of those two it is not. Switching collects that whole two-thirds.
The host's knowledge is doing the work. Tick the box that makes the host forget. Now he sometimes opens the car by accident, and those games are thrown out. In the games that survive, staying and switching each win half the time. Same doors, same goats, but the host's choice stopped carrying information.
When this puzzle appeared in a magazine in 1990, thousands of readers, including many with mathematics PhDs, wrote in to say the answer was wrong. The simulation settles it faster than the arguing did.