Switch doors, and you'll win the car twice as often
The Monty Hall Problem, live. Pick a door. I'll open one of the others to show you a goat. Then decide: stick with your door, or switch?
Run the numbers
Playing by hand only gets you a few rounds. Simulate thousands to see the real odds.
Why switching wins twice as often. When you first pick, there's a 1-in-3 chance you picked the car and a 2-in-3 chance you picked a goat. The host, who knows where the car is, always opens one of the other two doors to reveal a goat — he can always do this, no matter what you picked.
If you originally picked a goat (2 times out of 3), the host is forced to open the other goat door, which means the car is behind the one door left unopened — switching wins. If you originally picked the car (1 time out of 3), switching loses. So switching wins in exactly the 2-out-of-3 cases where your first guess was wrong, and staying only wins in the 1-out-of-3 case where it was right.
See also: the Birthday Paradox, another probability result that breaks most people's intuition on first read.
Source: first rigorously stated by Steve Selvin in a 1975 letter to The American Statistician; popularized (and initially disputed by many mathematicians before being confirmed correct) after Marilyn vos Savant answered a reader's question about it in Parade magazine, 1990.