Answer by Marcus Geduld:
The Monty Hall Problem is a good example of this:
George is on a game show, and he has to pick one of three doors. There's a car behind one of them and goats behind the other two. He picks door number one. The host, who knows what's behind all the doors, opens door number two, revealing a goat. He then says to George, "I'll let you switch doors if you want."
Should George stick with door number one or switch to door number three?
It seems obvious that it makes no difference. The car is equally likely to be behind door one or three, so whichever door George picks, he has a 50-50 chance of getting the car.
That's wrong, though. Switching will give him an advantage.
Just to be clear, I'm not saying the host always opens door number two. I'm saying he opens one of the two doors the guest didn't pick. So (1) guest picks a door; (2) host (who knows where the car is), opens one of the two doors the guest didn't pick, revealing a goat. (He always reveals a goat, never a car.)
At this point, the guest knows that there are two unopened doors left. There's a car behind one of them and a goat behind the other. One of the two unopened doors is the one he originally picked. It either has a car or a goat behind it and so does the other unpicked door. Should he stay with the door he originally picked or switch to the other unpicked door? Does it make any difference?
[1?] [2?] [3?] <– question marks means there could be a car or a goat.
[1?] [2?] [3?] <– guest picks the bold door.
[1-g-] [2?] [3?] <– host opens one of the doors, revealing goats (-g-)
Two out of three doors have goats behind them. The host (who knows where the car is) has just revealed one of the goat doors. Now, should the guest stuck with the bold door or switch to the other question-mark door?
UPDATE: Just for fun, I whipped up this simulation:
To change the number of trials, edit this:
var iterations = 20;
Even though I knew what the results would show, I was surprised. I figured I'd have to run 200 iterations or so to see a payoff, but with just 20, it's clear that switching is a better choice.