On the old American game show Let's Make a Deal, there was always a
segment in which the contestant had to pick one of three doors,
looking for the car which was behind one of them and hoping not to get
one of the goats that were behind the other two. The rules
were as follows: The contestant picks a door, but before the door is
opened, the host opens one of the doors that was not selected
and reveals a goat. Now there are still two doors left, the one
originally selected and another. The contestant again has the choice
of which of the two doors to pick!
The interesting thing about this situation is that most people think that it does not matter whether you switch doors or keep the same door at this point. (This is why this was mentioned on a recent episode of the TV show NUMB3RS.) "What difference does it make?" they might say. "After all, there's a 50% chance of the car being behind either of the two doors...right?"
WRONG! It doesn't take too much mathematics to see that you are twice as likely to win if you switch doors at that point. I will try to convince you of this fact in two ways: