Science Puzzle
The Three Doors
On a game show, there are three closed doors. Behind one is a car; behind the other two are goats. You pick door 1. The host, who knows where the car is, opens a different door, door 3, to show a goat. He always opens a goat door, never the one you picked. Then he asks: do you want to stick with door 1, or switch to door 2?
Should you stick or switch, or does it make no difference?
The Answer
Switch. Sticking wins the car one time in three; switching wins it two times in three.
Most people feel it must be 50:50, since two doors are left. But the host's choice isn't random. When you first picked, you had a 1 in 3 chance of being right and a 2 in 3 chance that the car was behind one of the other two doors. Nothing the host does changes the odds for your door. What he does is take the other two doors and, using what he knows, remove a goat from them. If the car was in that pair, it's now behind the one door he left closed.
Put another way, switching wins every time your first pick was wrong, and your first pick is wrong two times in three. It's even clearer with 100 doors. You pick one, and the host opens 98 goat doors, leaving yours and one other. Would you stick with your original 1 in 100 guess?
This is known as the Monty Hall problem, after the host of the American game show Let's Make a Deal. When the columnist Marilyn vos Savant gave the right answer in a magazine in 1990, she received thousands of letters telling her she was wrong, many of them from people with doctorates. The great mathematician Paul Erdős is said to have been convinced only after seeing a computer simulation. You can test it yourself with three cups, a coin and a friend: play 30 rounds sticking and 30 switching, then count the wins.
The principle: Conditional probability. New information changes the odds only in ways that depend on how it was produced; a host who knowingly avoids the prize gives you information that favours switching.