For anyone unfamiliar with it, here's the basic premise of the problem:
You're on a game show where you are presented with three closed doors. One, chosen at random, has a car behind it, while the other two have goats. You pick a door, trying to find the car. After you have picked, the host opens one door, always selecting one of the two that you did not choose, and always selecting one that has a goat behind it (since the host knows where the car is). You then are given the option of changing your selection from your original one to the other remaining unopened door. What do the odds say you should do?
It should be obvious enough that your initial chance of picking the correct door is 1/3, since there are three doors, each equally likely to hide the car, and you can only pick one. Let's label the doors A, B, and C, with A arbitrarily being the door that you picked. This leaves these three, equally likely, cases:
- A) You picked door A, the door that has the car.
- B) You picked door B, one of the two doors with a goat.
- C) You picked door C, the other one of the doors with a goat.
At this point, the host opens a door, revealing a goat. Opening the door didn't change anything. The goats and car didn't move around. So now there are only two doors left, one with the car and one with a goat. Either door could be correct, so it doesn't really matter whether you switch or not, right? There's a huge problem with that assumption, though.
There are still three cases.
- A) Your initial guess is still right. The host can open either of B or C; it doesn't really matter. If you switch, you lose.
- B) Your initial guess is still wrong. The host must open door C, so switching gives you door B and the win.
- C) Your initial guess is still wrong. The host must open door B, so switching gives you door C and the win.
It's easy to assume that, with two doors to pick from, you have a 50-50 chance either way. And that would be true, if you picked a door only after the host opened one. But you didn't! That's the trick. Your initial selection has a 1 in three 3 chance of being correct, period. Nothing the host does after you pick can alter that. The host knows something you don't, though, namely where the car is. The host never opens the door you picked, so not opening that door tells you nothing about what might be behind it. However, opening a door does affect the odds for the remaining unopened door, since the host never opens the door with the car. In two out of three cases, the unpicked closed door remains closed precisely because it hides the car.
The confusion comes from incorrectly conflating cases B and C by thinking of the situation as "the car is either behind the door I picked or behind the other unopened door". The thing is, there's only one way it can be behind the door you picked, but two different ways it can be behind the other door. It doesn't matter that you can express it as "either one or the other"; the probabilities simply aren't equal.