Statistics/Introduction/A confusing problem/Answer

Switching your choice to door number three doubles your chance of winning.

Why? This seems counter-intuitive, as many would assume it makes no difference if you switch or not. Many would think that with only two doors left, it's a 50-50 shot. So why is there actually a 2/3 chance that the prize is behind the other door you chose? The reason is that the information has changed in the two steps of the problem. Given two random doors, yes, there is a 50% chance that the prize is behind the door you choose. But in this problem, we have information from the first part, changing the odds. There are a few different ways to look at it.

The first way is to say that there is a 1/3 chance that the prize is behind the original door you choose. That leaves a 2/3 chance that the prize is behind one of the two doors you didn't choose. So say you choose door one, the chances that the prize is behind either door two or door three is 2/3. When the host opens door two, the probability that the prize is behind door two goes down to 0. The probability that the prize is behind door two or three is 2/3, and the probability that is behind door two is 0, thus making the probability that it is behind door three 2/3. Therefore, by switching from door one to door three, you double your chances at winning.

If you still don't believe that, a second way to look at it is this: For now, let's assume the prize is behind door one. If you choose door one in the first picking, then the host opens door two, showing nothing. In this case, if you switch doors, you will lose. Now, if you pick door two originally, the host will reveal door three (because he can't reveal door one, as that is where the prize is). Now, if you switch from door two to door one, you win. Along the same lines, if you choose door three at first, the host will reveal door two, and by switching you win. In short, if you pick the door with the prize in the first round, then switching will make you lose. If you pick a door without the prize, then by switching you will win. Since there is a 2/3 chance of picking a door without the prize in the first round, there is a 2/3 chance that switching your door choice will make you win.