Wisely Choosing A Door

The "three door puzzle" is an interesting and unusual probability question. Here's how it works. You are a contestant in a game show, and the game show host tells you there is a prize behind one of the three doors you face. You have to guess which door to open.

But when you make your guess, instead of opening the door you picked, the game show host opens a different door...one that he knows has nothing behind it. So now you're down to two doors. And the game show host says, "I'll let you change your choice, if you want to."

And the question is, do you change your guess? Or keep your original choice?

The natural assumption is that it makes no difference, and so people tend to keep their original guess - reasoning being that we all know you should never "second guess" yourself. But is that really a wise choice?

Here's the math behind the question.

When you pick your door, the probability that you picked the correct on is one-third (1/3). Thus, the probability that you chose incorrectly is two-thirds (2/3). Nothing that the game show host does at this point has ANY effect on that probability.

Want To Keep Your Guess?

Let's suppose that you guessed correctly. Then it makes no difference what the game show host does, the other door is always the wrong door. So in that case, by keeping your choice, the probability that you win is 1/3 x 1 = 1/3.

But let's suppose you guessed incorrectly. In that case, the remaining door is guaranteed to be the correct door. Thus, by keeping your choice, the probability of winning is 2/3 x 0 = 0.

Your total chances of winning by keeping your guess is: 1/3 + 0 = 1/3.

Want To Change Your Guess?

Again, let's suppose that you guessed correctly. By changing your guess the probability that you win is 1/3 x 0 = 0.

But let's suppose you guessed incorrectly. Again, this means that the remaining door must be the correct one. Therefore by changing your choice, the probability of winning is 2/3 x 1 = 2/3.

Your total chances of winning by changing your guess is: 2/3 + 0 = 2/3.

Do You Believe It?

It seems counter-intuitive, doesn't it? It seems that it shouldn't matter whether you keep your guess or change it. Do you need more convincing? There are a few ways you can convince yourself that this is true.

Get a friend and do a simulation. Have your friend pick a door without telling you, and go through the whole process. Always keep your guess, and see how often you come up a winner. If you do this enough times, you'll find that your probability settles right around 1/3.
Alternately, do it the other way around. Always change your guess. You'll find that you win about two thirds of the time.
Still not convinced? Okay, let's imagine another game, in which there are a million doors! You pick one. What are the chances that you picked correctly? 1/1,000,000. Now the game show host opens all but one of the other doors. What guess would you make about the location of the prize now?