IanG on Tap

This one does the rounds every so often, but it always tickles me. A friend recently blogged about it, so I thought I'd post my take on it.

The Problem

The problem, named after the host of a game show on which it sometimes appeared, is as follows:

There are three doors, behind one of which is a valuable prize, but you don't know which door. Choose a door. You are not told straight away whether you've made the right choice. Instead, the host of the game will then open one of the doors you did not pick, showing you that there is no prize behind it. You are now offered the chance to change your mind. This effectively narrows down your choice - the prize is behind one of two doors, either the one you picked, or the door that neither you nor the host picked.

What should you do to maximize the probability of winning the prize? Should you stick with your first choice, or switch to the other door? Or does it not matter?

The Solution v1 (The Wrong Answer)

There's an obvious and straightforward answer which, like so many obvious and straightforward answers turns out to be wrong: you know the prize is behind one of the two doors, so it doesn't matter which you pick - you're equally likely to win either way.

Sadly, if you actually try it, this turns out to be wrong. There has been a lot of resistance to this fact over the years from people who should know better. (E.g. a lot of statisticians and mathematicians wrote in to complain when Marilyn vos Savant wrote about this in her weekly column pointing out that the obvious answer was wrong - they 'corrected' her, saying that it makes no difference whether you switch or not.) But the experimental evidence is pretty compelling if you can be bothered to try it.

The Solution v3.1 (The Right Answer)

If you always choose to switch to the other door, you will win about 2/3 of the time. If you always stick with your first choice, you'll only win about 1/3 of the time.

Huh?

This often confuses people. They think that the only information they have is that the prize is not behind the door that the host opened. But that's not really true - the key is to understand that the host has complete knowledge of the state of the system, and you are exercising some control over what the host reveals with your initial choice.

There are several ways of thinking about this, many mentioned here. Here's how I think about it:

What Game Do You Want To Play?

By choosing whether to switch or not, you are making a choice as to which game you want to play, or if you prefer, which bet you want to place. You have two options:

1. You could place a bet that the prize is behind a specific door.
2. You could place a bet that the car is not behind a specific door.

Your chances of winning the second bet are obviously better - with three doors, picking one at random and betting that it's the one with the prize offers you only a 1/3 chance of winning, but betting that the prize is not behind it (i.e. that it's behind one of the other two) gives you a 2/3 chance of winning.

The rules of the game permit you to choose which of these two bets you want to place. By electing not to switch, you are placing the first bet. By electing to switch, you are placing the second bet.

If your game strategy is to bet that the prize is behind a specific door, then you can just ignore the door the host opens - you just pick one and stick with it. You can stick your fingers in your ears, shut your eyes and shout "La la la la la la la" while the host shows you what was behind the door - you're not planning to switch so you obviously don't need to have that information the host is offering you. So your chances of being correct at this stage are clearly one in three - there were three doors, and you picked one of them. (And wilfully ignored the extra bit of information the host was prepared to give you.)

If you want to bet that the prize is not behind a particular door, you pick that door first and then switch. If your bet is correct, the host has already ruled out one of the other two doors, meaning that the prize will be behind the remaining door. So if you switch to that door, the probability of winning is exactly that of winning your original bet that the prize was not behind the first door you picked, i.e. 2/3.