The Monty Hall Problem

Mon 11 December 2017 by A Silent Llama

Here’s a fun little logic problem:

You’re on a game show and the host (Monty Hall) tells you that there’s a prize behind one of three doors with the other two doors being empty. You need to choose a door to win the prize. After choosing a door the host will reveal which of the two remaining doors is empty and then he’ll offer you a choice: do you want to switch doors or stick with your original choice?

Should you switch?

The normal line of logic goes something like this: you have a 1 in 3 chance of getting the prize before the empty door is revealed and revealing that door doesn’t materially change anything which means the chances should still be 1 in 3. Therefore switching should make no difference.

This is wrong. You should switch.

What makes this such a great problem is that it sounds like magic. How on earth does switching make you more likely to win? It really is counter intuitive and the first time the question was answered thousands of people wrote in to say they didn’t agree. In fact some psychologists theorise that we’re psychologically predisposed not to agree. You can read about it here. There are a quite a few mathematical ways of proving it (that wikipedia link lists a few of them) but I’d like to suggest a line of thought that doesn’t require any math (ahem that is to say it doesn’t require any calculation. It technically is math of course).

Think of it this way - what are the chances that your first choice is empty? Well - since there are 2 empty doors out of three your chances of being wrong are 2 in 3. Now the game show host opens an empty door. You likely chose an empty door in the beginning (2 in 3 chance) so chances are the host had no choice but to select the door he did. That means that the remaining door is slightly more likely to have the prize: you probably chose the wrong door, the host had to show you the other wrong door - so the remaining door probably has the prize.

And if you work out the math it turns out that switching gives you a 2 in 3 chance of winning the prize - twice as much as your 1 in 3 starting point. Cool huh?

To really spell it out consider we need to consider three cases:

Case 1:

You choose the first empty door.

In this case the host has to open the other empty door and the prize is behind the remaining door. If you switch: you win!

Case 2:

You choose the prize.

In this case it doesn’t matter which of the remaining doors the host opens. If you switch you lose.

Case 3.

You choose the other empty door.

The host has to open the remaining empty door. If you switch: you win!.

Those are the only 3 possibilities. Switching wins in two of the three cases so - drumroll - switching gives you a 2 in 3 chance of winning.

Q.E.D.

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