Yesterday, after reading this interesting article, I think I finally gained a proper understanding of the Monty Hall problem. Although it’s a very famous problem, in case you haven’t seen it before, I’ll state the problem below. If it is new to you, before reading on, you can have a think about what the answer might be.
So, suppose you’re on a game show where there are three doors. One of the doors holds a prize behind it, the others don’t hold anything. Your goal is simply to open the door with the prize behind it. The interesting part is, when you pick a door, you don’t open it right away: instead your host (Monty Hall) opens another door, different from the one you picked. Monty knows which door has the prize behind it, and he never opens that door. So then there are only two unopened doors remaining, one of which you picked, and one of which has the prize behind it. Monty asks you if you’d like to pick the other door instead. If you do so, are you any more likely to open the door with the prize behind it?
I will state the answer in the next paragraph.
When you first pick from the three doors, you know nothing about which door has the prize behind it. All the doors are equally likely to have the prize behind them; the probability that you have the right door is, therefore, 1/3. This should be pretty obvious. What many people don’t realise is that after Monty opens one of the doors, you do know something about which door has the prize behind it. Let’s call the door you picked originally door 1, the door Monty opened door 3, and the remaining door door 2. Door 3 does not have the prize behind it, and in fact, Monty would never have picked it if it did have the prize behind it. Therefore, the fact that Monty didn’t open door 2 is evidence in favour of it having the prize behind it. On the other hand, Monty was never considering opening door 1, because he was never going to open the door you picked. So you have evidence in favour of door 2 having the prize behind it, and no evidence in favour of door 1 having the prize behind it. Therefore, you should switch to door 2. You might also wonder how much more likely are you to get the prize. Well, the probability that the prize is not behind door 1 is 2/3. And since Monty opened door 3, if the prize is not behind door 1, it must be behind door 2. So the probability that the prize is behind door 2 is 2/3.
This problem became well-known after a reader posed it to Marilyn vos Savant in a letter to Parade magazine in 1990. She gave the correct answer, but later received hundreds of letters from readers saying she was wrong (some of them rather rudely), and that it isn’t helpful to switch. The crucial thing which these readers failed to see is that Monty is giving you extra information by opening one of the doors. If you didn’t know that Monty would only open a door which you hadn’t picked and which didn’t have the prize behind it, then you would not have this extra information, and as far as you knew, switching would not increase your probability of being correct.