1) The Monty Hall problem. A quiz show. There are three doors. Behind one of them is a valuable prize. The contestant chooses one door. The host (Monty), who knows where the prize is, then opens one of the other doors revealing nothing.

The contestant is then given the chance to change their mind about which door to choose - ie either keep the door originally chose or swap to the other one that wasn't opened.

What should they do - and does it make any difference?

I'll leave some white space so you can't see the answer straight away, in case you want to think about it...

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Answer 1. The answer is that the contestant

*should*swap. For me - and I think for most people - this was really counter-intuitive when I first met it. However, after pondering it for a while, it switched from "no, I just don't believe it" to "yes, actually, it is pretty clear". The argument I used to understand it was as follows. After choosing a door, we now think about two possibilities:

(a) prize is behind the chosen door = win

(b) prize is behind one of the two not chosen = lose.

Since each door is equally likely to conceal the prize, the prob of (a) is 1/3 and the prob of (b) 2/3.

When Monty opens one of the doors this doesn't change these probabilities, but now possibility (b) applies only to one door. So if the contestant swaps they get a 2/3 prob of winning rather than 1/3.

It is more obvious if you imagine lots of doors - say 10. The prize is much more likely (9/10) to be behind one of the nine not chosen than the one chosen (1/10). If Monty then opens the eight others, it is probably behind the one he didn't open among the (b) doors, rather than one (a) door.

2. With the earthquake. Now, same as the original problem, but just before Monty opens a door, there is an earthquake that opens a door by chance. This happens to be one the doors that the contest did not choose, and the prize is not behind it. What should the contestant do now?

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

This time it makes no difference whether the contestant swaps or not! It is the same probability - 1/2 - either way. Again, I couldn't believe this at first, but, now, what do know, it seems perfectly reasonable. But no time to explain now. Will come back at a later date. [In the next post]

## No comments:

Post a comment