The excellent Skeptic’s Guide to the Universe podcast has a section called ‘science or fiction?’ in which the panelists must figure out which one of three science ‘facts’ presented by the host is false. Last week one of them was unsure, so guessed option 3. The host then started going through the facts, and revealed that option 1 was true. The panelist immediately said ‘I want to change my answer’, which was about the best thing I’d heard all week. Unfortunately I then started thinking about it and found I’d lost my grasp of the maths.
It’s the old Monty Hall problem. You’re shown three boxes, one of which contains £1000 and the other two a goat. Your guess is pure chance – let’s say you choose Box 1. I then open Box 3 and reveal a goat. Should you change your guess? Probability says you should, but it’s completely non-intuitive.
I absolutely despised this problem when I first heard it. I can’t stand not understanding logic puzzles, and remember devoting an hour to trying to get a correct and intuitive understanding. I eventually did, but it’s one of those things you have to top up every so often or you forget.
Essentially, what you’re being asked is “there’s £1000 in one of these boxes. Do you want to open Box 1, or Boxes 2 and 3? By the way, it’s not in Box 3.” In the first place, the chances of the money being in Box 1 was 1/3, but the chances of it being in one of Boxes 2 or 3 was 2/3. Being told that it’s not in Box 3 doesn’t change the fact that it was originally more likely to be in one of Box 2 or Box 3, and now you know it’s not Box 3! So Box 2 still has a 2/3 probability, and you should change your guess.
The obvious thought is that the probability will revert to 50/50 when there are only two boxes remaining, but this is unintuitive when you increase the number of boxes. If there were 10,000 boxes, and you randomly guessed Box 1, then I removed 9,998 others and left only Box 3345, you can see that the remaining box is somewhat conspicuous. The money was always more likely to be in one of the other 9999 boxes, and I’ve removed 9998 of them that definitely contained goats. My removal of the wrong boxes has given you information.
I’m not sure how clear that is, but it’s what I use to get a grasp on the problem. There are plenty of other aids on the Wikipedia entry.