Tuesday, April 8, 2008

The Problem that Refuses to Die

There are some problems that so strange that your brain almost seeks out the wrong answer. Perhaps the best known is the so-called Monty Hall problem.
It's been written up (yet again) in the NYT, and as usual I found myself thinking about it incorrectly before catching myself. So here it is:

You are presented with three closed doors. Behind one of them is a prize -- the other two get you nothing. A host asks you to pick a door, which you do hoping to win the prize. But the host doesn't open that door. He says he wants to give you a better shot at winning the prize, and he opens one of the remaining doors that does not have a prize. He then gives you two options. You can change your guess to the third door, or stick with your original guess. Should you switch or stick with your decision? Here are two ways of looking at the problem:

  1. There is no point switching since the host can always show you a door with no prize. He really isn't giving you any more information. Whether you switch or not, you win half the time. It all seems so obvious, and this is what most people will say.
  2. If you picked the right door initially, then switching will always make you lose. But if you picked the wrong door initially then you always win when you switch. But the chance that you picked the wrong door initially is 2/3. So if you switch you win the prize with probability 2/3. If you don't switch you win only if you had picked the correct door initially, and that happens with probability 1/3. So you are twice as likely to win if you switch.
The correct way to look at this problem is 2 NOT 1. The host DOES give you valuable information when he shows you the empty door, since he gives you a way to win 2/3 of the time.

But there are legions of people (including professional mathematicians) who will still argue 1 (or some variant of it). They can't see 2! Why?
Click the widget below to see more information on the problem, and to play the game yourself!