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!


dicky said...

That was coool!!!! I tried to ignore your logic, played the game and did not switch. What happened? 6 out of 8 times I lost. When I listened to your suggestion and switched when presented with the option, 9 times out of 10 I won.

Neat game. I think I'm ready to go on the game show!

michaelmross said...

Well I stuck with the same door and won 8 times out of 6. So a tiny sample means nothing. This is a coin toss. The coin does not have 3 sides. The 3rd door - the one that gets eliminated - simply does not figure in the equation. There are only 2 choices!

Abhay Parekh said...

Michael, use the "automate" feature on the game and wait for about 100 times. You'll see that you will win about 1/3 of the time.
If you switch you'll win about 2/3 of the time.

michaelmross said...

The odds are 50-50 for 2 doors. If you'd rather put your faith in some pseudorandom algorithm - or a psychologist - you are making a grave error of judgment. The only caveat one could reasonably make to that is if the program permits "Monty" to use a bait-and-switch tactic. It's quite foolish to make things more complicated than they actually are. However, the voluminous discussion in the Times forum says a lot about the fickleness of the human condition and the vanishingly small chances that people will agree on anything...!

Anonymous said...

Wow, no need to get huffy about the matter.

Take a look at this Wikipedia image-- http://en.wikipedia.org/wiki/Image:Monty_tree_door1.svg -- which displays the decision tree for the Monty Hall Game. (Decision trees are a excellent and illuminative tool when grappling with statistics/expected values)

Abhay Parekh said...

Well, in defense michaelmross I have to say that a LOT of people just don't seem to want to accept that switching is a much better strategy, but that's fine....one can only persuade so much :-).

Decision trees are indeed an excellent way to explain this. Thanks.

michaelmross said...

Quoting from the Wikipedia article in which that decision tree is used: 'Asked whether the contestant should switch, vos Savant correctly replied, "If the host is clueless, it makes no difference whether you stay or switch.(vos Savant, 2006).' This has been my position and assumption all along. I did not invest Monty with any intelligence whatsoever. However, if we assume that Monty has some interest or control over the outcome, yes a decision tree becomes relevant. However, this is then a discussion about game strategy in general, not probability in particular.

Abhay Parekh said...

ah ok.

So if Monty doesn't open the door but just points to it, AND he's clueless, then michael, you are right, there is no advantage to switching.

But if Monty ALWAYS OPENS a door with no prize then you agree that it is better to switch, right?

This is true even if he is clueless as long he OPENS the door to show you what's behind it. That's because if he opens up a door with a prize, you've clearly lost (whether you switch or not), but if he opens a door with no prize then switching is better.