Wednesday, June 21, 2006

The Monty Hall Problem

If you want a break from thinking about infinitely repeating decimals, why don't you work on the Monty Hall problem? Imagine that you are on Let's Make a Deal. You have three doors to choose from: A, B, and C. Behind one door is a bar of gold and donkeys are behind the other two.

You choose a door, say A, and Monty Hall opens one of the doors you did not choose, we'll say B, to reveal a donkey.

The question is: knowing that the gold is definitely not behind door B, should you switch from your original choice and now go with whatever is behind door C?

Intuitively, my first guess was that it doesn't matter: that there is now a fifty-fifty chance that the gold is behind either A or C. That's not how the numbers work, though. In reality, you vastly increase your chances of winning if you switch doors, in our case to door C.

The explanation is that you made your original guess without knowing anything about the three doors. There are two donkeys and one gold bar, so your odds of winning (if you don't switch) are 1/3. If you do switch, you're taking advantage of the fact that Monty has taken one of the donkey doors out of play. The only way you lose, if you switch, is if you happened to pick correctly the first time. Because the odds of that happening were 1/3, by switching, you've increased your odds of winning to 2/3. Those are fantastic odds.

Here's a link to a good representation of the Problem.


trace said...

can you apply this same probability theorem to "Deal or No Deal" as well? if so, please contact dave and carri

i said...

I saw this explained by Marilyn vos Savant a long while back, who reported that it was one of the most criticized puzzle responses she had put up. It's not intuitive, but it does make sense.

Anonymous said...

Looking at the linked explanation (the color wheel) it does statistically make sense (fighting my intuition). Of course, a key assumption that I missed initally is that Monty knows the location of the prize.

It reminds me of some of the families that come to L&D at the hospital. They think (errantly) that since they had 4 girls already, then this pregnancy will have a greater chance of yeilding a boy.

Thanks to you, I am now constructing a color wheel illustration to show to pregnant couples.


Anonymous said...

I do see your point, Matt. If Monty happens to open a door that has a donkey - regardless if he is in the dark like the contestant - you are in the same situation from that point onward.

Man, I really am having a difficult time accepting this. There must have been a million letters sent to the author...