Monte Hall Pass

Keywords: probability, uncertainty

Let's Make a Deal

Here is a puzzle that often stumps people that can illustrate some ideas about probability. It is the "Monte Hall" problem: There are 3 closed doors, behind one of the doors is a prize, and behind the other two are gags. You are asked to pick a door. You reason (correctly) that the prior probability of the prize behind each door is the same: 1/3. So you pick door 2 because you like its color.

Monte then opens door number 3 and reveals a gag. You are relieved, but not for long. He poses the question, "Do you want to change your choice from door 2 to door 1?"

Do you? (Try to establish what you think before reading on.)

You ask your two friends, Simplicio and Salviati for their advice. Simplicio says, "Monte eliminated one of the choices; We had three, now we have two. Now it is between door 1 and door 2. It's obvious that the prize could be behind either - its 50-50 which. You like door 2's color. It was your pick. It doesn't matter so you might as well stay."

Salviati frowns but being a good-natured fellow he tries not to criticize. He just clears his throat, smoothes his frock and slowly starts talking. "I think Monte has provided some very useful information. You see, Monte knows which door hides the prize even though you do not. Before Monte opened door 3, there was a 1/3 chance that you had hit the prize door with your guess, and a 2/3 chance that you missed the prize. Don't you agree, dear Simplicio?"

Simplicio nods.

"Now that Monte has opened a door, it really does not change the fact that you had and have a 1/3 chance of being right. Monte revealed no new information about whether the prize is behind door 2."

"If you do not switch, 1/3 is your probability to get the prize. However, if you have missed (and this with the probability of 2/3) then the prize is behind one of the remaining two doors. Furthermore, of these two, Monte has opened one, leaving the prize door closed. Therefore, if you have guessed incorrectly and now switch, you are certain to get the prize. Summing up, if you do not switch your chance of winning is 1/3 (the same as when you first picked door 2) whereas if you do switch your chance of winning is 2/3."

"Another way of putting this is: Suppose Monte instead of opening door 3 had offered to combine doors 1 and 3 and said you could keep the contents of the combined doors if you switched as long as you gave him back the gag. Then you would switch in an instant because the probability of the combination of doors 1 and 3 having the prize would be 2/3 whereas sticking with door 2 would have a probability of winning remaining 1/3."

Simplicio has learned something about conditional independence and how probabilities change when new information is obtained. This is the essence of probabilistic logic - correctly reasoning about probabilities under partial information.

And, by the way, cognitive scientists particularly like this problem. When people are asked whether they should switch, somewhere in the high 90s percentage report that it does not matter or believe that they should not switch regardless of their level of education. A sobering thought when you consider that life and death decisions are made by people who do not correctly reason under very simple circumstances. And, it should give some pause to anyone reading books and articles advocating that you "just blink" to make decisions.