06 October 2010

The world is wrong about the Monty Hall problem

Today, at school, we considered the probability theory paradox called the Monty Hall problem:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which he knows has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

The world says that by seeing that goat, you learn that the probability of the car being behind door number 2 is 2/3 and Wikipedia has various explanations.

I claim it's 1/2. My first argument is that there is full symmetry between doors 1 and 2. How could you lower the probability of door 1 being the winning one by picking it? Nonsense.

For my second argument, let's treat it like the conditional probability that it is. All probabilities will be from your, the contestant 's, point of view.

At the start, we've got 3 equally probable situations: I'll denote them as CGG, GCG and GGC hoping it's clear enough.

Now the host opens a door. You see the goat: you gain information. Probabilities change. The probability of the door number 3 being the winning one falls down to 0.

So far, no difference from the "official" solutions. However, those maintain that while the probability of the car being behind the first door stays the same, the probability of it being behind the second one increases. Why the incoherence?

An important remark here: with the new information, we now have new events. We know that GGC didn't happen, so we have 3 events that assume ~GGC, namely:
  1. if the first door wins, it's CGG | ~GGC and P(CGG | ~GGC) is 1/2
  2. if the second door wins, it's GCG | ~GGC and P(GCG | ~GGC) is 1/2
  3. if the third door wins, it's GGC | ~GGC and P(GGC | ~GGC) is 0,
the final results being basic conditional probability calculations.

According to Wikipedia, "even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer". That's comforting: I may be wrong, but at least I have excellent company.

[UPDATE] After reading all the comments, I can see that I was wrong. Boy, is that problem counterintuitive! Thanks everyone for contributing! [/UPDATE]