An article in the New York Times, And Behind Door No. 1, a Fatal Flaw, reports on a forthcoming paper that argues that the outcome of experiments dealing with monkey choice of M&M color are due to statistics rather than psychology. The article relates this to the Monty Hall Problem, which is a fun counterintuitive mental exercise.
Here’s how Monty’s deal works, in the math problem, anyway. (On the real show it was a bit messier.) He shows you three closed doors, with a car behind one and a goat behind each of the others. If you open the one with the car, you win it. You start by picking a door, but before it’s opened Monty will always open another door to reveal a goat. Then he’ll let you open either remaining door.It is easy to think that the choice boils down to a 50-50 shot. However, if you switch you will win 2/3 of the time. My way of thinking of it is this:
Suppose you start by picking Door 1, and Monty opens Door 3 to reveal a goat. Now what should you do? Stick with Door 1 or switch to Door 2?
By choosing one door, you only have a 1/3 chance of choosing the correct door. This means that there is a 2/3 chance that you are wrong. Now let's say that it is your policy to always switch when given the option. When you switch you are, in effect, choosing the other two doors--you just get to discard the incorrect door.
I wish I was perspicacious enough to have seen that when I first encountered the problem, but I fell into the 50-50 trap. Anyway, the monkeys and M&Ms are interesting too. There is a nice graphic that explains that issue quite clearly.
1 comments:
For me, a simpler way to think about this is to note that the choice of which door the host will open is not random 2/3 of the time, because he will presumably only open a door with a goat behind it, and he knows which door has the car. It is only random if you are initially correct (Pr=1/3). If you are initially wrong (with 2/3 probability), the host is telling you which door the car is behind when he opens a door with a goat behind it. Thus, the probability that you will be right when you switch doors is 2/3. The problem is about using the hosts knowledge to your advantage.
Changing the assumptions about the hosts knowledge and behavior changes the results of the problem. For example, if the host does not know which door has the car, picks randomly, and happens to hit one with a goat, your odds do not improve by switching doors.
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