The Three-Door Problem and Bayes’ Law
The three-door problem (as known as the Monty Hall problem) is an intriguing and counter-intuitive thought experiment of how Bayes’ Law works. It sets up an imaginary scene: you are on a TV show, and you want to get a super large prize. To get it, you have to guess which of the three doors on the stage has the prize behind it. When you select one door, the host would open a door that does not have a prize, and ask you to guess again. The question is: is it more likely to get the prize by switching your initial choice?
Before you learn about Bayes’ Law, you probably would say something like, “It would not affect the probability of me choosing the right door, because the remaining two both have the probability of 1/2 (1/3 before the host acted) of containing the prize.” It may sound valid, but in fact, it is not. Initially, the probabilities for each door to have the prize are the same: 1/3. However, by opening one door that certainly does not have a prize, the probability of getting the prize would increase if you switch. But why?
This is a problem that Bayes’ Law applies to:
Assume there are three doors: A, B, and C; assume you choose door A, event X = you chose correctly (the prize is behind A), and event B = host opens B (which means to eliminate B as a wrong choice);
P(X|Y) = P(Y|X)P(X)/P(Y), according to Bayes’ Law;
P(Y|X) = 1/2, because the prize could be either behind door A or C;
P(X) = 1/3, obviously;
P(Y) = (1/3)*(1/2)+(1/3)*1, because if you chose correctly, the probability of the host choosing B is 1/2; if you chose wrong, the host would have only one choice left.
Therefore, P(X|Y) = (1/2)*(1/3)/(1/2) = 1/3. This is the probability of getting the prize if you didn’t switch. Surprisingly, this also implies that the probability of getting the prize after switching is 1-1/3 = 2/3. The probability increased if you switch!
If we use less a mathematical way to solve the problem by listing all the possibilities, we would discover that the result coincides with the result calculated by Bayes’ Law. If we write a trivial computer program using random numbers to simulate the experiment and run a large number of times (say 10,000 times), the result would still be the same. This is probably the magic of Bayes’ Law, and a proof that Bayes’ Law is always more reliable than intuitions.
Source: http://vp5qw4uf5x.scholar.serialssolutions.com/?sid=google&auinit=T&aulast=Slembeck&atitle=Do+institutions+promote+rationality%3F:+An+experimental+study+of+the+three-door+anomaly&id=doi:10.1016/j.jebo.2003.03.002&title=Journal+of+economic+behavior+%26+organization&volume=54&issue=3&date=2004&spage=337&issn=0167-2681