Wednesday, February 6, 2008

The Monty Hall Problem

It's 1975 and you are dressed as a chicken attending Let's Make a Deal with Monty Hall. Monty picks you and offers you the choice of 3 doors - behind 1 door is a new car, behind the other 2 doors are goats. You choose your door and Monty, knowing what's behind the doors, opens a door you didn't choose revealing a goat (in the game, he always reveals a goat behind one of the unselected doors to raise suspense). He then asks you if you want to change your choice and switch to the other remaining door. He taunts you by calling you a chicken. Should you switch? Does it matter?

In fact, it does matter and you should switch doors. By switching, you double your chances of winning the new car. This may seem crazy, how could your odds be anything other than 50-50? That's the Monty Hall problem - the famous, confounding probability paradox.

Every time the Monty Hall problem is published, it generates lots of incredulous mail. Steve Selvin originated the paradox in Monty Hall form in 1975 but a 1990 letter to Marilyn vos Savant's Ask Marilyn column in Parade magazine made it famous. The resolution of the paradox was contested in thousands of letters to Parade including nearly one thousand letters from Ph.D.s. That's because the problem hits a blind spot in our intuitions. Recently Michael Shermer printed the problem in his review of Leonard Mlodinow's The Drunkard's Walk and he received a record number of letters. Here is Shermer's nice explanation:

The James Madison University mathematics professor Jason Rosenhouse, who has written an entire book on the subject—The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brainteaser (Oxford University Press, 2009)—explained to me that you double your chances of winning by switching doors when three conditions are met: (1) Monty never opens the door you chose initially; (2) Monty always opens a door concealing a goat; (3) When the first two rules leave Monty with a choice of doors to open (which happens in those cases where your initial choice was correct) he makes his choice at random. “Switching turns a loss into a win and a win into a loss,” says Rosenhouse, “and since my first choice is wrong 2/3rds of the time, I will win that often by switching.”

Here’s why: At the beginning of the game you have a 1/3rd chance of picking the car and a 2/3rds chance of picking a goat. Switching doors is bad only if you initially chose the car, which happens only 1/3rd of the time. Switching doors is good if you initially chose a goat, which happens 2/3rds of the time. Thus, the probability of winning by switching is 2/3rds, or double the odds of not switching (keeping in mind the three rules above). Analogously, if there are 10 doors, initially you have a 1/10th chance of picking the car and a 9/10ths chance of picking a goat. Switching doors is bad only if you initially chose the car, which happens only 1/10th of the time. Switching doors is good if you initially chose a goat, which happens 9/10ths of the time. Thus, the probability of winning by switching is 9/10ths, again, assuming that Monty has shown you 8 other doors with goats.

Here is a diagram from John de Pillis:

The lessons: 1) your intuitions are not reliable 2) never try to match wits with Monty Hall.

(image credit: wikipedia.org; diagram: John de Pillis)