## The Monty Hall Problem

| May 16th, 2008I assume that most of you have watched the movie ‘21‘. Well if you haven’t, I recommend that you watch it. Around 20 minutes into the movie there is a scene where the professor poses a problem in front of the protagonist:

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 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? [via Whitaker, Craig F. (1990). [Letter]. “Ask Marilyn” column, *Parade Magazine* p. 16]

The protagonist chooses to switch. It is not very intuitive as to why one should deviate from one’s original choice. It seems as if the probability of winning the car just bumped up to 1/2 from 1/3 after the host opens a door with a goat behind it. But it can be empirically proven that switching the choice is the rational strategy for the player. This ubiquitous game show dilemma is the Monty Hall problem.

Let me try to prove why is it so.