The Monty Hall Problem

The Monty Hall problem is named after it's similarity to some of the games that were played on the original Let's Make a Deal television show hosted by Monty Hall.  Imagine a room with three doors.  Behind two of the doors are goats and the third is a car.  You are asked to pick one of the doors.  Lets say you choose door #1.  The host, who knows what is behind each door then opens one of the other two doors which has a goat behind it.  In this case, lets say he opens door #3.  At this point the host then gives you an option.  Do you want to stay with door #1 or would you like to switch to door #2.  The question is, should you stick with your first choice, switch or does it even matter?

Most people (my self included) initially say that it doesn't matter one way or the other because intuitively it seems as if you have a 50/50 chance at getting the car whether you switch or not.  In reality, it is always advantageous to switch.  The easiest way to understand this is to break down it's probabilities.  When you originally chose door #1 there was a 1/3 chance of getting the car and a 2/3 chance of not getting the car.  Another way of saying this is that you have a 1/3 chance that the car is behind door #1 and a 2/3 chance that the car is behind door #2 or #3.  When the host opens door #3 to reveal the goat, you still have a 1/3 chance of the car being behind door #1 (not switching) but now a 2/3 chance that the car is behind door #2 (switching).

Here is a good video which explains it well



The story behind the problem
Another interesting part of this problem is the story behind it's rise to fame.  In her September 1990 column in Parade magazine, Marilyn vos Savant answered the following question posed by a reader:

"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 #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?"
Craig F. Whitaker
Columbia, Maryland

"Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?"

Many of the response letters that poured in disputing her answer were arrogant and pompous.  Often, the worst of these were from academics.  Here are a few examples:

"You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!"

Scott Smith, Ph.D 

University of Florida 

"Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful."

Robert Sachs, Ph.D. 

George Mason University 

Marilyn wrote another column providing more support to her answer but the negative responses kept coming:

"May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again?"

Charles Reid, Ph.D.
University of Florida

"I am sure you will receive many letters on this topic from high school and college students. Perhaps you should keep a few addresses for help with future columns."

W. Robert Smith, Ph.D.
Georgia State University 

"You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively towards the solution of a deplorable situation. How many irate mathematicians are needed to get you to change your mind?"

E. Ray Bobo, Ph.D. 

Georgetown University

But then, after some time had passed, the letters started to turn in favor of support:

"You are indeed correct. My colleagues at work had a ball with this problem, and I dare say that most of them, including me at first, thought you were wrong!" 

Seth Kalson, Ph.D 

Massachusetts Institute of Technology

"The teachers in my graduate-level mathematics classes, most of whom thought you were wrong, conducted your experiment as a class project. Each of the twenty-five teachers had students in their middle or high school classes play at least 400 games. In all, we had 14,800 samples of the experiment, and we’re convinced that you were correct —the contestant should switch!"

Eloise Rudy, Furman University
Greenville, South Carolina

In the end, after countless simulations were run, Marilyn vos Savant was proven right.  I'm sure her sense of feeling vindicated must have been quite sweet.

Update:
Here's another good site with easy to grasp explanations