Subj:    Monty Hall Problem
         From: Tom's Webpage on 9/2/2005
         At: http://planettom.home.mindspring.com/webtoys.htm
         Source: http://planettom.home.mindspring.com/vossavant.htm
         also called
         Marilyn Vos Savant's Monty Hall (3 doors - 2 goats) Problem

 
 


 

Picture from Grand Illusions		 This story is true, and comes from an American TV game show. Here is the situation. Finalists in a TV game show are invited up onto the stage, where there are three closed doors. The host explains that behind one of the doors is the star prize - a car. Behind each of the other two doors is just a goat. Obviously the contestant wants to win the car, but does not know which door conceals the car.

The host invites the contestant to choose one of
the three doors. Let us suppose that our contestant chooses door number 3. Now, the host does not initially open the door chosen by the contestant. Instead he opens one of the other doors - let us say it is door number 1. The door that the host opens will always reveal a goat. Remember the host knows what is behind every door! 

The contestant is now asked if they want to stick with their original choice, or if they want to change their mind, and choose the other remaining door that has not yet been opened. In this case number 2. The studio audience shout suggestions. What is the best strategy for the contestant? Does it make any difference whether they change their mind or stick with the original choice? 

The answer to this question is not intuitive. Basically, the theory says that if the contestant changes their mind, the odds of them winning the car double. And over many episodes of the TV show, the facts supported the theory - those people that changed their mind had double the chance of winning the car. 

Why should this be so? After all, the contestant doesn't know which door the car is behind, and so the chance of the car being behind any one particular door is one third, isn't it? So surely the chance of winning the car if they stick with their original choice is one third, and the chance of winning the car if they change their mind is also one third? How can the odds double?

To help you check this amazing answer, try playing a fair copy of the game
below.

Door #1 Door #2 Door #3
 
     
  Start from scratch
     
 
 
 
     
 
 
One way to understand that two-thirds is the correct answer is the following:

Let's say that you choose your door (out of 3, of course). Then, without showing what's behind any of the doors, Monty says you can stick with your first choice or you can have both of the two other doors. I think most everyone would then take the two doors collectively.  When Monty opens one of the two doors, shows you the goat, and lets you switch doors, he is effectively letting you choose both of the other two doors.

A second explanation for why you should switch doors can be found at the
site http://math.ucsd.edu/~crypto/Monty/montybg.html
Thanks Ricky for sending this solution.

I want to thank Anon Jr. for debugging the source code errors.