# Monty Hall Problem - Page 1

This brings quite a few people out in a hot flush. Before you start sending us rude e-mails telling us we have got it wrong, please bear in mind that we didn't make this up. This is an old mathematical 'chestnut' that has been published quite a few times in the mathematics literature over the last 100 years.

This version of the 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, our 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. Now the door that the host opens will always reveal a goat. Remember the host knows what is behind every door!

Lucky you didn't choose THAT door, says the host - as you can see there is just a goat there. 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 tension in the studio rises, and the studio audience shout out suggestions. But amid all this excitement 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, conditional probability theory says that if the contestant changes their mind, the odds of them winning the car will double. If they stick with their original choice, they have a 1 in 3 chance of winning the car, and if they switch they have a 2 in 3 chance of winning. And over many episodes of the tv show, the facts supported the mathematics - those people that changed their mind did indeed double their chances of winning the car.

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