The Game Show

Imagine a game show. It's the kind of show where you have to choose a box out of three options. You know one of them has a car and the other two contain goats. You also know that after you have made your choice, the host of the show will reveal a goat (not yours) and then give you the option of keeping your original box or switching to the other remaining box. My question is this: Would switching boxes increase your chances of winning the car, decrease it or make no difference?

Though most people insist it would make no difference, the remarkable truth is that by switching your chances of getting the car double.

At the start your box had a one in three chance of being the car. If this is the case switching is bad. It has a 2 in three chance of being a goat. In this situation, switching is good. So there's a two in three chance you win with a switch. And only one in three chance of winning if you stick.

Can you prove it using a tree diagram? Start off with the choice between car, goat and goat. Next branch off with which box the host opens. Then add up the probabilities that you get the car. Finally, see what the probability will be if you switch.

How did you do?

*This problem is a famous Maths puzzle called the Monty Hall problem. If you can get your head around this, then conditional probability should be no problem for you.