h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I've recently again read about a problem that baffles a huge number of people, including actual mathematicians and statisticians. I'd like to try it on my fellow-h2g2ers. Here it is.

It's a game show. There are three doors. Behind one of the doors there is a fancy car; behind two of the doors there are goats. The game show host knows what's behind each door but the contestant doesn't.

The contestant picks a door, any door.

The host then opens one of the two doors the contestant hasn't picked - *he* picks a door that has a goat behind it. So now the contestant knows at least that that particular door doesn't hide the car. But now he's left with a choice of two remaining 'unknown' doors: the one he originally picked, or the other one, that the host hasn't opened.

He now gets a chance: he can keep the door he picked first, or he can switch his choice to the other door, the one the host hasn't opened. Should he switch - does he improve his chances if he does?

Can you answer this - and if you can, can you give a simple and clearly understood explanation of why? Because apparently this baffles a heck of a lot of people, and I think I can give a clear and concise explanation of the solution.