Wednesday, 5 December 2007

The Monty Hall Problem

I stumbled on this paradox by accident. If it's been doing the rounds in an annoying email, then apologies. But it was new to me and I love it. It is so counter-intuitive that it does my head in, and I think it will yours too.

Imagine you're on a game show. For a prize, you have to choose one of three doors. There is a car behind one door, and a goat behind each of the other two. The game show host knows what is behind each door.

You tell the game show host which door you have chosen. He opens one of the other two, revealing a goat. He then gives you the opportunity to change your mind and choose a different door. Should you change your mind?

You know there is a goat behind the opened door. You therefore have a choice between two doors, one with a car behind and the other with a goat. That looks like a fifty-fifty chance, with no particular advantage in swapping. Right?

Not right. Actually, you should always switch doors. There was a two in three chance that you picked a goat in your original choice. Whether you picked the car or a goat, there was always going to be a spare goat door, and when the host opened it, it did not change the 2/3 odds that you had picked a goat. But once a goat has been revealed, switching to the other door gives you a better than fifty-fifty chance of picking the car. Since the door you picked is most likely to contain a goat, and a goat has been revealed behind another door, the last door is odds on to contain the car. Statistically, switching doors brings a win two out of three times.

When Marilyn vos Savant (then listed in The Guiness Book of Records as possessor of the world's highest IQ), included the correct solution to this problem in her 'Parade' column, 10,000 readers, including maths academics, wrote to the magazine claiming she was wrong. Most subsequently conceded she was right.

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