Monty Hall

Monty Hall

Picture the scene. You've made it through to the final round of a game show. There are three doors in front of you. Behind one of the doors is the star prize - a brand new car. There is a goat behind each of the other two doors. You make your choice, hoping to select the star prize. The game show host (who knows where the car is hidden) opens a different door to reveal a goat. The choice is now down to two doors. He asks whether you'd like to stick with your original choice or whether you'd like to switch doors... 

It is easy to assume that there is a fifty-fifty chance of picking the correct door first time round. In fact you are twice as likely to choose the car if you switch doors!

It's hard to believe, isn't it? After all, when the host opens one door there's a straight choice between two doors - one hiding a goat, the other the car. The argument that there's a fifty-fifty chance of winning is a very seductive one - but it's wrong.

The problem is based on a real-life scenario from the American TV gameshow Let's Make a Deal and gets its name from the show's host. The problem gained notoriety when it became the subject of the syndicated American newspaper column Ask Marilyn. Responding to a correspondent who posed the problem, Marilyn vos Savant provided a detailed explanation of the correct answer, explaining that there is a two in three chance of winning the car by switching doors.

The outcome is so different from our intuition that it is very hard to accept. Indeed, after her first explanation Marilyn received a vociferous postbag from a disbelieving public. Many letters came from indignant mathematicians who failed to agree with vos Savant.

The true result, however, is quite easy to verify. I've written a simulation of the Monty Hall Problem in Flash to try out for yourself.

Have a go! Choose a door, and decide whether to stick or switch when prompted. At the end of each trial you will be given a summary of the effectiveness of your strategy. Perform a large number of trials to get a more accurate result.

Not convinced? Here's my attempt at an explanation:

  • There is a one in three chance that the car is behind the door you originally picked.
  • There is a zero in three chance that the car is behind the door Monty picks (because he always picks a goat).
  • There must therefore be a two in three chance that the car is behind the remaining door.

But don't just take my word for it. There is a surprising number of sites dealing with the Monty Hall Problem, giving various explanations and providing CGI, Java and JavaScript simulations. Some sites also go into detail about the controversy the original article provoked. So there are several places where you can find a second, or third, opinion!