Sat, 30 Jul 2005

The Monty Hall Problem

I doubt many people think that probability can be the slightest bit interesting, but check out this little gem from the folks at Grand Illusions: the Monty Hall Problem.

The problem goes like this: you are a contestant on a game show and are playing for the grand prize. You have to choose one of three doors. Behind one door is a brand new car. Behind the other two doors is a goat.

Having made your choice, the host proceeds to play up the suspense. Instead of showing what's behind the door you selected, he decides to open a different door which, of course, reveals a goat. He then offers you "one last chance" to change your mind. The audience and enthusiastic people watching at home shout you advice, and you are left doubting and re-doubting and creating great TV drama.

So the question is, what should you do? Given that the host will always reveal a goat from behind one of the remaining doors (otherwise where's the suspense?), should you change your mind or stick with your original choice? Does it even matter?

Think about it before reading on...

Most people's initial response is something along the lines of "How can it make any difference? There is equal probability that it car is behind either door - the odds are 50/50.". My reaction was something along those lines. Given that many professional mathematicians and even the great Cecil Adams have also responded this way, I dont feel too bad about it...

If you said "Change your mind!" then you're doing really well. Theory says that if you change your mind then your chances of winning are doubled. Despite years of experience with actual game shows and the ease with which one can conduct experiments, this simple problem remains sure to stir up controversy!

It seems that our intuition about probability lets most of us down in questions like this. It is easy to see that you have a 1/3 chance of picking the winning door initially. It is also easy to see that the odds of it being the winning door cant magically change after the fact. The bit we seem to miss out on is that the game really only contains two options ("stay" or "switch") and one of them only has a 1/3 chance of success!

For a detailed explanation, check out the Monty Hall Problem page on Grand Illusions or Wikipedia. For a quick-and-dirty explanation, think about it like this:

  • If you initially picked the car, then changing your mind will cause you to lose.
  • If you initially picked a goat, then changing your mind will cause you to win.
  • There is a 2/3 chance that you initially picked the goat.
  • Therefore, there is a 2/3 chance that changing your mind will cause you to win!

This is a simple example of the usefulness of Conditional Probability, brought to you by We make learning fun!