A friend recently posed a question to me.  It went like this:

You’re on a game show.  The host reveals three doors.  He tells you that behind two of the doors are goats and behind the third is a beautiful new car.  You get to choose a door and whatever is behind it you get to keep.  At this point your chances of winning a car are one in three.  You choose a door but don’t open it.  At this moment, the host opens another door revealing a goat.  He then gives you the option of sticking with your original door or switching to the other unopened door.

You would think that your chances of winning the car are one in two or 50%.  You know that there are two doors, one hiding a goat and the other hiding a car.  The car is just as likely behind your door as behind the other: 50/50.

You would be wrong.

This is called the Monty Hall Problem and it, and other questions like it, are the subject of an entire field of economics called game theory which deals with questions of strategy and probability.  Usually the solutions are counter-intuitive.  They go against your common sense.  Indeed sometimes they make no sense at all, but they are mathematically proven to be true.

So how can choosing between two doors be anything other than a 50% probability?  Think of it this way: every two out of three times you will choose a goat on your first guess.  That means two out of three times the host will have to reveal the only other goat.  Therefore, if you switch when given the chance you will win a car two out of three times, or 66.6%.

All the proofs say the same thing: switching improves your chances of winning.  Yet people continue to debate the point.  They are looking at two doors.  They know one has a car behind it and the other a goat.  It really seems like there is a 50/50 chance of winning.  But there’s not.  This is another failure of common sense.

As I’ve put it before before: some say common sense isn’t all that common but I say it doesn’t make much sense.