The Game Show

The Game Show puzzle has bewildered some of the greatest mathematicians. It's solution was so counter-intuitive that it was not accepted by the academic community for quite some time. See if you can get your head around it:

You are on a game show. In front of you are three boxes, one of which contains the keys to a brand new Porsche. If you pick the box with the keys, the car is yours. The game show host gets you to point to one of the three boxes. Now, before checking the box you have chosen, the game show host removes one of the other two boxes, and this is a box which does not contain the keys. So, if you choose box A, then the host will remove either box B or box C, depending on which box does not contain the keys. If the keys are in box A, then the host will remove either B or C - it does not matter which.

Now comes the dilemma. After the host has removed a box, should you change your decision? Does this increase your chances of winning the car?

People well-learned in probability would instinctively claim that it doesn't matter whether you change your decision or not: the probability of choosing the keys is always 1/3. But this is in fact false. You actually increase your chances of winning if you change your decision. The probability of choosing goes from 1/3 to 2/3.

But why? The probabilities for all the boxes should be 1/3. Well, probabilities are not concrete properties of objects - they change with situations. After the host removes a box, he has given you a new piece of information. Before he takes it away, the probability that the keys were in one of the two boxes which you did not choose was 2/3. So when he removes the box, the probability that it is in the box that he does not take away becomes 2/3. So there you have it - logic works in funny ways.