You are shown three boxes, and told that one of them contains a prize. You are then asked to pick one box, and if that box is the one with the prize, you will win it. After picking a box, you are shown that one of the other two boxes is empty, and offered a chance to change your selection.

Should you do this? Would changing your choice to the other remaining box affect your odds of winning? Why or why not?

This is the only way to write a convincing solution:

There are three boxes. You pick one. Call this box A. The host shows you one of B or C (the remaining boxes). Here are the three posibilities.

1) Prize is in Box A
The host will show you one of B or C (it doesn't matter which). You win if you don't change.

2) Prize is in Box B
The host shows you box C. You win if you switch.

3) Prize is in Box C
The host shows you box B. You win if you switch.

Since each of these three events is equally likely to happen, and on two of them you win if you switch, you are twice as likely to win if you switch as if you don't switch.

This may seem absurd, but the reason it is not 50% is that you know that you picked a box and you know which of the other boxes is empty. If someone walked in after the box was opened, and was asked to pick a box, they would have a 1/2 chance of picking correctly. But since you are armed with the knowledge that you picked one of the boxes before hand, you have a 1/3 chance of winning if you switch.

Posted by Iain on 2004-05-09 14:24:09