In one of three boxes there is a prize, the other two boxes are empty. You do not know which box contains the prize, but the host does. You must point to one of the boxes, which you think contains the prize. After this, the host opens one of the two remaining boxes. Since he does not want to give away the prize immediately, he opens an empty box. After this, you are offered to finally choose a box. Can you win the prize with a probability greater than $1 / 2$?
Showing the box for the first time, you most likely (with a probability of 2/3) made a mistake.

## Solution

You can win the prize with a probability of 2/3. Suppose you pointed to box A first, and the host opened an empty box B. Then you should choose the third box (i.e., the box different from A and B). Indeed, you will only be wrong if the prize was in box A, but initially, you should assume that the prize is in each box with a probability of $1 / 3$.