There are three boxes and one has a prize inside. You are told to choose a box. One of the other boxes is then opened, showing that it is empty. You are given the option to switch your choice to the other remaining box. Should you switch? Why?

Let's think of the three boxes as A, B and C.
Say we choose box A.
There is a 1 in 3 chance that our box has the prize inside, and so a 2 in 3 chance that it does not.
When a box is opened and shown to be empty, this DOES NOT leave us with a 50/50 chance of choosing the prize box when given the choice of switching.
We started with a 2 in 3 chance of having not chosen correctly, and so we have a 2 in 3 chance of the switch being a good choice, as it will take us to the box with the prize, since that is the only remaining box. This means that the best choice is to switch our box.