A friend offers you a chance to win a prize. The game works as follows: There are three boxes, labelled 1, 2, 3. A single prize has been hidden by the friend inside one of them at random. You select one box, however it is not immediately opened. Instead your friend will open one of the other two boxes, and, knowing where the prize is, will do so in such a way as not to reveal the prize. At this point, you will be given a chance to change your choice of box: you can either stick with your first choice, or switch to the other closed box. All the boxes will then be opened and you will receive whatever is inside your final choice of box.

So let's say you first choose box 1; then your friend opens box 3, revealing nothing, as promised.

What should you do:

a. stick with box 1

b. switch to box 2

c. does it make no difference?

You may assume that initially, the prize is equally likely to be inside any of the 3 boxes. Prove that your answer is correct. If you had to pay $1 for a chance to play this game and you always played it optimally (if possible), how much would the prize have to be in order for you to, on average, make money by playing? Prove that your answer is correct. Hint: Start by defining two random variables, one which indicates which box the prize is inside and one which indicates which box the host opens. Then compute the probabilities of the prize being in each box, given the box that the host opened