At Pharaoh's Palace, a hypothetical Las Vegas casino, management has tampered with the dice. But management announced its tampering: It modified a standard six-sided die so that the probability of each side is proportional to the number of spots on the side. [That is, the probability of a 2 is twice the probability of a 1; the probability of a 3 is three times the probability of a 1; etc.]

1. What is the uncertainty distribution for the number of spots on the up side when the die is thrown? Write down all of the possible outcomes and their probabilities. [Hint: The probabilities must add up to 1.]

2. Calculate the mean of the uncertainty distribution.

3. Calculate the standard deviation of the uncertainty distribution.

4. Suppose that I offer to pay you $1,000 for each spot showing on the up side of a single throw of the die. But I will charge you $4,000 for the right to play the game with me. Would you accept my offer to play? Why or why not?