Y5

Suppose that you roll a die. If the number is even you win $10, if the number is odd you lose $10. a) Compute the expected value and variance of this lottery. (Hint: the probability that a die roll is even or odd is 0.5.)

b) Now consider a modification of this lottery: You roll two dice. For each roll, you win $5 if the number is even and lose $5 if the number is odd. Verify that this lottery has the same expected value but a smaller variance than the lottery with a single die. (Hint: the rolls are independent, so the probability that one die is even and the second is odd is 0.50.5=0.25, and a similar calculation applies to the other possibilities.) Why does the second lottery have a smaller variance?