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
0.5×0.5=0.25, and a similar calculation applies to the other possibilities.) Why does
the second lottery have a smaller variance?

 

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a For a single die roll Probability of even 05 Probability of odd 05 Expected value if ... View the full answer