15.2. Suppose that you flip a coin. If it comes up heads, you win $10; if it comes up tails, you lose $10.

a) Compute the expected value and variance of this lottery.

b) Now consider a modification of this lottery: You flip two fair coins. If both coins come up heads, you win $10.

If one coin comes up heads and the other comes up tails, you neither win nor lose—your payoff is $0. If both coins come up tails, you lose $10. Verify that this lottery has the same expected value but a smaller variance than the lottery with a single coin flip. (Hint: The probability that two fair coins both come up heads is 0.25, and the probability that two fair coins both come up tails is 0.25.)

Why does the second lottery have a smaller variance?