6.3 A roulette wheel has 38 slots: 18 are black, 18 are red, and 2 are green. A gambler bets that the ball will land in a black slot. He will win $10 if the ball lands in a black slot, otherwise he will lose $10. If the random variable X denotes his winnings on any spin of the wheel, the probability distribution of X is clearly x : -10 10 P(X = x): 20 38 18 38 Suppose the gambler plays 20 times. Let X denote his mean winnings for the 20 spins. (a) On average, how much does the gambler expect to win for each play? In other words, find the mean of X, = E(X). (b) Find the standard deviation of X, o = SD(X). (c) What is the expected value of his mean winnings for the 20 plays? In other words, find the mean of X, ux = E(X). (d) Find the standard deviation of his mean winnings X, ox = SD(X). (e) On average, how much is his expected total winnings (or loss) for the 20 spins?