Problem: Playing the numbers: the house has a business. Unlike Joe (see the previous exercise) the operators of the numbers racket can rely on the law of large numbers. It is said that the New York City mobster Casper Holstein took as many as 25,000 bets per day in the Prohibition era. That's 150,000 bets in a week if he takes Sunday off. Casper's mean winnings per bet are $0.40 (he pays out 60 cents of each dollar bet to people like Joe and keeps the other 40 cents.) His standard deviation for single bets is about $18.96, the same as Joe's.
Required:
(i) What are the mean and standard deviation of Casper's average winnings x on his 150,000 bets?
(ii) According to the central limit theorem, what is the approximate probability that Casper's average winnings per bet are between $0.30 and $0.50? After only a week, Casper can be pretty confident that his winnings will be quite close to $0.40 per bet.