The law of large numbers tells us what happens in the long run. Like many games of chance, the number racket has outcomes so variable -- one three-digit number wins $600 and all others win nothing -- that gamblers never reach "the long run." Even after many bets, their average winnings may not be close to the mean. For the numbers racket, the mean payout for single bets is $0.60 (60 cents) and the standard deviation of payouts is about $18.96. If Joe plays 350 days a year for 40 years, he makes 14,000 bets.

(a) What are the mean and standard deviation of the average payout x-bar that Joe receives from his 14,000 bets?

(b) The central limit theorem says that his average payout is approximately Normal with the mean and standard deviation you founding part (a). What is the approximate probability that Joe's average payout per bet is between $0.50 and $ 0.70? You see that Joe’s average may not be very close to the mean $0.60 even after 14,000 bets.