he numbers racket is a well‑entrenched illegal gambling operation in most large cities. One version works as follows: you choose one of the 1000 three‑digit numbers 000 to 9999and pay your local numbers runner a dollar to enter your bet. Each day, one three‑digit number is chosen at random and pays off $600.off $6. The law of large numbers tells us what happens in the long run. Like many games of chance, the numbers racket has outcomes that vary considerably—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
he numbers racket is a well‑entrenched illegal gambling operation in most large cities. One version works as follows: you choose one of the 1000 three‑digit numbers 000 to 9999and pay your local numbers runner a dollar to enter your bet. Each day, one three‑digit number is chosen at random and pays off $600.off $6.
The law of large numbers tells us what happens in the long run. Like many games of chance, the numbers racket has outcomes that vary considerably—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 350days a year for 40 years, he makes 14,000 bets.
Unlike Joe, 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 is 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.
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? Use Table A to answer the question. Give your answer to four decimal places.
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