Question

In: Math

The numbers racket is a well‑entrenched illegal gambling operation in most large cities. One version works...

The 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 999 and 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 . 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 350 days 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'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.

(a) What is the mean of Casper's average winnings ?¯ on his 150,000 bets? (Enter your answer as dollars rounded to two decimal places.)

mean of average winnings=

What is the standard deviation of Casper's average winnings ?¯ on his 150,000 bets? (Enter your answer as dollars rounded to three decimal places.)

standard deviation=$

(b) 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 ? (Enter your answer rounded to four decimal places.)

approximate probability=

Expert Solution

(a) What is the mean of Casper's average winnings ?¯ on his 150,000 bets? (Enter your answer as dollars rounded to two decimal places.)

mean of average winnings= is 0.40 , as per the central limit theorem , the mean of the sample will be close to the population mean , when sufficient large number of samples are taken

What is the standard deviation of Casper's average winnings ?¯ on his 150,000 bets? (Enter your answer as dollars rounded to three decimal places.)

standard deviation=$

is given as SD/sqrt(n) = 18.96/sqrt(150000) = 0.0489 cents

(b) 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 ? (Enter your answer rounded to four decimal places.)

approximate probability=

now we know that the z score is

z = (x - mean)/sd

= (0.30 - 0.40)/0.0489 = -2.044

and

z = (x - mean)/sd

= (0.50 - 0.40)/0.0489 = 2.044

please keep the z tables ready now

Left-tailed p-value: P(Z < z) = 0.0204768
Right-tailed p-value: P(Z > z) = 0.0204768

area between the region is thus

1 - 2*0.0204768 = 0.9590

milcah answered 7 months ago

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