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 x¯ on his 150,000 bets? (Enter your answer as dollars rounded to two decimal places.)

mean of average winnings=$

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

standard deviation=$

(c) 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=

Check Explanation

Step-by-step explanation:

For the numbers' racket. It was given that the expected amount of winnings per bet of the owner of the game = $0.4 with a standard deviation of $18.96.

So, E(X) = $0.4

σ(X) = $18.96

where X is the random variable for the amount of winnings per bet.

a) Mean of Casper's average winnings x¯ on his 150,000 bets  

Expected average amount of total winnings is E(150,000X)

E(150,000X) = 150,000 × E(X) = 150,000 × 0.4

E(150,000X) = $60,000.00

On a per bet basis, Expected amount of winnings is = $0.40

b) Standard deviation of Casper's average winnings x¯ on his 150,000 bets

Standard deviation on the average winnings for the total winnings of the 150,000 bets = σ(150,000X)

σ(150,000X) = √150,000 × σ(X)

= √150,000 × 18.96 = $7343.176

On a per bet basis, the standard deviation = $18.96

But for the distribution of average winnings per bet for his 150,000 bets, standard deviation of Casper's average winnings per bet on his 150,000 bets is given by

σₓ = (σ/√n) = (18.95/√150,000) = $0.049

c) The approximate probability that Casper's average winnings per bet are between $0.30 and $0.50.

Using the central limit theorem, we can say that the distribution of average winnings per bet approximates a normal distribution.

Mean = μ = $0.40

Standard deviation of this distribution = σₓ = $0.049

To obtain the required probability,

P(0.30 < x < 0.50)

We obtain the standardized scores of these amounts.

The standardized score for any value is the value minus the mean then divided by the standard deviation.

For $0.30,

z = (x - μ)/σ = (0.30 - 0.40)/0.049 = -2.04

For $0.50

z = (x - μ)/σ = (0.50 - 0.40)/0.049 = 2.04

To determine the required probability

We'll use data from the normal probability table for these probabilities

P(0.30 < x < 0.50) = P(-2.04 < z < 2.04)

= P(z < 2.04) - P(z < -2.04)

= 0.97932 - 0.02068

= 0.95864

Hope this Helps!!!

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