The law of large numbers tells us what happens in the long run. Like many games of chance, the numbers 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 payoutx-bar that Joe receives from his 14,000 bets?

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

(a) The mean and standard deviation of the average payout that Joe receives from his 14,000 bets are $0.60 and $0.1602 respectively.

(b) The probability that Joe's average payout per bet is between $0.50 and $0.70 is 0.4647.

Step-by-step explanation:

According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

Then, the mean of the distribution of sample mean is given by,

And the standard deviation of the distribution of sample mean is given by,

(a)

Let X = average payout of a bet.

The mean of the random variable X is, μ = $0.60.

The standard deviation of the random variable X is, σ = $18.96.

The number of total bets made is, n = 14000.

Since n = 14000 > 30, the Central limit theorem can be used to approximate the sampling distribution of sample mean.

The mean of the distribution of sample mean is given by,

And the standard deviation of the distribution of sample mean is given by,

Thus, the mean and standard deviation of the average payout that Joe receives from his 14,000 bets are $0.60 and $0.1602 respectively.

(b)

Compute the probability that Joe's average payout per bet is between $0.50 and $0.70 as follows:

Thus, the probability that Joe's average payout per bet is between $0.50 and $0.70 is 0.4647.

answer: the answer is 127 degrees