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 is the mean of the average payout x that Joe receives from his 14,000 bets?
(b)What is the standard deviation of the average payout x that Joe receives from his n=14,000 bets?
(c)The central limit theorem says that his average payout is approximately Normal with the mean and standard deviation you found above.
What is the approximate probability that Joe's average payout per bet is between $0.50 and $0.70?
(a) To find the mean of the average payout x that Joe receives from his 14,000 bets, we can use the formula:
Mean = (Total Payouts) / (Number of Bets)
In this case, the total number of bets Joe has made is 14,000, and since each individual bet has a mean payout of $0.60, the total payout would be (14,000 * $0.60) = $8,400.
Therefore, the mean of the average payout x is $8,400 / 14,000 = $0.60.
(b) To find the standard deviation of the average payout x that Joe receives from his 14,000 bets, we can use the formula:
Standard Deviation = (Standard Deviation of Individual Payouts) / sqrt(Number of Bets)
In this case, the standard deviation of individual payouts is approximately $18.96, and the number of bets is 14,000.
Therefore, the standard deviation of the average payout x is $18.96 / sqrt(14,000) = $0.1605.
(c) According to the central limit theorem, Joe's average payout per bet is approximately normally distributed with the mean and standard deviation calculated above.
To find the approximate probability that Joe's average payout per bet is between $0.50 and $0.70, we can standardize the values using the formula:
Z = (X - Mean) / Standard Deviation
where X is the variable of interest, in this case, the average payout per bet.
For $0.50:
Z1 = ($0.50 - $0.60) / $0.1605
For $0.70:
Z2 = ($0.70 - $0.60) / $0.1605
We can then use a standard normal table or calculator to find the probabilities associated with the calculated Z-scores.
Once we have the probabilities for Z1 and Z2, we can subtract the probability associated with Z1 from the probability associated with Z2 to find the approximate probability that Joe's average payout per bet is between $0.50 and $0.70.
To find the answers to these questions, we need to use the concepts of the law of large numbers and the central limit theorem.
(a) To find the mean of the average payout x that Joe receives from his 14,000 bets, we can use the formula:
Mean = (Total Sum of Payouts) / (Number of Bets)
Since the mean payout for single bets is $0.60, we can multiply this by the number of bets to get the total sum of payouts:
Total Sum of Payouts = ($0.60) * (14,000)
Then divide this by the number of bets (14,000) to get the mean:
Mean = ($0.60 * 14,000) / (14,000)
Simplifying this expression, we find that the mean of the average payout x is $0.60.
(b) To find the standard deviation of the average payout x that Joe receives from his 14,000 bets, we can use the formula for the standard deviation of a sample mean:
Standard Deviation = (Standard Deviation of Individual Bets) / sqrt(Number of Bets)
Given that the standard deviation of payouts is about $18.96, we can substitute this into the formula along with the number of bets:
Standard Deviation = $18.96 / sqrt(14,000)
Calculating this expression, we find that the standard deviation of the average payout x is approximately $0.16.
(c) The central limit theorem states that the distribution of the average of a large number of independent and identically distributed random variables, regardless of the shape of the original distribution, is approximately Normal.
Since Joe's average payout per bet is approximately normally distributed with a mean of $0.60 (as found in part a) and a standard deviation of $0.16 (as found in part b), we can use the properties of the normal distribution to estimate probabilities.
To find the approximate probability that Joe's average payout per bet is between $0.50 and $0.70, we can standardize the values and use a z-table to find the corresponding probabilities.
First, we need to standardize the values:
Standardized lower value = ($0.50 - $0.60) / $0.16
Standardized upper value = ($0.70 - $0.60) / $0.16
Then, we can use a z-table (or a software/calculator that provides such functionalities) to find the probabilities associated with these standardized values. The probability we are interested in is the difference between the two probabilities.
For example, if we find that the standardized lower value corresponds to a probability of P1 and the standardized upper value corresponds to a probability of P2, then the approximate probability that Joe's average payout per bet is between $0.50 and $0.70 would be P2 - P1.
Note: To get more precise results, it is recommended to use a z-table or a normal distribution calculator.