91Ó°ÊÓ

Playing the Numbers: A Gambler Gets Chance Outcomes. The law of large numbers tells us what happens in the long run. Like many games of chance, the numbers racket described in the previous exercise 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. a. What are the mean and standard deviation of the average payout \(x\) that Joe receives from his 14,000 bets? b. The central limit theorem says that his average payout is approximately Normal with the mean and standard deviation you found in part (a). What is the approximate probability that Joe's average payout per bet is between \(\$ 0.50\) and \(\$ 0.70\) ? You see that Joe's average may not be very close to the mean \(\$ 0.60\) even after 14,000 bets.

Step by step solution

01

Understand the Exercise

The exercise involves calculating the probability of the average payout per bet for Joe, who makes a large number of bets (14,000). We need to use the mean and standard deviation of payouts to find the distribution of the average payout.

02

Central Limit Theorem Application

The Central Limit Theorem (CLT) states that the distribution of sample means will be approximately normal, given a large enough sample size. We are given 14,000 bets, which satisfies this criterion.

03

Calculate Mean of Average Payout

The mean of a single bet payout is \(\mu = \\(0.60\). According to the CLT, the mean of the average payout of 14,000 bets is also \(\mu = \\)0.60\).

04

Calculate Standard Deviation of Average Payout

The standard deviation for the payout of a single bet is \(\sigma = \\(18.96\). For the average payout of 14,000 bets, the standard deviation \(\sigma_{\bar{x}}\) is calculated by dividing the single bet standard deviation by the square root of the number of bets: \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{18.96}{\sqrt{14000}} \approx \\)0.1601\).

05

Apply Normal Distribution

With the mean and standard deviation of the average payout known (\(\mu = 0.60\), \(\sigma_{\bar{x}} = 0.1601\)), we use the normal distribution to find the probability \(P(0.50 < x < 0.70)\).

06

Normalize the Range

Convert the range \(0.50\) to \(0.70\) to standard normal form (z-scores): \(z = \frac{x - \mu}{\sigma_{\bar{x}}}\). For 0.50, \(z = \frac{0.50 - 0.60}{0.1601} = -0.6237\), and for 0.70, \(z = \frac{0.70 - 0.60}{0.1601} = 0.6237\).

07

Use Z-table for Probability

Use a standard normal distribution table (z-table) to find the probabilities associated with the z-scores. \(P(Z < 0.6237) \approx 0.734\) and \(P(Z < -0.6237) \approx 0.267\). The probability \(P(-0.6237 < Z < 0.6237) = 0.734 - 0.267 = 0.467\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Law of Large Numbers

The Law of Large Numbers is a fundamental concept in probability and statistics that describes what happens when an experiment is repeated many times. Simply put, it states that as the number of trials of a random event increases, the average of the outcomes will converge to the expected value. This means that while individual results may be unpredictable or erratic, their average behavior over many trials becomes more predictable. For instance, in gambling, each single bet's outcome may vary wildly, but with enough bets, the average winnings tend to stabilize around the expected value. This is critical for understanding games of chance and helps explain why casinos and lotteries always expect to make a profit in the long run.

Central Limit Theorem

The Central Limit Theorem (CLT) is a powerful statistical theorem that helps us understand how sample means behave. It says that when you take a large enough sample size from a population, the distribution of the sample means will approximate a normal distribution, regardless of the shape of the original population distribution. This is why the CLT is so important - it allows us to use normal distribution to make inferences about sample means, even when the individual outcomes are not normally distributed.

In Joe's scenario, despite the fact that each gamble might have extreme variability due to the high standard deviation, the average payout over 14,000 bets follows a normal distribution. This is because 14,000 is a sufficiently large sample size, allowing us to apply the CLT and predict outcomes more reliably with the bell curve of normal distribution.

Normal Distribution

Normal distribution, often called the bell curve, is a continuous probability distribution that is symmetrical on both sides of the mean, illustrating that data near the mean are more frequent in occurrence than data far from the mean. This distribution is a key concept because it allows statisticians to use a variety of tools and techniques for in-depth data analysis, thanks to its consistent properties.

The mean, indicated by the highest point of the curve, gives a central measure. The spread of the curve is determined by the standard deviation, which impacts how "wide" the bell appears. In Joe's case, the average payout's probability of lying between two specific values is determined using this model. With a calculated mean and adjusted standard deviation from the CLT, the normal distribution paints a clearer picture of Joe's potential outcomes over many bets.

Standard Deviation

Standard deviation is a measure of how spread out numbers in a data set are. It quantifies the amount of variation or dispersion in a set of values. Low standard deviation means the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a wider range. In our gambling scenario, the standard deviation helps us understand how much Joe's payouts fluctuate around the mean payout.

Calculating the standard deviation of Joe’s average payout involves adjusting the standard deviation of individual bets by the sample size. This adjustment is crucial because it tells us about the variability of Joe's average payout across many trials. With this information, we can apply the normal distribution to calculate the likelihood of various average payouts, which makes understanding risks and predictions in gambling more feasible and less of a gamble!