91Ó°ÊÓ

Playing the Numbers: The House Has a Business. Unlike Joe (see the previous exercise), 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 an average of 60 cents per 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 are the mean and standard deviation of Casper's average winnings \(x\) on his 150,000 bets? b. According to the central limit theorem, what is the approximate probability that Casper's average winnings per bet are between \(\$ 0.30\) and \(\$ 0.50\) ? After only a week, Casper can be pretty confident that his winnings will be quite close to \(\$ 0.40\) per bet.

Step by step solution

01

Define Known Parameters

We know that Casper takes 150,000 bets per week, his mean winnings per bet are \(\mu = 0.40\) dollars, and the standard deviation per single bet is \(\sigma = 18.96\) dollars.

02

Calculate Mean of Average Winnings

The mean of Casper's average winnings after 150,000 bets is the same as the mean per single bet. Therefore, \( \mu_x = \mu = 0.40 \) dollars.

03

Calculate Standard Deviation of Average Winnings

The standard deviation of the average winnings over 150,000 bets can be calculated using the formula for the standard deviation of a sample mean: \( \sigma_x = \frac{\sigma}{\sqrt{n}} = \frac{18.96}{\sqrt{150,000}} \). Computing this, we get \( \sigma_x \approx 0.0489 \) dollars.

04

Use Central Limit Theorem to Find Probability

The Central Limit Theorem suggests that the distribution of the sample mean will be approximately normal with a mean of \(0.40\) and a standard deviation of \(0.0489\). To find the probability that \(\bar{x}\) (average winnings) is between \(0.30\) and \(0.50\), we will standardize these values using the Z-score formula: \( Z = \frac{x - \mu_x}{\sigma_x} \).

05

Calculate Z-Scores

Calculate the Z-scores for \(x = 0.30\) and \(x = 0.50\):1. For \(x = 0.30\), \(Z_1 = \frac{0.30 - 0.40}{0.0489} \approx -2.04\).2. For \(x = 0.50\), \(Z_2 = \frac{0.50 - 0.40}{0.0489} \approx 2.04\).

06

Find Probability from Z-Scores

Using a standard normal distribution table, find the probabilities corresponding to \(Z_1 = -2.04\) and \(Z_2 = 2.04\). These probabilities are approximately 0.0207 for \(Z = -2.04\) and 0.9793 for \(Z = 2.04\).

07

Calculate Probability of Average Winnings Range

Determine the probability that \(\bar{x}\) is between \(0.30\) and \(0.50\) by calculating \( P(-2.04 < Z < 2.04) \). This is approximately \(0.9793 - 0.0207 = 0.9586\).

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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 principle in statistics and probability theory. It states that as the number of trials or observations increases, the average of the results gets closer to the expected value. In the case of Casper Holstein and his numbers game, this law is at play.

Imagine taking only a few bets; the results might be all over the place. However, with 150,000 bets, the randomness averages out to show us a more predictable and consistent pattern. This is because the actual average winnings will be at or very close to the mean of $0.40 per bet.

This law is what makes businesses like casinos profitable over time. Even though any individual bet is unpredictable, the law ensures that they achieve their expected profit margin as they continue dealing with a large number of bets.

Probability

Probability quantifies the likelihood of an event occurring. For Casper's situation, the question is: "What are the odds that his average winnings per bet will fall between specific figures, say $0.30 and $0.50?"

To determine this, statisticians use the probability distribution of sample means, which is derived via the Central Limit Theorem. This allows for predictions about the distribution of the average winnings in a sample size (here, 150,000 bets).

• The goal is to calculate the likelihood, or probability, of the winnings falling within a given range.
• This range is expressed using the concept of probabilities, which are values between 0 and 1.

Thus, in Casper's case, the probability that his average winnings fall between $0.30 and $0.50 per bet is found to be around 0.9586, which means there’s a 95.86% chance of this occurring.

Standard Deviation

Standard deviation measures the amount of variation or dispersion in a set of values. For Casper's winnings, the variation per single bet is \\(18.96\. This value captures the typical deviation from the mean gain of \\)0.40\ per bet.

However, when Casper takes 150,000 bets, we calculate the standard deviation of averages to understand the expected consistency of weekly winnings. This standard deviation of the sample mean is calculated using:\[\sigma_x = \frac{\sigma}{\sqrt{n}}\]where \(\sigma\) is 18.96, and \(n\) is the number of bets (150,000).
This gives a much smaller standard deviation of roughly \$0.0489\. This smaller number reflects how the average winnings stabilize when averaged over a large number of bets, emphasizing the confidence in expected profit predictions.

Z-score

The Z-score is a statistical measure that tells us how many standard deviations a data point is from the mean. It is crucial when comparing occurrences within a normal distribution. In the context of Casper's average winnings, we use Z-scores to find how unusual a particular average would be.

To find the probability of having average winnings between \(0.30 and \)0.50, we convert these dollar amounts into Z-scores. The Z-score formula is applied as follows:\[Z = \frac{x - \mu_x}{\sigma_x}\]

• For \(0.30, the Z-score is calculated to be approximately \(-2.04\).
• For \)0.50, it is about \(2.04\).

These Z-scores help us look up probabilities in a standard normal distribution table, which reveals how likely or unlikely the specified average winnings are. The broader the coverage between these scores, the higher the probability that his average per bet falls within desired parameters.