91Ó°ÊÓ

Trevor is interested in purchasing the local hardware/sporting goods store in the small town of Dove Creck, Montana. After examining accounting records for the past several years, he found that the store has been grossing over 850 per day about 60 of the business days it is open. Estimate the probability that the store will gross over 850 (a) at least 3 out of 5 business days. (b) at least 6 out of 10 business days. (c) fewer than 5 out of 10 business days. (d) fewer than 6 out of the next 20 business days. Interpretation If this actually happened, might it shake your confidence in the statement \(p=0.60 ?\) Might it make you suspect that \(p\) is less than \(0.60 ?\) Explain. (e) more than 17 out of the next 20 business days. Interpretation If this actually happened, might you suspect that \(p\) is greater than \(0.60 ?\) Explain.

Step by step solution

01

Understand the Problem

We are given a scenario where we need to estimate the probability of an event occurring based on historical data. Specifically, we know that the store grosses over 850 dollars 60% of the time. This probability (\( p \)) of each business day is \( 0.60 \). The goal is to find the probability of the store grossing over 850 dollars on certain numbers of days within different sets of business days.

02

Define the Binomial Distribution Formula

The situation can be modeled using a binomial distribution where each business day is a trial, the event of grossing over 850 dollars is a success with probability \( p = 0.60 \), and we are counting the number of successes in a given number of trials (business days). The probability of \( k \) successes in \( n \) trials is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( \binom{n}{k} \) is the binomial coefficient.

03

Calculate Probability for Part (a)

For part (a), we calculate the probability that the store grosses over 850 dollars at least 3 out of 5 days. This means we need \( P(X \geq 3) \) for \( n = 5 \) and \( p = 0.60 \). Calculate \( P(X = 3) \), \( P(X = 4) \), and \( P(X = 5) \) and then sum them: \[ P(X = 3) = \binom{5}{3} (0.60)^3 (0.40)^2 \] \[ P(X = 4) = \binom{5}{4} (0.60)^4 (0.40)^1 \] \[ P(X = 5) = \binom{5}{5} (0.60)^5 (0.40)^0 \] Add them to get the total probability.

04

Calculate Probability for Part (b)

For part (b), calculate the probability of grossing over 850 dollars at least 6 out of 10 days, i.e., \( P(X \geq 6) \) for \( n = 10 \) and \( p = 0.60 \). Calculate \( P(X = 6) \), \( P(X = 7) \), \( P(X = 8) \), \( P(X = 9) \), and \( P(X = 10) \) and sum them. Use the formula as before for each probability value and then sum them to get the total probability.

05

Calculate Probability for Part (c)

For part (c), we find the probability that the store grosses over 850 dollars fewer than 5 out of 10 days, i.e., \( P(X < 5) \) for \( n = 10 \) and \( p = 0.60 \). Calculate \( P(X = 0) \), \( P(X = 1) \), \( P(X = 2) \), \( P(X = 3) \), and \( P(X = 4) \). Sum these probabilities to get the final probability value.

06

Calculate Probability for Part (d) and Analyze

For part (d), the probability of fewer than 6 out of 20 days is calculated as \( P(X < 6) \) for \( n = 20 \) and \( p = 0.60 \). It requires calculating individual probabilities for \( X = 0 \) through \( X = 5 \) and summing them. If the resulting probability is significantly low, it could suggest that the true \( p \) might be less than \( 0.60 \), thus potentially shaking our confidence in the initial estimate of \( p \).

07

Calculate Probability for Part (e) and Analyze

For part (e), calculate the probability that the store grosses over 850 dollars more than 17 out of 20 days, i.e., \( P(X > 17) \). Consider \( P(X = 18) \), \( P(X = 19) \), and \( P(X = 20) \). If this probability is extremely low, yet it occurs, it might suggest that the actual probability (\( p \)) of grossing over 850 dollars is greater than 0.60, indicating a strong performance above the given estimate.

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Key Concepts

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

Probability Calculation

Estimating the likelihood of specific outcomes based on given data is a crucial part of probability calculation. In the context of binomial distribution, we focus on whether a particular event occurs or not. Here, for Trevor's store, the event of interest is the store grossing over $850. Given that this happens 60% of the time, the probability of exceeding $850 on any particular day is

• p = 0.60.

The probabilities in parts (a) to (e) of the original problem are calculated using the binomial distribution formula. This formula takes into account the number of trials (days) and the desired number of successes (days exceeding $850). More specifically, it calculates the likelihood of achieving 'k' successes in 'n' trials. Understanding how to apply this formula is key to solving problems of this nature.

Success Trials

Success trials refer to the attempts in which the desired outcome occurs, in this case, when the store grosses over $850 in a single day. In binomial terms, each day the store is open can be considered a trial, with a success happening when the store exceeds the target gross. The sum of these successful outcomes over multiple days gives us the probability we're trying to determine.
Within Trevor's problem:

• The success probability per day, p, is 0.60.
• We are interested in the number of successful trials over a specified period.
• For example, scenarios assess situations like at least 3 out of 5 days or more than 17 out of 20 days being successful.

Breaking down each scenario into its respective success and failure trials helps visualize the calculations needed to derive the desired probabilities.

Binomial Coefficient

The binomial coefficient is a key part of the binomial probability formula. It measures how many combinations of successes can occur among the given trials. It's denoted as \( \binom{n}{k} \), which reads as 'n choose k'. Here:

• 'n' is the total number of trials (days the store is open).
• 'k' is the number of successful trials (days exceeding \(850).

For instance, calculating \( \binom{5}{3} \) reveals how many ways more than \)850 can be grossed on exactly 3 out of 5 days. The binomial coefficient thus plays a vital role in computing the probability of different success scenarios, ensuring we consider all possible configurations of success across the specified number of days.

Business Forecasting

Business forecasting, in this case, involves predicting the financial performance of Trevor's store using probability models. By applying historical data and binomial distribution, you can make educated guesses about future sales patterns.
Understanding these probabilities helps

• Evaluate business risks and opportunities.
• Plan for contingencies based on the likelihood of certain income levels being met.
• Assess the reliability of initial data estimates and adjust business strategies accordingly.

For example, if the estimated probability suggests that the store frequently exceeds the expected gross, Trevor might conclude that the store’s performance is better than forecasted, leading to optimistic business planning. On the contrary, if the probability is lower than expected, it could be a signal of potential concerns that need addressing.