91Ó°ÊÓ

Franchise Stores: Profits Wing Foot is a shoe franchise commonly found in shopping centers across the United States. Wing Foot knows that its stores will not show a profit unless they gross over \(\$ 940,000\) per year. Let A be the event that a new Wing Foot store grosses over \(\$ 940,000\) its first year. Let \(B\) be the event that a store grosses over \(\$ 940,000\) its second year. Wing Foot has an administrative policy of closing a new store if it does not show a profit in either of the first 2 years. The accounting office at Wing Foot provided the following information: \(65 \%\) of all Wing Foot stores show a profit the first year; \(71 \%\) of all Wing Foot stores show a profit the second year (this includes stores that did not show a profit the first year); however, \(87 \%\) of Wing Foot stores that showed a profit the first year also showed a profit the second year. Compute the following: (a) \(P(A)\) (b) \(P(B)\) (c) \(P(B | A)\) (d) \(P(A \text { and } B)\) (e) \(P(A \text { or } B)\) (f) What is the probability that a new Wing Foot store will not be closed after 2 years? What is the probability that a new Wing Foot store will be closed after 2 years?

Step by step solution

01

Calculate Probability for First Year Profit

Given: 65\% of the stores make a profit in the first year, so \(P(A) = 0.65\).

02

Calculate Probability for Second Year Profit

Given: 71\% of the stores make a profit by the second year, so \(P(B) = 0.71\).

03

Conditional Probability for Second Year Given First Year

Given: 87\% of stores that showed a profit the first year also made a profit the second year, therefore \(P(B|A) = 0.87\).

04

Joint Probability of First Year and Second Year Profits

Using the formula \(P(A \text{ and } B) = P(A) \times P(B|A)\), we have \(P(A \text{ and } B) = 0.65 \times 0.87 = 0.5655\).

05

Probability of Profit in Either First or Second Year

Using the formula \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\), we have \(P(A \text{ or } B) = 0.65 + 0.71 - 0.5655 = 0.7945\).

06

Probability of Not Being Closed After 2 Years

A store will not be closed if it makes a profit in either the first or second year, which is \(P(A \text{ or } B)\). So, \(P(\text{Not Closed}) = 0.7945\).

07

Probability of Being Closed After 2 Years

A store will be closed if it doesn't make a profit either year. Thus, \(P(\text{Closed}) = 1 - P(A \text{ or } B) = 1 - 0.7945 = 0.2055\).

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Key Concepts

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

Conditional Probability

Conditional probability helps us determine the likelihood of an event occurring, given that another event has already happened. It's denoted as \( P(B | A) \), which reads as "the probability of \( B \) given \( A \)." In our exercise, \( A \) represents the event that a Wing Foot store makes a profit in its first year, while \( B \) signifies a profit in the second year. Here, we know that 87% of stores that succeed in their first year also succeed in the second one. Therefore, the conditional probability \( P(B | A) = 0.87 \). By using conditional probability, we can better forecast outcomes based on specific conditions or preceding occurrences. It is an essential tool in data analysis, helping businesses make informed decisions.

To compute conditional probability, use the formula:

• \( P(B | A) = \frac{P(A \text{ and } B)}{P(A)} \)

In our case, we are given \( P(B | A) \) directly, streamlining the process.

Joint Probability

Joint probability relates to the likelihood of two events happening at the same time. In probability theory, it's essential to assess complex scenarios where multiple conditions or events can occur simultaneously. For the Wing Foot stores, we are interested in these two specific events: the store making a profit in its first year \( A \), and again in its second year \( B \).

To find the joint probability \( P(A \text{ and } B) \), we use the formula that combines the individual probability of \( A \) and the conditional probability of \( B \) given \( A \):

• \( P(A \text{ and } B) = P(A) \times P(B|A) = 0.65 \times 0.87 = 0.5655 \)

This result tells us that the likelihood of a store being profitable in both the first and second years is approximately 56.55%. Such insights are crucial for companies to evaluate risks and set strategic goals.

Event Probability

When we talk about event probability, we are discussing the likelihood of a single event occurring. It could be a straightforward assessment without considering any conditions or influences. In our situation, event probabilities include the probabilities of a store making a profit in its first or second year without any conditions.

From the problem:

• \( P(A) = 0.65 \), meaning a 65% chance of profitability in the first year.
• \( P(B) = 0.71 \), a 71% chance for the second year.

Understanding individual event probabilities is foundational. This basic step lets you compute more complex scenarios, such as joint or conditional probabilities. It gives a clear snapshot of likely outcomes, aiding businesses in predicting future performance.

Statistical Analysis

Statistical analysis allows us to make informed decisions by dealing with uncertainties and variations in data. This process involves calculating probabilities to evaluate future events or tendencies.

For instance, whether a store will stay open or be closed after two years is determined by the probability of it making a profit in either year. Using the formula for probability of union, add the probabilities of profit in first year \( P(A) \) and second year \( P(B) \), then subtract the joint probability \( P(A \text{ and } B) \):

• \( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) = 0.65 + 0.71 - 0.5655 = 0.7945 \)

This means there's a 79.45% chance that a store will not be closed after two years. To find the probability of closure, subtract this result from 1:

• \( P(\text{Closed}) = 1 - 0.7945 = 0.2055 \)

This suggests there's a 20.55% chance of closure, highlighting risk and aiding in decision-making processes.