Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The personnel department of Franklin National Life Insurance Company compiled the accompanying data regarding the income and education of its employees: $$\begin{array}{lcc}\hline & \text { Income } & \text { Income } \\ & \mathbf{\$ 5 0 , 0 0 0} \text { or Below } & \text { Above } \mathbf{\$ 5 0 , 0 0 0} \\ \hline \text { Noncollege Graduate } & 2040 & 840 \\ \hline \text { College Graduate } & 400 & 720 \\\\\hline\end{array}$$ Let \(A\) be the event that a randomly chosen employee has a college degree and \(B\) the event that the chosen employee's income is more than \(\$ 50,000\). a. Find each of the following probabilities: \(P(A), P(B)\), \(P(A \cap B), P(B \mid A)\), and \(P\left(B \mid A^{c}\right)\). b. Are the events \(A\) and \(B\) independent events?

Step-by-Step Solution

Answer

The probabilities are as follows: \(P(A) = 0.28\), \(P(B) = 0.39\), \(P(A \cap B) = 0.18\), \(P(B \mid A) = 0.643\), and \(P\left(B \mid A^{c}\right) = 0.292\). The events A and B are not independent since \(P(A \cap B) \neq P(A)P(B)\).

Step 1: Create a table with total frequencies for each category

First, we will create a table with income and education categories, calculating the total frequencies for each category (rows and columns). $$\begin{array}{|c|c|c|c|} \hline & \text {Income <= \$50,000} & \text {Income >\$50,000} & \\ \hline \text {Noncollege Graduate} & 2040 & 840 & 2880 \\ \hline \text {College Graduate} & 400 & 720 & 1120 \\ \hline & 2440 & 1560 & 4000 \\ \hline \end{array}$$ The total number of employees is 4000.

Step 2: Find the probabilities P(A) and P(B)

Now, we will find the probabilities of event A (employee has a college degree) and event B (employee's income is more than \(50,000\)). $$ P(A) = \frac{\text{Number of College Graduates}}{\text{Total Number of Employees}} = \frac{1120}{4000} = 0.28 $$ $$ P(B) = \frac{\text{Number of Employees with Income > \$50,000}}{\text{Total Number of Employees}} = \frac{1560}{4000} = 0.39 $$

Step 3: Find the probability P(A ∩ B)

Next, we will find the probability of the intersection of event A and event B, i.e., the employee both has a college degree and an income more than \(50,000\). $$ P(A \cap B) = \frac{\text{Number of College Graduates with Income > \$50,000}}{\text{Total Number of Employees}} = \frac{720}{4000} = 0.18 $$

Step 4: Find the conditional probabilities P(B|A) and P(B|Ac)

Now, we will find the conditional probabilities P(B|A) (probability of having an income more than \(50,000\) given that the employee has a college degree) and P(B|A^c) (probability of having an income more than \(50,000\) given that the employee does not have a college degree). $$ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.18}{0.28} = 0.643 $$ $$ P\left(B \mid A^{c}\right) = \frac{\text{Number of Noncollege Graduates with Income > \$50,000}}{\text{Total Number of Noncollege Graduates}} = \frac{840}{2880} = 0.292 $$

Step 5: Check if events A and B are independent

Finally, we will check if the events A and B are independent by checking if \(P(A \cap B) = P(A)P(B)\). $$ P(A \cap B) = 0.18 $$ $$ P(A)P(B) = 0.28 \times 0.39 = 0.1092 $$ Since \(P(A \cap B) \neq P(A)P(B)\), the events A and B are not independent.

Key Concepts

Conditional Probability

Conditional probability is a significant concept in probability theory, explaining how likely an event is to occur given that another event has already happened. It helps us understand the relationship between events by updating our belief about an event when certain information becomes known. In our scenario with the employees of Franklin National Life Insurance Company, we calculate the probability of having an income greater than \(50,000 given the employee is a college graduate. This is represented mathematically as \( P(B | A) \).
The formula for conditional probability is given by:

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

Here, \( P(A \cap B) \) is the probability of both events A (having a college degree) and B (earning more than \)50,000) happening at the same time, which we calculated as 0.18. \( P(A) \) is the probability that the employee has a college degree, which is 0.28. Plugging in these values, we find the conditional probability \( P(B | A) \) to be approximately 0.643.
This means that if you randomly pick an employee and find out they are a college graduate, there is roughly a 64.3% chance that they earn more than $50,000. Understanding conditional probability allows us to make better informed decisions based on new evidence or conditions.

Independent Events

Independent events are a crucial concept in probability theory, describing two events that have no influence on each other's outcomes. If two events are independent, the occurrence of one does not change the probability of the occurrence of the other. In other words, knowing whether one event has occurred does not affect our knowledge of the other.
To check if the events A (college degree) and B (income > \(50,000) are independent in this problem, we compare:\( P(A \cap B) \) and \( P(A)P(B) \).

• Calculated \( P(A \cap B) = 0.18 \)
• Calculated \( P(A)P(B) = 0.28 \times 0.39 = 0.1092 \)

Since \( P(A \cap B) eq P(A)P(B) \), we conclude that these events are not independent. The disparity in these values indicates a relationship or dependence between earning above \)50,000 and having a college degree. This insight helps in understanding the dynamics and dependencies between educational attainment and income levels among employees.

Intersection of Events

The intersection of events in probability theory refers to the event where two or more events happen simultaneously. It is denoted by \( A \cap B \) for the events A and B. In this context, the intersection represents employees who both have a college degree and earn more than \(50,000. The intersection provides valuable information, particularly in compound scenarios where two conditions are simultaneously considered.
The probability of the intersection \( P(A \cap B) \) was found to be 0.18, using the formula:

• \( P(A \cap B) = \frac{\text{Number of College Graduates with Income > \\)50,000}}{\text{Total Number of Employees}} \)

Here, out of 4000 employees, 720 fit the criteria of being college graduates and earning over $50,000. Therefore, \( P(A \cap B) = \frac{720}{4000} = 0.18 \).
This indicates that 18% of the employees meet both of these conditions. By examining intersections, we better understand combined outcomes and how various attributes interact. In business contexts, these insights can help in developing strategies or evaluating relationships between different factors, like education and income in this case.