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When one event's occurrence depends on another, we use conditional probability to measure the likelihood of the first event given that the latter event has occurred. In our example, assessing whether an employee earns more than \(50,000, given they are a college graduate, is an application of conditional probability.

To calculate the conditional probability of event B given event A, denoted as P(B | A), we use the formula: \[\begin{equation}P(B | A) = \frac{P(A \cap B)}{P(A)}\end{equation}\]
Using data from our exercise, we found that P(B | A) ≈ 0.643, which tells us that, within the group of college graduates, approximately 64.3% of employees earn more than \)50,000. An interesting comparison can be made against the conditional probability of event B given the complement of A, notated as P(B | Aᶜ), which was approximately 0.292. This stark contrast enables us to clearly see the impact of higher education on income within this company's workforce.

The joint probability refers to the likelihood of two events occurring simultaneously. Our objective was to determine the probability of an employee being a college graduate and earning over \(50,000 a year, labeled as event A and B respectively, or P(A ∩ B) in probability notation.

For joint probability, we found: \[\begin{equation}P(A \cap B) = \frac{\text{Number of employees with college degree and income > \)50,000}}{\text{Total number of employees}}\end{equation}\]
Upon calculation, the joint probability is 0.18, meaning there is an 18% chance that a randomly selected employee meets both criteria. This measure is especially important when we look to understand the overlap between two variables—in this case, education and income levels within the company.

Two events are independent if the occurrence of one does not impact the probability of occurrence of the other. Independence is a fundamental concept in probability theory, as it significantly simplifies the analysis of events. To test for independence between two events, A and B, we check if the equation P(A ∩ B) = P(A) * P(B) holds true.

In our analysis, we revealed that P(A ∩ B) is not equal to P(A) * P(B), as 0.18 does not equal approximately 0.109. As a result, we conclude that having a college degree and earning more than $50,000 a year are not independent events for the employees at Franklin National Life Insurance Company. This information can impact company policy, salary structures, and education incentives.