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Understanding the concept of conditional probability is essential for students tackling problems in applied mathematics, especially those involving uncertainty and prediction. Simply put, conditional probability is the likelihood of an event occurring, given that another event has already happened.

Let's delve into the exercise which is a practical example. In the given scenario, we want to find out the probability that a randomly selected full/associate professor at a U.S. medical school is female, given that we already know they are a full/associate professor. This is a classic case of seeking a conditional probability, notated as P(B|A), where event B is 'being female' and event A is 'being a full/associate professor'.

To calculate this, we used the formula:
\(P(B|A) = \frac{P(A \cap B)}{P(A)}\)
where \(P(A \cap B)\) represents the intersection of events A and B, and P(A) represents the probability of event A. The solution to this computation provided us with the probability of a full/associate female professor, which is approximately 22.23%.

Moving on to another vital topic, the law of total probability. This concept helps us break down complex probabilities into simpler parts. When we have a set of events that are exhaustive and mutually exclusive, we can use this law to find the overall probability of a particular event.

Referring back to our exercise, we used the law of total probability to determine the likelihood of a faculty member being a full/associate professor, regardless of gender. This involved adding the probability of a female professor to the probability of a male professor, which is mathematically represented by:
\(P(A) = P(A|B) \cdot P(B) + P(A|\sim B) \cdot P (\sim B)\)
Here, the probabilities on the right side represent the weighted likelihood of being a full/associate professor for both women and men, respectively. With the calculated probability of event A, we could then move on to find the conditional probability as described in the previous section.

The concept of the intersection of events plays a crucial role when evaluating the likelihood of two or more events occurring simultaneously. In probability terms, an intersection is represented by 'and'—we're interested in the odds of event A and event B happening together.

In the context of our example, we calculated \(P(A \cap B)\), which is the probability of a faculty member both being female (event B) and being a full/associate professor (event A). We find this intersection by multiplying the probability of one event by the conditional probability of the other:
\(P(A \cap B) = P(A | B) \cdot P(B)\)
After computing \(P(A \cap B) = 0.0992\), this value was instrumental for finding our final conditional probability. Understanding intersections is crucial for grasping more complex probability concepts and working through multifaceted probability exercises.