Problem 12 - A First Course in Probability

Question

A recent college graduate is planning to take the first three actuarial examinations in the coming summer. She will take the first actuarial exam in June. If she passes that exam, then she will take the second exam in July, and if she also passes that one, then she will take the third exam in September. If she fails an exam, then she is not allowed to take any others. The probability that she passes the first exam is \(.9 .\) If she passes the first exam, then the conditional probability that she passes the second one is \(.8,\) and if she passes both the first and the second exams, then the conditional probability that she passes the third exam is \(.7 .\) (a) What is the probability that she passes all three exams? (b) Given that she did not pass all three exams. what is the conditional probability that she failed the second exam?

Step-by-Step Solution

Answer

The probability of the student passing all three exams is 0.504. Given that she did not pass all three exams, the conditional probability that she failed the second exam is approximately 0.3629.

Step 1: Define the events and probabilities

We are given the following probabilities: - P(A) = probability of passing the first exam = 0.9 - P(B|A) = probability of passing the second exam given passing the first = 0.8 - P(C|A∩B) = probability of passing the third exam given passing the first two exams = 0.7

Step 2: Find the probability of passing all three exams (P(A∩B∩C))

To find P(A∩B∩C), we can use the conditional probabilities. We know that P(A∩B∩C) = P(C|A∩B) * P(A∩B). Since P(A∩B) = P(B|A) * P(A), we can substitute and get: P(A∩B∩C) = P(C|A∩B) * P(B|A) * P(A) Now, plug in the given probabilities: P(A∩B∩C) = 0.7 * 0.8 * 0.9 = 0.504 So, the probability of the student passing all three exams is 0.504.

Step 3: Find P(B'|A∩B'∪A'∩B')

To find this conditional probability, we will use the formula: P(B'|A∩B'∪A'∩B') = P(B'∩(A∩B'∪A'∩B')) / P(A∩B'∪A'∩B') First, we need to find P(A∩B'∪A'∩B'). This is equal to 1 - P(A∩B∩C), which is the complement of passing all three exams. We have already calculated P(A∩B∩C) = 0.504 in Step 2. So, P(A∩B'∪A'∩B') = 1 - 0.504 = 0.496 Next, we need to find P(B'∩(A∩B'∪A'∩B')). We know that A'∩B' is the event of not passing the first exam, and in this case, the student cannot take the second exam. Therefore, P(A'∩B') = 0. So, we only need to find P(B'∩A∩B'). Since P(B'|A) = 1 - P(B|A) = 1 - 0.8 = 0.2, we can say that: P(B'∩A∩B') = P(B'|A) * P(A) Now, plug in the given probabilities: P(B'∩A∩B') = 0.2 * 0.9 = 0.18 Finally, we can calculate the conditional probability: P(B'|A∩B'∪A'∩B') = P(B'∩(A∩B'∪A'∩B')) / P(A∩B'∪A'∩B') = 0.18 / 0.496 = 0.3629 So, given that the student did not pass all three exams, the conditional probability that she failed the second exam is approximately 0.3629.

Key Concepts

Understanding Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. Think of it as updating your beliefs about the chances of something happening in light of new information. To calculate the conditional probability of an event B occurring given that event A has happened, we use the formula:

\[\begin{equation} P(B|A) = \frac{P(A \cap B)}{P(A)}\end{equation}\]
Where:

• P(B|A) is the conditional probability of B given A
• P(A \cap B) is the joint probability of A and B occurring together
• P(A) refers to the probability of A occurring

In our actuarial exam problem, we calculated the conditional probability that the student passes the second exam given that she has passed the first by using the provided joint and individual probabilities. It's essential to correctly apply this concept, as it forms the basis of understanding complex probabilistic scenarios often encountered in actuarial science as well as in other fields that involve risk and uncertainty.

Joint Probability and its Significance

Joint probability comes into play when we are concerned with the likelihood of two events occurring simultaneously. This is expressed as:

\[\begin{equation} P(A \cap B)\end{equation}\]
The actuarial exam problem provides an excellent illustration of joint probability. To find the probability that our graduate passes all three exams, we calculate the joint probability of passing each consecutive exam, based on the conditional probabilities. As we are dealing with dependent events — where the outcome of one influences another — understanding joint probability enables actuaries to predict the likelihood of an individual facing various risks all at once. Always keep in mind that for dependent events, the joint probability is found by multiplying the probability of the first event by the conditional probability of the second event, given the first.

The Complement Rule Explained

The complement rule is a straightforward yet powerful concept in probability theory. It states that the probability of an event not occurring (the complement of the event) is 1 minus the probability of the event occurring.

\[\begin{equation} P(A') = 1 - P(A)\end{equation}\]
In terms of our actuarial exam scenario, we applied the complement rule to determine the probability of the student not passing all three exams — the complement of our previous calculation for passing all three exams. By subtracting the joint probability of passing all exams from 1, we arrived at the probability of at least one failure. This illustrates how the complement rule allows us to switch perspectives from success to failure (or vice-versa), which can be particularly useful in the field of insurance and risk management where understanding the full spectrum of outcomes is crucial.