A student feels that 70 % of her college courses have been...

A student feels that $70 \%$ of her college courses have been enjoyable and the remainder have been boring. This student has access to student evaluations of professors and finds out that professors who had previously received strong positive evaluations from their students have taught $60 \%$ of his enjoyable courses and $25 \%$ of his boring courses. Next semester the student decides to take three courses, all from professors who have received strongly positive student evaluations. Assume that this student's reactions to the three courses are independent of one another. a. What is the probability that this student will find all three courses enjoyable? b. What is the probability that this student will find at least one of the courses enjoyable?

A student feels that $70 \%$ of her college courses have been enjoyable and the remainder have been boring. This student has access to student evaluations of professors and finds out that professors who had previously received strong positive evaluations from their students have taught $60 \%$ of his enjoyable courses and $25 \%$ of his boring courses. Next semester the student decides to take three courses, all from professors who have received strongly positive student evaluations. Assume that this student's reactions to the three courses are independent of one another.
a. What is the probability that this student will find all three courses enjoyable?
b. What is the probability that this student will find at least one of the courses enjoyable?

Key Concepts

Law of Total Probability

The Law of Total Probability helps compute the overall probability of an event by considering all possible ways that the event can occur. In the context of this problem, it is used to calculate the probability that a course is taught by a professor with a strong evaluation by summing over the mutually exclusive cases (course enjoyable and course boring) weighted by their respective probabilities.

Complement Rule

The Complement Rule is used to simplify probability calculations by noting that the probability of an event occurring is one minus the probability that it does not occur. In this problem, rather than summing the probabilities for one or more enjoyable courses directly, it is easier to calculate the probability of the opposite event (no enjoyable courses) and then subtract that value from one to find the desired probability of at least one enjoyable course.

Independent Events

When events are independent, the outcome of one does not affect the outcome of another. Here, the student’s reaction to each course is stated to be independent. This means that the probability of experiencing a particular outcome in multiple courses can be calculated by multiplying the individual probabilities, a principle that is essential for solving parts (a) and (b).

Bayes' Theorem

Bayes' Theorem is a key tool for updating probabilities when new information is available. It relates the conditional probability of an event given a condition to the reverse conditional probability. In a context like this, it is used to determine the probability that a course is enjoyable given that the professor had a strong evaluation, by combining the probability of course enjoyment overall with the likelihood of a professor’s evaluation occurring in enjoyable versus boring courses.

Conditional Probability

This concept refers to the probability of an event occurring given that another event has already occurred. In many problems, including this one, we are interested in the probability of an outcome (such as enjoying a course) once some condition (such as taking a course from a professor with a strongly positive evaluation) is met. It allows us to update the likelihood of events based on additional information.

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