Final answer:
This mathematics question involves calculating probabilities based on provided data about a college's students, with a focus on part-time students and those who have taken distance learning classes.
Step-by-step explanation:
The question revolves around the application of basic probability and set theory concepts. We have two events of interest:
- D = event that a student takes a distance learning class
- E = event that a student is a part-time student
The college data shows that 10 percent of students (which we can represent as P(D) = 0.10) have taken a distance learning class. Additionally, 40 percent of students are part-time students, meaning P(E) = 0.40. Furthermore, among part-time students, 20 percent have taken a distance learning class, which tells us P(D|E) = 0.20, where P(D|E) represents the conditional probability of a student taking a distance learning class given that they are a part-time student.
Using these probabilities, various other probabilities can be calculated such as the probability of a student being either a distance learner or a part-time student (P(D ∪ E)), using the formula P(D) + P(E) - P(D ⋅ E). The joint probability P(D ⋅ E) represents students who are both part-time and are taking a distance learning class, which could be derived from the given percentages.