Two students, Alice and Bob, occasionally miss their morning lectures. Let event AAA be the event that Alice misses the lecture and event BBB be the event that Bob misses the lecture. The probabilities are given as:
P(A)=0.40,P(A∩B)=0.12andP(A′∩B′)=0.55 P(A) = 0.40, \quad P(A \cap B) = 0.12 \quad \text{and} \quad P(A' \cap B') = 0.55 P(A)=0.40,P(A∩B)=0.12andP(A′∩B′)=0.55Find the probability that on a randomly chosen day:
At least one of Alice or Bob misses the lecture.
Bob misses the lecture.
Given that Alice missed the lecture, find the probability that Bob also missed it.
The professor believes that Bob missing the lecture might be related to Alice missing the lecture. Determine whether or not events AAA and BBB are statistically independent.
Comment on the professor's belief in light of your answer to part (d).