Final answer:
The question is about determining the relationship between two events, A and B, given their individual probabilities and their joint probability. Events A and B are not independent since P(A ∩ B) does not equal P(A) * P(B), and they are not mutually exclusive as P(A ∩ B) is not zero.
Step-by-step explanation:
The problem you've presented involves calculating the probability of two events, A and B, occurring together (also known as the joint probability) and determining their relationship, specifically whether they are independent or mutually exclusive events.
Given that P(A) = 0.50, P(B) = 0.60, and P(A ∩ B) = 0.40, we can use these values to analyze the events. The probability of both A and B occurring, denoted P(A AND B), is P(A ∩ B).
To determine if events A and B are independent, one condition that should be met is that P(A ∩ B) should equal P(A) * P(B). If A and B were independent, P(A ∩ B) would be (0.50)(0.60) = 0.30, which is different from the given P(A ∩ B) of 0.40.
This difference indicates that the events are not independent. The fact that P(A ∩ B) is not zero also tells us that the events are not mutually exclusive since for mutually exclusive events, P(A ∩ B) would be zero.