Final answer:
The correct statement is D) A and B are complementary events. This is because P(A ∪ B) = 1 signifies that either A or B or both must occur, whereas P(A ∩ B) = 0 means A and B cannot occur simultaneously, making them complementary.
Step-by-step explanation:
If P(A ∪ B) = 1 and P(A ∩ B) = 0, we are dealing with some fundamental probability concepts associated with a sample space. The expressions refer to the probability of either event A or B occurring (the OR event), denoted by P(A ∪ B), and the probability of both events A and B occurring together (the AND event), denoted by P(A ∩ B). In our case, P(A ∪ B) = 1 indicates that event A or event B or both will certainly occur. In contrast, P(A ∩ B) = 0 indicates that events A and B cannot occur at the same time. These two events are therefore mutually exclusive and complementary. The correct answer is D) A and B are complementary events.
When A and B are mutually exclusive, it means they have no outcomes in common, which aligns with P(A ∩ B) = 0. Events are complementary when they cover all possible outcomes in the sample space without overlapping, which aligns with P(A ∪ B) = 1.
Thus, the notion that these events are complementary is the true statement reflecting the relationship between events A and B given P(A ∪ B) = 1 and P(A ∩ B) = 0.