Final answer:
Complementary events in probability theory are two events that together cover all possible outcomes without overlap, and their probabilities sum to 1. The provided statements fail to confirm that the events mentioned are complementary, since they either don't sum up to 1 or suggest the existence of additional outcomes.
Step-by-step explanation:
In probability theory, complementary events are pairs of events that are mutually exclusive and exhaustive, meaning that one of the events must occur, and both cannot occur at the same time. The sum of the probabilities of complementary events is always 1. In response to the provided options:
- (a) is incorrect because the probability of drawing a red marble is not necessarily the complement of choosing a blue marble unless there are only red and blue marbles.
- (b) indicates that the sum of probabilities is 3/5, which does not satisfy the requirement for complementary events, so it is incorrect in this context.
- (c) If the number of blue and red marbles does not equal the total number of outcomes, then they cannot be complements.
- (d) Suggests that there are additional outcomes, implying that the two events mentioned might not encompass all possible outcomes and therefore cannot be complemented.
To find complementary probabilities, we need to ensure there are no other possible outcomes besides the two events we are considering. If there are only two events, and those events cover all possible outcomes without overlap, their probabilities will sum to 1. For example, if event A is getting a head on a coin flip, then the complement, A', is getting a tail, and indeed P(A) + P(A') = 1.