The events A and B are independent. The probability that event A occur

The events A and B are independent. The probability that event A occurs is p and the probability that event B occurs is q. Which of the following is equal to the probability that exactly one of the events A and B occurs?

A. p − 2pq
B. q − pq
C. p + q − 2pq
D. p + q − pq
E. p + q

KAPLAN OFFICIAL EXPLANATION

Let's say that P(A) is the probability that event A occurs, P(B) is the probability that event B occurs, P(not A) is the probability that event A does not occur, P(not B) is the probability that event B does not occur, P(A and not B) is the probability that event A occurs and event B does not occur, and P(not A and B) is the probability that event A does not occur and event B occurs. Then P(A) = p and P(B) = q. The probability that an event does not occur is equal to 1 minus the probability that the event occurs. So P(not A) = 1 − p and P(not B) = 1 − q.

The probability that exactly one of the events A and B occur is P(A and not B) + P(not A and B). The events A and B are independent, so P(A and not B) = P(A)P(not B) = p(1 − q) = p − pq. Again, the events A and B are independent, so P(not A and B) = P(not A)P(B) = (1 − p)q = q − pq.

Thus, P(A and not B) = p − pq and P(not A and B) = q − pq.

So the probability that exactly one of the events A and B occurs is P(A and not B) + P(not A and B) = p − pq + q − pq = p + q − 2pq.

Choice (C) is correct.