Bunuel wrote:
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
Because events A and B are independent, the probability that both A and B occur is the product of their individual probabilities, so we have P(A and B) = pq.
Now, note that the event “A only” means that the part of A that includes both A and B happening is not part of this event. Thus, we have:
P(A only) = P(A) - P(A and B) = p - pq.
Similarly, for the event “B only” we see that
P(B only) = P(B) - P(A and B) = q - pq.
Thus, the probability that exactly one event happens (i.e., A only or B only) is equal to
p - pq + q - pq = p + q - 2pq.
Answer: C
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