An event X can happen with probability p and event Y can happen with probability q. Further, X and Y are independent events. Which of the following statements is/are correct? I. The probability that exactly one of the events happens is p + q - pq. II. The probability that at least one of the events happens is p + q - 2pq. Select the answer using the code given below.
D. Neither I nor II
Given P(X) = p and P(Y) = q. Since X and Y are independent events, P(X Y) = P(X)P(Y) = pq. The probability that exactly one of the events happens is given by P(X Y') + P(X' Y) = P(X) + P(Y) - 2P(X Y) = p + q - 2pq. Statement I claims this probability is p + q - pq, which is incorrect. The probability that at least one of the events happens is given by P(X Y) = P(X) + P(Y) - P(X Y) = p + q - pq. Statement II claims this probability is p + q - 2pq, which is also incorrect. Therefore, neither Statement I nor Statement II is correct. Answer: Neither I nor II
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