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.

Step-by-step solution

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 Sta