2024 ACJC P1 Q8

Events \(A\) and \(B\) are such that \(P(A | B) = \frac{7}{10}\), \(P(B) = \frac{1}{3}\) and \(P(A \cup B) = \frac{3}{5}\).

1. Find \(P(A \cap B)\). Hence show that \(P(A) = \frac{1}{2}\).[4]
2. Find the probability that exactly one of \(A\) and \(B\) occurs.[2]
3. Determine whether the events \(A\) and \(B\) are independent.[1]