1. **Stating the problem:** We are given two events A and B with probabilities: - $P(A) = 0.4$ - $P(B|A) = 0.5$ - $P(A \cup B) = 0.75$ We need to find: (a) $P(A \cap B)$ (b) $P(B)$ 2. **Formula and rules:** - Conditional probability formula: $$P(B|A) = \frac{P(A \cap B)}{P(A)}$$ - Union of two events: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ 3. **Find $P(A \cap B)$:** Using the conditional probability formula: $$P(B|A) = \frac{P(A \cap B)}{P(A)} \implies P(A \cap B) = P(B|A) \times P(A)$$ Substitute values: $$P(A \cap B) = 0.5 \times 0.4 = 0.2$$ 4. **Find $P(B)$:** Using the union formula: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ Rearranged to solve for $P(B)$: $$P(B) = P(A \cup B) - P(A) + P(A \cap B)$$ Substitute known values: $$P(B) = 0.75 - 0.4 + 0.2 = 0.55$$ **Final answers:** - (a) $P(A \cap B) = 0.2$ - (b) $P(B) = 0.55$