Question

If \[ P(A) = 0.4 \, \text{and} \, P(B) = 0.5, \, \text{also, A and B are independent events, then find} \] (i) \( P(A \cup B) \) and (ii) \( P(A \cap B) \).

Hint:

For independent events, \( P(A \cap B) = P(A) \cdot P(B) \). Also, use the formula for the union of events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \]

Answer 1

(i) To find \( P(A \cup B) \), we use the formula for the probability of the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \] Since A and B are independent events, \[ P(A \cap B) = P(A) \cdot P(B). \] Substituting the given values: \[ P(A \cap B) = 0.4 \times 0.5 = 0.2. \] Now, substitute into the formula for \( P(A \cup B) \): \[ P(A \cup B) = 0.4 + 0.5 - 0.2 = 0.7. \] (ii) We already know that \[ P(A \cap B) = 0.2. \] Conclusion: (i) The value of \( P(A \cup B) \) is \[ \boxed{0.7}. \] (ii) The value of \( P(A \cap B) \) is \[ \boxed{0.2}. \]