Question

(c) If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A \mid B) = \frac{2}{5} \), then find \( P(A \cap B) \).

Hint:

For conditional probability, always use the formula \(P(A / B) = \frac{P(A \cap B)}{P(B)}\).

Answer 1

Use the formula for conditional probability: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \quad\Rightarrow\quad \frac{2}{5} = \frac{P(A \cap B)}{\frac{5}{13}}. \] Simplify: \[ P(A \cap B) = \frac{2}{5} \times \frac{5}{13} = \frac{2}{13}. \]