Question

Evaluate \( P(A \cup B) \) if \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A|B) = \frac{2}{5} \):

Hint:

Use conditional probability formula \( P(A|B) = \frac{P(A \cap B)}{P(B)} \) and the addition rule \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).

Answer 1

Given: \[ P(B) = \frac{5}{13}, \quad 2P(A) = P(B) \implies P(A) = \frac{5}{26} \] and \[ P(A|B) = \frac{2}{5} \] Recall: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \implies P(A \cap B) = P(A|B) \times P(B) = \frac{2}{5} \times \frac{5}{13} = \frac{2}{13} \] Now use the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{5}{26} + \frac{5}{13} - \frac{2}{13} = \frac{5}{26} + \frac{3}{13} \] Convert to common denominator 26: \[ = \frac{5}{26} + \frac{6}{26} = \frac{11}{26} \] % Final Answer Answer: \( P(A \cup B) = \frac{11}{26} \)