Question

If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A | B) = \frac{3}{10} \), find \( P(A \cup B) \).

Hint:

To find \( P(A \cup B) \), use the formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), and remember that \( P(A \cap B) = P(A | B) \times P(B) \).

Answer 1

Step 1: Use the formula for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Step 2: We are given \( P(A) = \frac{5}{13} \), \( P(B) = \frac{5}{13} \), and \( P(A | B) = \frac{3}{10} \). The probability \( P(A \cap B) \) is given by: \[ P(A \cap B) = P(A | B) \times P(B) = \frac{3}{10} \times \frac{5}{13} = \frac{15}{130} = \frac{3}{26} \] Step 3: Substitute the values into the formula for the union: \[ P(A \cup B) = \frac{5}{13} + \frac{5}{13} - \frac{3}{26} = \frac{10}{13} - \frac{3}{26} = \frac{20}{26} - \frac{3}{26} = \frac{17}{26} \]