Solution — 2023 Question 8 Probability

(a)

By the conditional probability formula,

\begin{matrix} P (B | A) & = \frac{P (A \cap B)}{P (A)} \\ \frac{3}{5} & = \frac{P (A \cap B)}{x} \\ P (A \cap B) & = \frac{3}{5} x \end{matrix}

By the union formula,

\begin{matrix} P (A \cup B) & = P (A) + P (B) - P (A \cap B) \\ \frac{13}{24} & = x + P (B) - \frac{3}{5} x \\ = \frac{2}{5} x + P (B) \\ P (B) & = \frac{13}{24} - \frac{2}{5} x ∎ \end{matrix}

(b)

\begin{matrix} P (A^{'} \cap B) & = P (B) - P (A \cap B) \\ = \frac{13}{24} - \frac{2}{5} x - \frac{3}{5} x \\ = \frac{13}{24} - x \end{matrix}

Since $P (A \cap B) = 2 P (A^{'} \cap B),$

\begin{matrix} \frac{3}{5} x & = 2 (\frac{13}{24} - x) \\ = \frac{13}{12} - 2 x \\ \frac{13}{5} x & = \frac{13}{12} \\ x & = \frac{5}{12} ∎ \end{matrix}