Solution — 2018 Question 8 Probability

(a)

By the conditional probability formula,

\begin{matrix} P (A | B) & = \frac{P (A \cap B)}{P (B)} \\ \frac{30}{100} & = \frac{P (A \cap B)}{2 p} \\ \frac{3}{10} & = \frac{P (A \cap B)}{2 p} \\ P (A \cap B) & = \frac{3}{5} p \end{matrix}

By the union formula,

\begin{matrix} P (A \cup B) & = P (A) + P (B) - P (A \cap B) \\ \frac{80}{100} & = p + 2 p - \frac{3}{5} p \\ \frac{4}{5} & = \frac{12}{5} p \\ p & = \frac{1}{3} ∎ \end{matrix}

(b)

$P (A^{'} \cap B)$ is the probability that event $B$ occurs and event $A$ does not occur $∎$

\begin{matrix} P (A^{'} \cap B) & = P (B) - P (A \cap B) \\ = 2 (\frac{1}{3}) - \frac{3}{5} (\frac{1}{3}) \\ = \frac{7}{15} ∎ \end{matrix}

Hint: draw a Venn diagram to help calculate this last quantity