Solution — 2022 Question 7 Probability

(a)

By the conditional probability formula,

\begin{matrix} P (B | A) & = \frac{P (A \cap B)}{P (A)} \\ 0.1 & = \frac{P (A \cap B)}{0.4} \\ 0.040 000 & = P (A \cap B) \\ P (A \cap B) & = 0.04 \end{matrix}

By the conditional probability formula,

\begin{matrix} P (A | B) & = \frac{P (A \cap B)}{P (B)} \\ 0.25 & = \frac{0.04}{P (B)} \\ 0.25 P (B) & = 0.04 \\ P (B) & = 0.16 ∎ \end{matrix}

(b)

\begin{matrix} P (A \cup B) & = P (A) + P (B) - P (A \cap B) \\ = 0.4 + 0.16 - 0.04 \\ = 0.52 ∎ \end{matrix}

(c)

We use the values we have (in particular $P (A) = 0.4$ and $P (A \cap B) = 0.04$ ) in a Venn diagram

Venn diagram for 2022 Q7

\begin{matrix} P (A^{'} \cup B) \\ = 1 - 0.36 \\ = 0.64 ∎ \end{matrix}