Solution — 2021 Question 8 Probability

(a)

By the conditional probability formula

\begin{matrix} P (A | B) & = \frac{P (A \cap B)}{P (B)} \\ 0.8 & = \frac{0.2}{P (B)} \\ 0.8 P (B) & = 0.2 \\ P (B) & = 0.25 \end{matrix}

By the union formula,

\begin{matrix} P (A \cup B) & = P (A) + P (B) - P (A \cap B) \\ 0.9 & = P (A) + 0.25 - 0.2 \\ P (A) & = 0.85 ∎ \end{matrix}

(b)

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

\begin{matrix} P (A^{'} \cap B^{'}) & = 1 - P (A \cup B) \\ = 1 - 0.9 \\ = 0.1 ∎ \end{matrix}

A Venn diagram is useful to visualize this.

(c)

We have $P (A \cap B) = 0.2$ and

\begin{matrix} P (A) P (B) \\ = 0.85 \times 0.25 \\ = 0.85 (0.25) \\ = 0.2125 \\ \neq P (A \cap B) \end{matrix}

Since $P (A \cap B) \neq P (A) P (B),$ $A$ and $B$ are not independent $∎$

(d)

Venn diagram