Solution — 2019 Question 7 Binomial Distribution

(a)

(i)

Let $X$ represent the number of members who received the bonus out of $8.$

$X \sim B (8, 0.32)$

\begin{matrix} P (X = 4) & = 0.156 94 \\ = 0.157 ∎ (to 3 s.f.) \end{matrix}

probability of X = 4

(ii)

\begin{matrix} P (X \geq 5) & = 1 - P (X \leq 4) \\ = 1 - 0.925 04 \\ = 0.074 964 \\ = 0.0750 ∎ (to 3 s.f.) \end{matrix}

probability of X \geq 5

(b)

\begin{matrix} Mean & = E (X) \\ = n p \\ = 8 (0.32) \\ = 2.56 ∎ \end{matrix}

\begin{matrix} Variance & = Var (X) \\ = n p (- p) \\ = 1.7408 \\ = 1.74 ∎ (to 3 s.f.) \end{matrix}