Solution — 2019 Question 7 Binomial Distribution

(a)

(i)

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

$ ParseError: Can't use function '$' in math mode at position 1: $̲ X \sim \mathrm{B} \left( 8, 0.32 \right)

\begin{align*} \mathrm{P}\left(X = 4\right) &= 0.156\,94 \\ &= 0.157\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.

[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)

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

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