Solution — 2025 Question 9 Binomial Distribution

(a)

$\begin{matrix} E (X) & = 18 \\ n p & = 18 & (1) \end{matrix}$

$\begin{matrix} Var (X) & = 1.8 \\ n p (1 - p) & = 1.8 & (2) \end{matrix}$

Substituting (1) into (2),

$\begin{matrix} 18 (1 - p) & = 1.8 \\ 1 - p & = 0.1 \\ p & = 0.9 \end{matrix}$

$\begin{matrix} n & = \frac{18}{0.9} \\ = 20 \end{matrix}$

$ ParseError: Can't use function '$' in math mode at position 1: $̲ X \sim \mathrm{B} \left( 20, 0.9 \right)

\begin{align*} \mathrm{P}\left(X \leq 19\right) &= 0.878\,42 \\ &= 0.878\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.

[probability of X \leq 19]

(b)

$\begin{matrix} P (X \geq 17 | X \leq 19) \\ = \frac{P (X \geq 17 \cap X \leq 19)}{P (X \leq 19)} \\ = \frac{P (17 \leq X \leq 19)}{0.878 42} \\ = \frac{P (X \leq 19) - P (X \leq 16)}{0.878 42} \\ = \frac{0.878 42 - 0.132 95}{0.878 42} \\ = 0.849 ∎ (to 3 s.f.) \end{matrix}$