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}

$X \sim B (20, 0.9)$

\begin{matrix} P (X \leq 19) & = 0.878 42 \\ = 0.878 ∎ (to 3 s.f.) \end{matrix}

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}