Solution — 2025 Question 9 Binomial Distribution

$$\begin{array}{cccc}& \mathrm{E}(X)& =18& \\ & np& =18& \text{(1)}\end{array}$$

$$\begin{array}{cccc}& \mathrm{Var}(X)& =1.8& \\ & np(1-p)& =1.8& \text{(2)}\end{array}$$

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

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

$$\begin{array}{cc}\mathrm{P}(X\le 19)& =0.87842\\ & =0.878∎\text{ (to 3 s.f.)}\end{array}$$

$$\begin{array}{cc}& \mathrm{P}(X\ge 17|X\le 19)\\ & =\frac{\mathrm{P}(X\ge 17\cap X\le 19)}{\mathrm{P}(X\le 19)}\\ & =\frac{\mathrm{P}(17\le X\le 19)}{0.87842}\\ & =\frac{\mathrm{P}(X\le 19)-\mathrm{P}(X\le 16)}{0.87842}\\ & =\frac{0.87842-0.13295}{0.87842}\\ & =0.849∎\text{ (to 3 s.f.)}\end{array}$$