Solution — 2023 Question 6 Binomial Distribution

(a)

Let $X$ represent the random variable with a binomial distribution.

$X \sim B (8, p)$

$\begin{matrix} P (X = 3) & = (\binom{8}{3}) p^{3} {(1 - p)}^{8 - 3} \\ = 56 p^{3} {(1 - p)}^{5} ∎ \end{matrix}$

(b)

$X \sim B (8, 0.35)$

$\begin{matrix} P (X \geq 4) & = 1 - P (X \leq 3) \\ = 1 - 0.706 40 \\ = 0.293 60 \\ = 0.294 ∎ (to 3 s.f.) \end{matrix}$

[probability of X \geq 4]

(c)

$\begin{matrix} E (X) & = n p \\ = 8 (0.35) \\ = 2.8 ∎ \end{matrix}$

$\begin{matrix} Var (X) & = n p (1 - p) \\ = 8 (0.35) (1 - 0.35) \\ = 1.8200 \end{matrix}$

$\begin{matrix} σ & = \sqrt{1.8200} \\ = 1.3491 \\ = 1.35 ∎ (to 3 s.f.) \end{matrix}$