Solution — 2016 Question 9 Binomial Distribution

(a)

(i)

Let $X$ represent the number of batteries with lifetime less than two years out of $8.$

$X \sim B (8, 0.6)$

$\begin{matrix} P (X = 8) & = 0.016 796 \\ = 0.0168 ∎ (to 3 s.f.) \end{matrix}$

[probability of X = 8]

(ii)

$\begin{matrix} P (X \geq 4) & = 1 - P (X \leq 3) \\ = 1 - 0.173 67 \\ = 0.826 33 \\ = 0.826 ∎ (to 3 s.f.) \end{matrix}$

[probability of X \geq 4]

(b)

Let $Y$ represent the number of packs where at least half of the batteries have lifetime less than two years, out of $4$ packs.

$Y \sim B (4, 0.826 33)$

$\begin{matrix} P (Y \leq 2) & = 0.141 79 \\ = 0.142 ∎ (to 3 s.f.) \end{matrix}$

[probability of Y \leq 2]

(c)

Out of syllabus.