Solution — 2016 Question 9 Binomial Distribution

(a)

(i)

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

$ ParseError: Can't use function '$' in math mode at position 1: $̲ X \sim \mathrm{B} \left( 8, 0.6 \right)

\begin{align*} \mathrm{P}\left(X = 8\right) &= 0.016\,796 \\ &= 0.0168\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.

[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.

$ ParseError: Can't use function '$' in math mode at position 1: $̲ Y \sim \mathrm{B} \left( 4, 0.826,33 \right)

\begin{align*} \mathrm{P}\left(Y \leq 2\right) &= 0.141\,79 \\ &= 0.142\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.

[probability of Y \leq 2]

(c)

Out of syllabus.