Solution — 2020 Question 7 Binomial Distribution

(a)

Let $X$ represent the number of oranges that are not ripe in a bag of $12.$

$ ParseError: Can't use function '$' in math mode at position 1: $̲ X \sim \mathrm{B} \left( 12, 0.15 \right)

\begin{align*} \mathrm{P}\left(X = 1\right) &= 0.301\,22 \\ &= 0.301\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.

[probability of X = 1]

(b)

$\begin{matrix} P (X > 3) & = 1 - P (X \leq 3) \\ = 1 - 0.907 79 \\ = 0.092 206 \\ = 0.0922 ∎ (to 3 s.f.) \end{matrix}$

[probability of X > 3]

(c)

$\begin{matrix} P (both bags at reduced price) \\ = 0.092 206 (0.092 206) \\ = 0.00850 ∎ \end{matrix}$

(d)

Let $Y$ represent the number of bags sold at the reduced price out of $20.$

$ ParseError: Can't use function '$' in math mode at position 1: $̲ Y \sim \mathrm{B} \left( 20, 0.092,206 \right)

\begin{align*} \mathrm{P}\left(Y < 4\right) &= \mathrm{P}\left(Y \leq 3\right) \\ &= 0.893\,66 \\ &= 0.894\; \QED \text{ (to 3 s.f.)} \end{align*}$$ ParseError: {align*} can be used only in display mode.

[probability of Y < 4]