Solution — 2017 Question 7 Binomial Distribution

(a)

Let $X$ represent the number of cartridges that last one week or more out of $8.$

$X \sim B (8, 0.7)$

\begin{matrix} P (X = 5) & = 0.254 12 \\ = 0.254 ∎ (to 3 s.f.) \end{matrix}

probability of X = 5

(b)

Let $Y$ represent the number of cartridges that last less than one week out of $8.$

$Y \sim B (8, 0.300 00)$

\begin{matrix} P (Y \geq 4) & = 1 - P (Y \leq 3) \\ = 1 - 0.805 90 \\ = 0.194 10 \\ = 0.194 ∎ (to 3 s.f.) \end{matrix}

probability of Y \geq 4

(c)

Let $B$ represent the number of boxes where at least half of the cartridges last less than one week, out of $6$ boxes.

$B \sim B (6, 0.194 10)$

\begin{matrix} P (B \leq 2) & = 0.908 23 \\ = 0.908 ∎ (to 3 s.f.) \end{matrix}

probability of B \leq 2