Solution — 2022 Question 11 Probability| Binomial Distribution

(a)

Let $S$ represent a successfully attempt and $F$ represent a unsuccessfully attempt.

(b)

\begin{matrix} P (completes challenge) \\ = 0.4 + 0.6 (0.2) + 0.6 (0.8) (0.2) \\ = 0.4 + 0.12 + 0.096 \\ = 0.616 ∎ \end{matrix}

(c)

\begin{matrix} P (completes | not successful at first) \\ = \frac{P (completes \cap not successful at first)}{P (not successful at first)} \\ = \frac{0.6 (0.2) + 0.6 (0.8) (0.2)}{0.6} \\ = \frac{0.12 + 0.096}{0.6} \\ = \frac{0.216}{0.6} \\ = 0.36 ∎ \end{matrix}

(d)

Let $X$ represent the r.v. of the number of challenges completed successfully out of $10.$

$X \sim B (10, 0.616)$

\begin{matrix} P (X = 6) & = 0.249 48 \\ = 0.249 ∎ (to 3 s.f.) \end{matrix}

probability of X = 6

(e)

\begin{matrix} P (X \geq 4) & = 1 - P (X \leq 3) \\ = 1 - 0.043 794 \\ = 0.956 21 \\ = 0.956 ∎ (to 3 s.f.) \end{matrix}

probability of X \geq 4

(f)

Let $Y$ represent the number of students who completed the challenge out of $12.$

$Y \sim B (12, 0.956 21)$

\begin{matrix} P (Y < 9) & = P (Y \leq 8) \\ = 0.001 371 0 \\ = 0.00137 ∎ (to 3 s.f.) \end{matrix}

probability of Y < 9