Solution — 2021 Question 10 Probability

(a)

Tree diagram for 2021 Q10

(b)

\begin{matrix} P (accepted) \\ = 0.8 (0.9) (0.9) + 0.8 (0.9) (0.1) + 0.8 (0.1) (0.4) + 0.2 (0.4) (0.9) \\ = 0.824 ∎ \end{matrix}

(c)

\begin{matrix} P (not accepted | passed first test) \\ = \frac{P (passed first test and not accepted)}{P (passed first test)} \\ = \frac{0.8 (0.1) (0.6)}{0.8} \\ = 0.06 ∎ \end{matrix}

(d)

Let $X$ be the number of applicants accepted out of 3.

$X \sim B (3, 0.824)$

\begin{matrix} P (X = 2) & = 0.358 ∎ \end{matrix}