Solution — 2021 Question 10 Probability

$$\begin{array}{cc}& \mathrm{P}\left(\text{accepted}\right)\\ & =0.8\left(0.9\right)\left(0.9\right)+0.8\left(0.9\right)\left(0.1\right)+0.8\left(0.1\right)\left(0.4\right)+0.2\left(0.4\right)\left(0.9\right)\\ & =0.824∎\end{array}$$

$$\begin{array}{cc}& \mathrm{P}(\text{not accepted}|\text{passed first test})\\ & =\frac{\mathrm{P}\left(\text{passed first test and not accepted}\right)}{\mathrm{P}\left(\text{passed first test}\right)}\\ & =\frac{0.8\left(0.1\right)\left(0.6\right)}{0.8}\\ & =0.06∎\end{array}$$