Andrew、Ben 和 Caleb 通过驾驶考试的概率分别为 $\frac{3}{5}$、$\frac{2}{3}$ 和 $\frac{3}{4}$。

1. Find the probability that Andrew passes while Ben and Caleb fail

$P(\text{only Andrew passes})=\dfrac{3}{5}\times\left(1-\dfrac{2}{3}\right)\times\left(1-\dfrac{3}{4}\right)$

$=\dfrac{3}{5}\times\dfrac{1}{3}\times\dfrac{1}{4}=\dfrac{1}{20}$

1. Find the probability that all three fail the driving test

Probability Andrew fails $=1-\dfrac{3}{5}=\dfrac{2}{5}$.

Probability Ben fails $=1-\dfrac{2}{3}=\dfrac{1}{3}$.

Probability Caleb fails $=1-\dfrac{3}{4}=\dfrac{1}{4}$.

So,
$P(\text{all fail})=\frac{2}{5}\times\frac{1}{3}\times\frac{1}{4}=\frac{1}{30}.$

2. Use the complement to find the probability that at least one passes

$P(\text{at least one passes})=1-P(\text{all fail})$
$=1-\frac{1}{30}=\frac{29}{30}.$