Question

In a race of $12$ cars, the probability that car $A$ will win is $\frac{1}{5}$ and of car $B$ is $\frac{1}{6}$ and that of car $C$ is $\frac{1}{3}$. Find the probability that only one of them won the race.

Answer 1

Since all the events are mutually exclusive.

Probability that only one of them won the race$=P\left(A\right)+P\left(B\right)+P\left(C\right)$

$=\frac{1}{5}+\frac{1}{6}+\frac{1}{3}$

$=\frac{1}{5}+\frac{1+2}{6}$

$=\frac{1}{5}+\frac{3}{6}$

$=\frac{1}{5}+\frac{1}{2}=\frac{5+2}{5\times 2}=\frac{7}{10}$