Question

$A$ is one of $6$ horses entered for a race, and is to be ridden by one of two jockeys $B$ and $C$. It is $2$ to $1$ that $B$ rides $A$, in which case all the horses are equally likely to win; if $C$ rides $A$, his chance is trebled; what are the odds against his winning?

• A. $13:5$
• B. $13:18$
• C. $18:13$
• D. $5:13$

Answer 1

Answer: (A)

$13:5$

Let $E_1$ be the event that $B$ rides $A$, $E_2$, the event that $C$ rides $A$ and $E$ the event that $A$ wins.

Then according to the question, $P\left(E_1\right)=\frac{2}{3},P\left(E_2\right)=1-\frac{2}{3}=\frac{1}{3}P\left(E/E_1\right)=\frac{1}{6}$

(since all the $6$ horses are equally likely to win when $B$ rides $A$)

$P\left(E/E_2\right)=3\times \frac{1}{6}=\frac{1}{2}$

(since $A$'s chance of winning is trebled when $C$ rides $A$)

$\therefore P\left(E\right)=P\left(E_1\right)P\left(E/E_1\right)+P\left(E_2\right)P\left(E/E_2\right)=\frac{2}{3}\cdot \frac{1}{6}+\frac{1}{3}\cdot \frac{1}{2}=\frac{58}{18}$

so that odds against $A$'s win are as $(18-5):5$, that is $13:5$.