Question

$H$ is one of the six horses entered the race and is to be ridden by one of two jokeys $A$ and $B.$ It is $2$to$1$ that $A$ rides $H$ in which case all horses are likely to win. If $B$ rides $H$ his chance is trebled. Then the odds against $H$ winning are

• A. $13:8$
• B. $13:5$
• C. $18:5$
• D. $5:18$

Answer 1

The correct option is B $13:5$

Let $E_1$ be the event horse is ridden by A.
$E_2$ be the event horse is ridden by B.
And $E$ be the event H wins.
$P\left(E_1\right)=\frac{2}{3},P\left(E_2\right)=\frac{1}{3}$
$P\left(\frac{E}{E_1}\right)=\frac{1}{6},P\left(\frac{E}{E_2}\right)=3\times \frac{1}{6}=\frac{1}{2}$
$P\left(E\right)=P\left(E_1\right).P\left(E/E_1\right)+P\left(E_2\right).P\left(E/E_2\right)$
$P\left(E\right)=\frac{2}{3}\times \frac{1}{6}+\frac{1}{3}\times \frac{1}{2}=\frac{5}{18}$
The odds against $H$ winning are $(18-5):5$ = $13:5$