Question

A is one of the $6$ horses entered for a race and is to be ridden by one of two jockeys B or C. It is $2:1$ that B rides A, in which case all the horses are equally likely to win; if C rides A, his chance is gets tripled. If the odds against his winning is $\frac{p}{q}$, where p and q are co-primes then find the value of $p+q$.

Answer 1

To find the odd against $A$'s winning, first we are to find the probability of $A$'s winning. And this can happen in one of the two mutual exclusive ways:

i) $B$ rides and $A$ wins.

ii) $C$ rides and $A$ wins.

The chance for $B$ to ride the horse $A$ is $2$ to $1$, i.e. $P\left(B\right)=\frac{2}{3}\Rightarrow P\left(C\right)=\frac{1}{3}$.

The chance for $A$ to win is equally likely if $B$ rides, i.e. $\frac{1}{6}$.

The probability for $A$ to win if $C$ rides, is trippled i.e.$\frac{3}{6}$.

The probability of $A$'s win $=\frac{2}{3}\times \frac{1}{6}+\frac{1}{3}\times \frac{3}{6}=\frac{5}{18}$.

The probability of $A$'s losing is $\frac{13}{18}$.

The odds against $A$'s winning are $13:5$.

According to the problem $p=13,q=5$.

$\therefore p+q=18$.