Question

In a race, $3$ horses $A,B,C$ are participating. The probability of $A$ winning is twice the probability of $B$. The probability of winning $B$ is twice the probability of $C$ winning. Then find the probabilities of winning $A,B$ and $C$.

Answer 1

$P\left(A\right)$ is the probability of $A$ winning the race, $P\left(B\right)$ is the probability of $B$ winning the race and $P\left(C\right)$ is the probability of $C$ winning the race.

According to the given conditions, $P\left(A\right)=2P\left(B\right)$ and $P\left(B\right)=2P\left(C\right)$

Also, $P\left(A\right)+P\left(B\right)+P\left(C\right)=1$ since either $A,B$ or $C$ has to win the race.

$\Rightarrow 4P\left(C\right)+2P\left(C\right)+P\left(C\right)=1$

$\therefore P\left(C\right)=\frac{1}{7}$

$\Rightarrow P\left(B\right)=\frac{2}{7},P\left(A\right)=\frac{4}{7}$