$A, B, C$  are three horses in a race. The probability of $A$ to win the race is twice that of $B$ , and probability of $B$ is twice that of $C$ . What are the probabilities of $A, B$  and $C$ to win the race?

$A,B,C$  are three horses in a race

Given

$P\left(A\right)=2P\left(B\right)\mathrm{\&}P\left(B\right)=2P\left(C\right)\Rightarrow P\left(C\right)=\frac{1}{2}P\left(B\right)$

A,B,C are mutually exclusive and exhaustive events then 

$$\begin{gathered} P(A\cup B\cup C)=1 \\ \Rightarrow P\left(A\right)+P\left(B\right)+P\left(C\right)=1 \end{gathered}$$

$2P\left(B\right)+P\left(B\right)+\frac{1}{2}P\left(B\right)=1$

$$\begin{gathered} P\left(B\right)\left(2+1+\frac{1}{2}\right)=1 \\ \Rightarrow P\left(B\right)\left(\frac{4+2+1}{2}\right)=1 \end{gathered}$$

$$\begin{gathered} \Rightarrow P\left(B\right)=\frac{2}{7} \\ \Rightarrow P\left(A\right)=2.P\left(B\right)=2\times \frac{2}{7}=\frac{4}{7} \end{gathered}$$

$P\left(C\right)=\frac{1}{2}\cdot P\left(B\right)=\frac{1}{2}\times \frac{2}{7}=\frac{1}{7}$

$\therefore P\left(A\right)=\frac{4}{7},P\left(B\right)=\frac{2}{7},and P\left(C\right)=\frac{1}{7}$