Question

In a competition A, B and C are participating. The probability that A wins is twice that of B, the probability that B wins is twice that of C. Find the probability that A losses.

Answer 1

$$\begin{gathered} P\left(\text{A wins}\right)=P\left(A\right) \\ P\left(\text{B wins}\right)=P\left(B\right)=\frac{P\left(A\right)}{2} \\ P\left(\text{C wins}\right)=P\left(C\right)=\frac{P\left(B\right)}{2}=\frac{P\left(A\right)}{4} \\ \mathrm{Now}, \\ P\left(A\right)+P\left(B\right)+P\left(C\right)=1 \\ \Rightarrow P\left(A\right)+\frac{P\left(A\right)}{2}+\frac{P\left(A\right)}{4}=1 \\ \Rightarrow P\left(A\right)\left[\frac{4+2+1}{4}\right]=1 \\ \Rightarrow P\left(A\right)=\frac{4}{7} \\ P\left(\text{A loses}\right)=P\left(\overline{A}\right) \\ =1-\frac{4}{7} \\ =\frac{3}{7} \end{gathered}$$