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 to 1 that B rides A, in which case all the horses are equally likely to win, if C rides A, his chance is trebled, then the probability A wins the race, is 

Consider, the events E_1, E₂and A. 

$E_1=$  jockey B rides the horse A 

 $E_2=$ jockey C rides the horse B 

  A= A's win the race

  $\begin{array}{c}P\left(E_1\right)=2/3,P\left(E_2\right)=1/3\\ P\left(A/{\mathrm{E}}_1\right)=1/6,P\left(A/{\mathrm{E}}_2\right)=3/6\end{array}$

 $\therefore$     Required probability  

  $\begin{array}{l}P\left(A\right)=P\left(E_1\right)\times P\left(A/{\mathrm{E}}_1\right)+P\left(E_2\right)\times P\left(A/{\mathrm{E}}_2\right)\\ =\frac{2}{3}\times \frac{1}{6}+\frac{1}{3}\times \frac{3}{6}\\ =\frac{2+3}{18}=\frac{5}{18}\end{array}$

Hence, option (b) is correct.