The probability of India winning a test match against West Indies is $\frac{1}{2}$. Assuming independence from match  to match, the probability that in a five match series India's second win occurs at third test is

Given that

$P\left(\text{ India wins }\right)=p=1/2$

$\therefore P\left(\text{ India loses }\right)=p^{\mathrm{\prime }}=1/2$

Out of 5 matches, India's second win occurs at third test. Hence, India wins third test and simultaneously it has won one match from first two and lost the other. Therefore, the required probability is 

$\begin{array}{c}\mathrm{P}\left(\mathrm{LWW}\right)+\mathrm{P}\left(\mathrm{WLW}\right)={\left(\frac{1}{2}\right)}^3+{\left(\frac{1}{2}\right)}^3\\ =\frac{1}{4}\end{array}$