Management Aptitude Test 2015 Paper

(c) Required probability = P (A) + $\mathit{P}\stackrel{-}{\left(\mathit{A}\right)}\cdot \mathit{P}\stackrel{-}{\left(\mathit{B}\right)}$ P (A) + ...
= $\frac{1}{2}+{\left(\frac{1}{2}\right)}^3+{\left(\frac{1}{2}\right)}^5+{\left(\frac{1}{2}\right)}^7$+ ....... ∞
Which an infinite term of geometric progression.
Here, first term = $\frac{1}{2}$ and common ratio = $\frac{1}{4}$
∴ ${\mathit{S}}_{\mathit{n}}=\frac{\mathit{a}\left(1-{\mathit{r}}^{\mathit{n}}\right)}{1-\mathit{r}}$
= $\frac{\frac{1}{2}\left\{{\left(1-\frac{1}{4}\right)}^{\mathit{\infty }}\right\}}{1-\frac{1}{4}}$
= $\frac{\frac{1}{2}}{\frac{3}{4}}=\frac{1}{2}\times \frac{4}{3}$
∴ P(winning of A) = $\frac{2}{3}$
and P(winnig of B) = $1-\frac{2}{3}=\frac{1}{3}$