AP ICET 2016 Paper

Given : $\mathit{p}\left(\mathit{A}\right)=\frac{7}{12},\mathit{p}\left(\mathit{B}\right)=\frac{9}{17}$
$\Rightarrow \mathit{P}\left(\overline{\mathit{A}}\right)=\frac{5}{12},\mathit{p}\left(\overline{\mathit{B}}\right)=\frac{8}{17}$
Probability that A and B can’t solve the problem
$=\mathit{p}\left(\overline{\mathit{A}}\right)\times \mathit{p}\left(\overline{\mathit{B}}\right)$
Probability that the problem is solved.
= 1 – probability that A and B can’t solve the problem.
$=1\mathit{–}\left[\mathit{p}\left(\overline{\mathit{A}}\right)\times \mathit{p}\left(\overline{\mathit{B}}\right)\right]$
$=1-\left(\frac{5}{12}\times \frac{8}{17}\right)=1-\frac{10}{51}=\frac{41}{51}$
Aliter : Problem can be solved in either of the following three cases :(i) A solves but not B (ii) B solves but not A (iii) Both A and B solve.
∴ Required probability
$=\frac{56+45+63}{12\times 17}=\frac{164}{12\times 17}=\frac{41}{53}$