Of the students in a college, it is known that 60\% reside in hostel and 40\% are day scholars (not residing in hostel). Previous year results results report that 30\% of all students who reside in hostel attain A grade and 20 of day scholars attain $A$ grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is a hostlier?

.: Let ${E_1}$ : `A student is hostlier' and

${E_2}$ : `A student is day scholar'

$\Rightarrow$ $P\left( {{E_1}} \right) = \cfrac{{60}}{{100}} = \cfrac{3}{5}$

and $P\left( {{E_2}} \right) = \cfrac{{40}}{{100}} = \cfrac{2}{5}$

Let $E$ : student attains A grade'

then, $P\left( {E/{E_1}} \right) = \cfrac{{30}}{{100}} = \cfrac{3}{{10}}$

and $P\left( {E/{E_2}} \right) = \cfrac{{20}}{{100}} = \cfrac{2}{{10}}$

Hence, required probability is
$P\left( {{E_1}|E} \right) = \cfrac{{P\left( {E|{E_1}} \right)P\left( {{E_1}} \right)}}{{P\left( {E|{E_1}} \right)P\left( {{E_1}} \right) + P\left( {E|{E_2}} \right)P\left( {{E_2}} \right)}}$

$= \cfrac{5}{{10}} \times \cfrac{7}{{12}} + \cfrac{5}{{10}} \times \cfrac{5}{{12}} = \cfrac{{35 + 25}}{{120}} = \cfrac{{60}}{{120}} = \cfrac{1}{2}$

$= \cfrac{{\cfrac{3}{1} \times \cfrac{3}{5}}}{{\cfrac{3}{{10}} \times \cfrac{3}{5} + \cfrac{2}{{10}} \times \cfrac{2}{5}}} = \cfrac{9}{{9 + 4}} = \cfrac{9}{{13}}$