Question

Of the students in a college, it is known that $60$% reside in hostel and $40$% are day scholars .Previous year 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 chose at random from the college and he has an 'A' grade, what is the probability that the student is a hostlier?

Answer 1

Let $E_1,E_2$ and $A$ be events such that

$E_1$ = students is a hostlier

$E_2$ = students is a day scholar

$A$ = getting A grade

Now from equation,

$P\left(E_1\right)=\frac{60}{100}=\frac{6}{10}$

$P\left(E_2\right)=\frac{40}{100}=\frac{4}{10}$

$P\left(\frac{A}{E_1}\right)=\frac{30}{100}=\frac{3}{10}$

$P\left(\frac{A}{E_2}\right)=\frac{20}{100}=\frac{2}{10}$

We have to find $P\left(\frac{E_1}{A}\right)$

Now, $P\left(\frac{E_1}{A}\right)=\frac{P\left(E_1\right).P\left(\frac{A}{E_1}\right)}{P\left(E_1\right).P\left(\frac{A}{E_1}\right)+P\left(E_2\right).P\left(\frac{A}{E_2}\right)}$

$=\frac{\frac{6}{10}.\frac{3}{10}}{\frac{6}{10}.\frac{3}{10}+\frac{4}{10}.\frac{2}{10}}$

$=\frac{\frac{18}{100}}{\frac{18}{100}+\frac{8}{100}}\Rightarrow \frac{18}{100}\times \frac{100}{26}\Rightarrow \frac{18}{26}\Rightarrow \frac{9}{13}$