Question

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 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 hosteller?

Answer 1

Let $E_1$ and $E_2$ be the events that the student is a hostler and a day scholar respectively and $A$ be the event that the chosen student gets grade $A$.

$\therefore P\left(E_1\right)=60$% $=\frac{60}{100}=0.6$

$P\left(E_2\right)=40$% $=\frac{40}{100}=0.4$

$P\left(A|E_1\right)=P\left(studentgettinganAgradeisahostler\right)=30$% $=0.3$
$P\left(A|E_2\right)=P\left(studentgettinganAgradeisadayscholar\right)=20$% $=0.2$

The probability that a randomly chosen student is a hostler, given that he has an $A$ grade, is given by $P\left(E_1|A\right)$.
By using Baye's theorem, we obtain

$P\left(E_1|A\right)=\frac{P\left(E_1\right)\cdot P\left(A|E_1\right)}{P\left(E_1\right)\cdot P\left(A|E_1\right)+P\left(E_2\right)\cdot P\left(A|E_2\right)}$

$=\frac{0.6\times 0.3}{0.6\times 0.3+0.4\times 0.2}$

$=\frac{0.18}{0.26}$

$=\frac{18}{26}$

$=\frac{9}{13}=0.69$