Question

Of the students in a college,it is known that $60\%$ reside in hostel and $40\%$ are day scholers.(not residing in hostel).Previous year result 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 choosen at random from the college and he has an A grade,What is the probability that the student is a hostler?

Answer 1

The correct answer is: $\frac{9}{13}$
Let $E_1$=Students residing in the hostel, $E_2$=day scholars(not residing in the hostel) and A=Student who attain grade A,
$P(E_1)=\frac{60}{100}, P(E_2)=\frac{40}{100}$
$P(A|E_1)=0.3,P(A|E_2)=0.2$
The probability that a randomly chosen student is a hostler, given that he has an A grade, is given by $P(E_1|A)$
Therefore,by Bayes'theorem,
$P(E_1|A)=\frac{P(E_1)P(A|E_1)}{P(E_1)P(A|E_1)+P(E_2)P(A|E_2)}$
$=\frac{0.6\times 0.3}{0.6\times 0.3+0.4\times 0.2}$
$=\frac{0.18}{0.26}$
$=\frac{18}{26}$
$=\frac{9}{13}$