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,
\(P(E_1|A)=\frac{P(E_1)P(A|E_1)}{P(E_1)P(A|E_1)+P(E_2)P(A|E_2)}\)
\(=\frac{0.6\times 0.3}{0.6\times 0.3+0.4\times 0.2}\)
\(=\frac{0.18}{0.26}\)
\(=\frac{18}{26}\)
\(=\frac{9}{13}\)