40% students of a college reside in hostel and the remaining reside outside. At the end of the year, 50% of the hostelers got A grade while from outside students, only 30% got A grade in the examination. At the end of the year, a student of the college was chosen at random and was found to have gotten A grade. What is the probability that the selected student was a hosteler ?

#### Solution

Let E_{1} and E_{2} be the events that the student is a hosteller or an outside student, respectively and A be the event that the chosen student gets A grade.

∴P(E_{1})=40%=40/100=0.4

P(E_{2})=(100−40)%=60%=60/100=0.6

P(A|E_{1})=P(Student getting A grade is a hosteller)=50%=0.5

P(A|E_{2})=P(Student getting A grade is an outside student)=30%=0.3

The probability that a randomly chosen student is a hosteller, given that he got A grade, is given by P(E_{1}|A).

Using Bayes’ theorem, we get

`P(E_1|A)=(P(E1)⋅P(A|E_1))/(P(E_1)⋅P(A|E_1)+P(E_2)⋅P(A|E_2))`

`=(0.4xx0.5)/(0.4xx0.5+0.6xx0.3)`

`=0.20/0.38 `

`=10/19`