Question

Of the students in a college, it is known that 60% reside in a hostel and 40% do not reside in hostel. Previous year results report that 30% of students residing in hostel attain A grade and 20% of ones not residing in hostel attain A grade in their annual examination. At the end of the year, one students is chosen at random from the college and he has an A grade. What is the probability that the selected student is a hosteler?

Answer 1

Let A, E₁ and E₂ denote the events that the selected student attains grade A, resides in a hostel and does not reside in a hostel, respectively.

$$\begin{gathered} \therefore P\left(E_1\right)=\frac{60}{100} \\ P\left(E_2\right)=\frac{40}{100} \\ \mathrm{Now}, \\ P\left(A/E_1\right)=\frac{30}{100} \\ P\left(A/E_2\right)=\frac{20}{100} \\ \mathrm{Using}\mathrm{Bayes}\text{'}\mathrm{theorem},\mathrm{we}\mathrm{get} \\ \mathrm{Required}\mathrm{probability}=P\left(E_1/A\right)=\frac{P\left(E_1\right)P\left(A/E_1\right)}{P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)} \\ =\frac{{\displaystyle \frac{60}{100}}\times {\displaystyle \frac{30}{100}}}{{\displaystyle \frac{60}{100}}\times {\displaystyle \frac{30}{100}}+{\displaystyle \frac{40}{100}}\times {\displaystyle \frac{20}{100}}} \\ =\frac{18}{18+8}=\frac{9}{13} \end{gathered}$$