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.

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