Question

Of the students in college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous years results 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 chosen at random from the college and he has an A grade, what is the probability that the student is a hostler?

Answer 1

Let $E_1$ : the event that the student is residing in hostel and $E_2$ : the event that the student is not residing in the hostel.

Let E: a student attains A grade,

Then, $E_1$ and $E_2$ are mutually exclusive and exhaustive. Moreoever,

P$E_1$ =60%= $\frac{60}{100}=\frac{3}{5}$ and P$E_2$ =40%=$\frac{40}{100}=\frac{2}{5}$

Then, $P\left(\frac{E}{E_1}\right)=30$ and $P\left(\frac{E}{E_2}\right)=20$

By using Baye's theorem, we obtain

$P\left(\frac{E}{E_1}\right)=\frac{P\left(\frac{E}{E_1}\right)P\left(E_1\right)}{P\left(\frac{E}{E_1}\right)P\left(E_1\right)+P\left(\frac{E}{E_2}\right)P\left(E_2\right)}=\frac{\frac{3}{10}\times \frac{3}{5}}{\frac{3}{10}\times \frac{3}{5}+\frac{2}{10}\times \frac{2}{5}}=\frac{9}{9+4}=\frac{9}{13}$