Question

Of the students in a college, it is known that 60% reside in hostels and 40% are day scholars. Previous year results report that 30% of all the students who reside in hostel 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

Hint: For solving this question, firstly we will let the student selected is a hostler be H, the student selected is a day scholar be ‘D’ and student has attained ‘A’ grade be ‘A’. After that we will be finding the probability that the selected student is a hostler by finding the probability that the student is a hostler, probability that the student is a day scholar, probability that the student gets an “A” grade if hostler and probability that the students gets an “A” grade if day scholar.

Complete step by step answer:
We have been provided that out of the students in a college 60% of the college students live in hostels. So, let the student selected as a hostler be H.
Now we will be finding the probability that student is a hostler using the formula:
Probability= number of possible outcomes $ \div $total number of outcomes
Probability that student a hostler P(H)= $60\% = \dfrac{{60}}{{100}} = 0.6$
 40% of the college students are day scholars. So, let the student selected as a day scholar be D.
Now we will be finding the probability that student is a day scholar using the formula:
Probability= number of possible outcomes $ \div $total number of outcomes
Probability that student is a day scholar P(D)= $40\% = \dfrac{{40}}{{100}} = 0.4$
30% of the college students attain an ‘A’ grade. So, let the student selected have an ‘A’ grade be ‘A’.
Now we will be finding the probability that student gets ‘A’ grade, if hostler using the formula:
Probability= number of possible outcomes $ \div $total number of outcomes
Probability that student is a hostler $P\left( {\dfrac{A}{H}} \right) = 30\% = \dfrac{{30}}{{100}} = 0.3$
Now we will be finding the probability that students get ‘A’ grade, if day scholar by using the formula:
Probability= number of possible outcomes $ \div $total number of outcomes

Probability that student gets ‘A’ grade, if day scholar $P\left( {\dfrac{A}{D}} \right) = 20\% = \dfrac{{20}}{{100}} = 0.2$
We need to find the probability that the student selected is a hostler, if he has an ‘A’ grade that is
$P\left( {\dfrac{H}{A}} \right) = \dfrac{{P(H).P\left( {\dfrac{A}{H}} \right)}}{{P(D).P\left( {\dfrac{A}{D}} \right) + P(H).P\left( {\dfrac{A}{H}} \right)}}$
Now we will be keeping the values in the above equation:
$P\left( {\dfrac{H}{A}} \right) = \dfrac{{0.6 \times 0.3}}{{0.4 \times 0.2 + 0.6 \times 0.3}}$
Now simplifying the above equation:
$P\left( {\dfrac{H}{A}} \right) = \dfrac{{0.18}}{{0.08 + 0.18}}$
So, the probability comes out to be:
$P\left( {\dfrac{H}{A}} \right) = \dfrac{9}{{13}}$
So, this is the solution for this question.

Note:
In this question we should be aware about the formula for finding probability. While using this formula: $P\left( {\dfrac{H}{A}} \right) = \dfrac{{P(H).P\left( {\dfrac{A}{H}} \right)}}{{P(D).P\left( {\dfrac{A}{D}} \right) + P(H).P\left( {\dfrac{A}{H}} \right)}}$ do mention the probability of the student being a day scholar that is P(D). Also, the sign in the denominator must be positive, i.e terms must be added.