Question

Of the students in a college, it is known that 60% reside in hostels and 40% are day scholars (not residing in hostels). Previous year results report that 30% of all students who reside in hostels 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 resides in a hostel?
(a) $\dfrac{5}{13}$
(b) $\dfrac{9}{13}$
(c) $\dfrac{8}{13}$
(d) None of these

Answer 1

Hint: We will assume that there are 100 students in the college. We will find the number of students living in hostels and day scholars using the percentage given. Then we will use the information given in terms of the percentage of students attaining A grade to calculate the number of students living in hostels who attain A grade and the number of day scholars who attain A grade. After that, we will calculate the probability that a randomly chosen student having an A grade resides in the hostel.

Complete step-by-step solution
We know that 60% of the college students reside in hostels and 40% are day scholars. That means 60 out of 100 students reside in the hostel and 40 out of 100 students are day scholars. So, for convenience, let us assume that there are 100 students in the college. Hence, we know that 60 students reside in the hostel and 40 students are day scholars.
Now, we are given that according to the previous year results report, 30% of all students who reside in the hostel attain A grade. So, to find the number of students residing in hostels who obtained A grade, we will calculate 30% of 60. The calculation is as follows,
$\begin{align}
 & \text{30 }\% \text{ of 60 }=\dfrac{30}{100}\times 60 \\
 & =3\times 6 \\
 & =18
\end{align}$
Therefore, there are 18 students who reside in hostels that have attained a grade.
Next, we are given that 20% of day scholars attain A grade. There are 40 day scholars. So, to find the number of day scholars who obtained A grade, we have to calculate 20% of 40. The calculation is as follows,
$\begin{align}
 & \text{20 }\%\text{ of 40 }= \dfrac{20}{100}\times 40 \\
 & =2\times 4 \\
 & =8
\end{align}$
Therefore, we have 8 day scholars who obtained A grade.
We have to calculate the probability that a randomly chosen student having an A grade resides in a hostel. Let us denote this probability by $P$. So, we have the following expression,
$P=\dfrac{\text{Number of students that reside in the hostel who have attained A grade}}{\text{Total number of students who attained A grade}}$
The total number of students who attained A grade is $18+8=26$. Substituting the required values in the above expression, we get
$P=\dfrac{18}{26}$
Reducing the above fraction, we have $P=\dfrac{9}{13}$.
Since this answer is not in any of the choices, the correct option is (d).

Note: It is essential to understand the concept of percentage in this question. We were able to assume the number of students in the college to be a hundred because we were given information about students in percentage. This problem can also be solved by assuming the number of students to be $x$ instead of 100.