In a school, 40% of the students live in the hostel and the rest are day scholars. 30% of the hostel students and 20% of the day scholars get an A grade. A student chosen at random has got an A grade. The probability that the student lives in the hostel is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{2}\)
Let \(E_1\) be the event that the student lives in the hostel, \(E_2\) that the student is a day scholar, and A that the student gets an A grade.
\(P(E_1)P(A|E_1) = 0.4\times 0.3 = 0.12\)
\(P(E_2)P(A|E_2) = 0.6\times 0.2 = 0.12\)
By Bayes’ theorem:
\( \displaystyle P(E_1|A) = \frac{0.12}{0.12+0.12} = \frac{1}{2} \)
The two groups give the same number of A grades, so an A-grade student is equally likely to be from either group.
Why the other options are incorrect
Option (1) \(\displaystyle\frac{2}{5}\) — Incorrect. This is the share of hostel students before the grade is known.
Option (2) \(\displaystyle\frac{3}{10}\) — Incorrect. This is \(P(A|E_1)\), the chance that a hostel student gets an A grade; the condition is the wrong way round.
Option (3) \(\displaystyle\frac{6}{25}\) — Incorrect. This is P(A) = 0.12 + 0.12 = 0.24, the chance that a student gets an A grade.