In a college, 50% of the students are in Science, 30% in Commerce and 20% in Arts. The probabilities that a student of Science, Commerce and Arts qualifies an entrance test are 0.6, 0.5 and 0.3 respectively. A student of the college is chosen at random. Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | Probability that the student qualifies the test | (I) | \(\displaystyle\frac{5}{17}\) |
| (B) | Probability that the student is from Science, given that the student qualifies | (II) | \(\displaystyle\frac{2}{7}\) |
| (C) | Probability that the student is from Commerce, given that the student qualifies | (III) | \(\displaystyle\frac{51}{100}\) |
| (D) | Probability that the student is from Arts, given that the student does not qualify | (IV) | \(\displaystyle\frac{10}{17}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
Let \(E_1, E_2, E_3\) be the events that the student is from Science, Commerce and Arts, and Q the event that the student qualifies.
\(P(E_1)P(Q|E_1) = 0.5\times 0.6 = 0.30\)
\(P(E_2)P(Q|E_2) = 0.3\times 0.5 = 0.15\)
\(P(E_3)P(Q|E_3) = 0.2\times 0.3 = 0.06\)
(A) Probability that the student qualifies the test
This matches (III) \(\frac{51}{100}\).
By the theorem of total probability:
\(P(Q) = 0.30 + 0.15 + 0.06 = 0.51\)
(B) Probability that the student is from Science, given that the student qualifies
This matches (IV) \(\frac{10}{17}\).
By Bayes’ theorem:
\( \displaystyle P(E_1|Q) = \frac{0.30}{0.51} = \frac{10}{17} \)
(C) Probability that the student is from Commerce, given that the student qualifies
This matches (I) \(\frac{5}{17}\).
\( \displaystyle P(E_2|Q) = \frac{0.15}{0.51} = \frac{5}{17} \)
(D) Probability that the student is from Arts, given that the student does not qualify
This matches (II) \(\frac{2}{7}\).
An Arts student fails with probability 0.7, and \(P(Q') = 1-0.51 = 0.49\):
\( \displaystyle P(E_3|Q') = \frac{0.2\times 0.7}{0.49} = \frac{0.14}{0.49} = \frac{2}{7} \)