In a coaching class, 40% of the students attend regularly and the rest do not. A regular student passes the final test with probability 0.9, and a student who does not attend regularly passes it with probability 0.4. A student chosen at random is found to have failed the test. The probability that the student attended regularly is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{10}\)
Let \(E_1\) be the event that the student attended regularly, \(E_2\) that the student did not, and F that the student failed. The chances of failing are \(1-0.9 = 0.1\) and \(1-0.4 = 0.6\).
\(P(E_1)P(F|E_1) = 0.4\times 0.1 = 0.04\)
\(P(E_2)P(F|E_2) = 0.6\times 0.6 = 0.36\)
By the theorem of total probability:
\(P(F) = 0.04 + 0.36 = 0.40\)
By Bayes’ theorem:
\( \displaystyle P(E_1|F) = \frac{0.04}{0.40} = \frac{1}{10} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{9}{10}\) — Incorrect. This is \(P(E_2|F)\), the chance that the failed student did not attend regularly.
Option (2) \(\displaystyle\frac{2}{5}\) — Incorrect. This is the share of regular students before the result is known (and also P(F) = 0.40), not the conditional chance.
Option (3) \(\displaystyle\frac{3}{5}\) — Incorrect. This is \(\frac{0.36}{0.60}\), the chance that a student who passed attended regularly.