In a competitive examination, an examinee either guesses, or copy or…

The correct answer is Option (2) → \( \displaystyle\frac{24}{29} \)

Let E₁, E₂, E₃ be the events that he guesses, copies and knows the answer, and A the event that his answer is correct. The three cases cover every possibility, so

    \( \displaystyle P(E_1)=\frac{1}{3},\quad P(E_2)=\frac{1}{6} \)

    \( \displaystyle P(E_3)=1-\frac{1}{3}-\frac{1}{6}=\frac{1}{2} \)

A guess picks one of four choices, and a known answer is always correct:

    \( \displaystyle P(A|E_1)=\frac{1}{4},\quad P(A|E_2)=\frac{1}{8},\quad P(A|E_3)=1 \)

By Bayes’ theorem,

    \( \displaystyle P(E_3|A)=\frac{P(E_3)P(A|E_3)}{P(E_1)P(A|E_1)+P(E_2)P(A|E_2)+P(E_3)P(A|E_3)} \)

    \( \displaystyle =\frac{\frac{1}{2}\times 1}{\frac{1}{3}\times\frac{1}{4}+\frac{1}{6}\times\frac{1}{8}+\frac{1}{2}\times 1} \)

    \( \displaystyle =\frac{\frac{1}{2}}{\frac{1}{12}+\frac{1}{48}+\frac{1}{2}} \)

    \( \displaystyle =\frac{\frac{24}{48}}{\frac{4+1+24}{48}} \)

    \( \displaystyle =\frac{24}{29} \)

Shortcut: Multiply the three products \( \frac{1}{12},\ \frac{1}{48},\ \frac{1}{2} \) by 48 to get the ratio 4 : 1 : 24 for guess : copy : know among correct answers. The share of “know” is \( \frac{24}{4+1+24}=\frac{24}{29} \).

Why the other options are incorrect

• Option (1) \( \displaystyle\frac{5}{29} \) — Incorrect. \( \frac{5}{29}=\frac{4+1}{29} \) is the probability that he guessed or copied, given a correct answer, which is the complement of the required event.

• Option (3) \( \displaystyle\frac{22}{29} \) — Incorrect. The share of “know” in the ratio 4 : 1 : 24 is 24 parts out of 29, not 22.

• Option (4) \( \displaystyle\frac{7}{29} \) — Incorrect. The share of “know” in the ratio 4 : 1 : 24 is 24 parts out of 29, not 7.