These days competitive examinations are held in online mode. During the examination student either guesses or copies or knows the answer of a multiple choice question (MCQ) with four choices each. The probability that a student makes guesses is \(\displaystyle\frac{1}{3}\) and that he/she copies \(\displaystyle\frac{1}{6}\). The probability of marking answer correctly when he/she copied is \(\displaystyle\frac{1}{8}\) and probability of making answer correctly when he/she guesses is \(\displaystyle\frac{1}{4}\). It is also assumed that a student marked answer correctly if he/she knows the answer.
By using the above mentioned information answer following question.
The probability that he copied it given that his answer is correct :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{1}{29}\)
Let A be the event that the answer is correct. By the theorem of total probability,
\( \displaystyle P(A) = \frac{1}{3} \times \frac{1}{4} + \frac{1}{6} \times \frac{1}{8} + \frac{1}{2} \times 1 = \frac{29}{48} \)
Here \(\frac{1}{2} = 1 - \frac{1}{3} - \frac{1}{6}\) is the probability that the student knows the answer. The copying term is
\( \displaystyle P(\text{copies})\,P(A|\text{copies}) = \frac{1}{6} \times \frac{1}{8} = \frac{1}{48} \)
By Bayes’ theorem,
\( \displaystyle P(\text{copies}|A) = \frac{1/48}{29/48} = \frac{1}{29} \)