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 of answering a question correctly, is :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{29}{48}\)
A student guesses, copies or knows, with probabilities
\( \displaystyle \frac{1}{3},\quad \frac{1}{6},\quad 1 - \frac{1}{3} - \frac{1}{6} = \frac{1}{2} \)
and the answer is then correct with probability \(\frac{1}{4}\), \(\frac{1}{8}\) and 1. By the theorem of total probability,
\( \displaystyle P(\text{correct}) = \frac{1}{3} \times \frac{1}{4} + \frac{1}{6} \times \frac{1}{8} + \frac{1}{2} \times 1 \)
\( \displaystyle = \frac{4}{48} + \frac{1}{48} + \frac{24}{48} \)
\( \displaystyle = \frac{29}{48} \)