UPSC NDA 2 2017 Mathematics Solved Paper

Concept:Bayes' theorem is used to find the probability that the examinee knows the answer, given that the answer is correct.Explanation:Let event $K$ be that the examinee knows the answer.Let event $G$ be that the examinee guesses.Let event $C$ be that the answer is correct.We are given:$P(K) = p$, $P(G) = 1-p$.If the examinee guesses, the probability of a correct answer is $P(C|G) = \frac{1}{m}$.If the examinee knows the answer, $P(C|K) = 1$.First, compute the total probability of a correct answer using the law of total probability:$P(C) = P(K) \cdot P(C|K) + P(G) \cdot P(C|G) = p \cdot 1 + (1-p) \cdot \frac{1}{m} = p + \frac{1-p}{m}$.We require the conditional probability $P(K|C)$: the probability that the examinee knows the answer given that the answer is correct.By Bayes' theorem:$P(K|C) = \frac{P(K) \cdot P(C|K)}{P(C)} = \frac{p}{p + \frac{1-p}{m}}$.Simplify the expression:$P(K|C) = \frac{p}{\frac{mp + 1 - p}{m}} = \frac{mp}{mp + 1 - p} = \frac{mp}{1 + (m-1)p}$.Answer:$\frac{mp}{1 + (m-1)p}$, which matches option B.