What is the probability that a student knows the answer given that he has answered it correctly,....?

Question

A large class in stochastic processes at at a school is taking a multiple choice test. For one particular question with m proposed multiple choice answers, the fraction of students who know the answer is p; the others will guess. The probability of answering the question correctly is 1 for the students who know the answer and 1/m for the ones who guess. What is the probability that a student knows the answer given that he has answered it correctly?

Answer 1

$p\left[p\left(1-n\right)+n\right]^{-1}$

Assuming that $n$ is the probably of getting the question correct for those who guess. With reasonable assumption, $n=m^{-1}$.

Answer 2

If $A$ means "The student knows the answer" and $B$ means "The student has answered correctly", we have $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$. If we consider that those who guess give the correct answer $1$ out of $m$ times, $P(B|A)=1$, $P(A)=p$ and $P(B)=p+(1-p)/m$.

So in the end, $P(A|B) = \frac{mp}{mp-p+1}$