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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?

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    $\begingroup$ You are missing the probability of guessing the answer. $\endgroup$ – John Habert Feb 18 '14 at 14:00
  • $\begingroup$ Well, if the word "guess" is used, I presume we can consider it is $1/m$. Or maybe I am extrapolating too much ? $\endgroup$ – Traklon Feb 18 '14 at 14:19
  • $\begingroup$ It have multiple choice answers how we can presume 1/m? $\endgroup$ – peeyush.cray Feb 18 '14 at 14:21
  • $\begingroup$ Because there are $m$ answers for the question. $\endgroup$ – Traklon Feb 18 '14 at 14:21
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    $\begingroup$ Please delete all your comments that just restate the question! $\endgroup$ – TonyK Feb 18 '14 at 20:42
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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, $P(A|B) = \frac{mp}{mp-p+1}$

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  • $\begingroup$ A large class in stochastic processes at BUI 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? $\endgroup$ – Hassan Feb 18 '14 at 20:11
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$p\left[p\left(1-n\right)+n\right]^{-1}$

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

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  • $\begingroup$ A large class in stochastic processes at BUI 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? $\endgroup$ – Hassan Feb 18 '14 at 20:11
  • $\begingroup$ @Hassan, your question has already been answered twice. $\endgroup$ – user129714 Feb 19 '14 at 15:04

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