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Suppose a class of probability and statistics is taking a multiple-choice test. Suppose for a certain question on the test, the fraction of students who know the answer is p, and 1 − p is the fraction that will guess. The probability of answering a question correctly is unity for a student who knows the answer and 1/m for the guessee; m is the number of multiple-choice alternatives. Compute the probability that a student knew the answer to a question given that he or she correctly answered it.

F.A.
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Technical Gen.
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  • Possibly helpful: https://math.stackexchange.com/questions/2279851/applied-probability-bayes-theorem/2279888#2279888 – Ethan Bolker Apr 14 '18 at 13:42

1 Answers1

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Pick out a student.

Let $C$ denote the event that the answer given by the student is correct.

Let $K$ denote the event that the student has knowledge of the answer.

Then to be found is $P(K\mid C)$ and we can make use of:$$P(K\mid C)P(C)=P(K\cap C)=P(C\mid K)P(K)$$

This equation tells us that it is enough to find $P(C),P(C\mid K)$ and $P(K)$.

It is not difficult to find $P(C\mid K)$ and $P(K)$ in this context.

In order to find $P(C)$ you can use:$$P(C)=P(C\mid K)P(K)+P(C\mid K^{\complement})P(K^{\complement})$$

drhab
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  • Can you please explain your answer. Sorry I am very very very weak in probability. – Technical Gen. Apr 14 '18 at 10:41
  • Have you found $P(K)$ and $P(C\mid K)$ already? – drhab Apr 14 '18 at 10:44
  • No I didn't find it yet – Technical Gen. Apr 14 '18 at 16:24
  • "the fraction of students who know the answer is $p$..." So what is $P(K)$ (i.e. the probability that the student knows the answer)? Further: what is - under the condition that the student knows the answer - the probability that the student gives correct answer? This is $P(C\mid K)$. – drhab Apr 14 '18 at 16:26