Please show in detail A multiple-choice question, with choices (a), (b), (c), (d

Question

Please show in detail

A multiple-choice question, with choices (a), (b), (c), (d) and (e), is given on an to a class of a large number of students. Suppose that only 45% of the students know the correct answer to this question; the rest of the class have absolutely no clue and are just randomly guessing. If the teacher randomly selects a completed exam, and if he sees that the right choice was circled for this question, what is the conditional probability that the student who handed in this particular exam actually knew the correct answer?

Explanation / Answer

Let the students who knows the answer be represented as X and rest be represented as Y. Then we are given that:

P(X) = 0.45 and therefore P(Y) = 0.55

P( correct | X) = 1 because they know the correct answer.

P( correct | Y) = 0.2 as they are randomly guessing and there are total of 5 options to guess from.

Therefore now we get:

P( correct ) = P( correct | X) P(X) + P( correct | Y) P(Y) = 0.45 + 0.55*0.2 = 0.56

Now we have to find the conditional probability that given the answer was correct what was the probability that the students actually knew the answer:

P( X | correct ) = ?

From Bayes thorem we get:

P ( X | correct ) P(correct ) = P( correct | X) P(X)

Putting all the values we get:

P ( X | correct )*0.56 = 1*0.45

Therefore we get:

P ( X | correct ) = 0.45 / 0.56 = 0.8036

Therefore 0.8036 is the required probability here.

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