In answering a question on a multiple-choice test, a student either knows the an

Question

In answering a question on a multiple-choice test, a student either knows the answer or guesses. Let p be the probability that the student knows the answer and 1 - p be the probability that the student guesses. Suppose there are 5 multiple-choice alternatives so a student who guesses at the answer will be correct with probability 1/5. (a) Show that the probability that a student knew the answer to a question given that he or she answered it correctly is 5p/(1 + 4p). (b) What is the probability that a student actually guessed the answer to a question given that he or she answered it correctly?

Explanation / Answer

Given information,

P(guess) =1- p, P(know the answer) = p

P(correct | guess) = 1/5 , P(incorrect | guess) = 0.8

P(correct | know the answer) = 1, P(incorrect | know the answer) = 0

(a)

By the law of the probability, the probability that student give the correct answer is

P(correct) = P(correct | know the answer) P(know the answer) + P(correct | guess) P(guess) = 1 * p + (1/5) * (1-p) = (1+4p) /5

By the Baye's theorem, the probability that a student actually guessed the answer to a question given that he or she answered it correctly is

P( know the answer | correct) =[ P(correct | know the answer) P(know the answer) ] / P(correct) = 5p / (1+4p)

Hence, proved

(b)

P( guess | correct) =[ P(correct | guess) P(guess) ] / P(correct) = (1-p) / (1+4p)

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