7) 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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