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Question
In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability 1/4 What is the probability that the student knows the answer given that he answered it correctly?
Solution
Let E1: know the answer; E2: be the placement of students.
E: The student gives the correct answer.
Then P(E1) = `3/4`, P(E2) = `1/4` or P(E|E1) = 1, P(E|E2) = `1/4`
Hence, from Baye's theorem
P(E1|E) = `(P(E_1) xx P(E|E_1))/(P(E_1) xx P(E|E_1) + P(E_2) xx P(E|E_2)`
= `(3/4 xx 1)/(3/4 xx 1+ 1/4 xx 1/4)`
= `(3/4)/(3/4 + 1/4)`
= `(3/4)/(13/16)`
= `3/4 xx 16/13`
= `12/13`
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