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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?

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Solution

## Let E1 and E2 be the respective events that the student knows the answer and he guesses the answer.Let A be the event that the answer is correct.∴P(E1)=34P(E2)=14The probability that the student answered correctly, given that he knows the answer, is 1.∴P(A|E1)=1Probability that the student answered correctly, given that he guessed, is 14∴P(A|E2)=14The probability that the student knows the answer, given that he answered it correctly, is given by P(E1|A)By using Baye's theorem, we obtainP(E1|A)=P(E1)⋅P(A|E1)P(E1)⋅P(A|E1)+P(E2)⋅P(A|E2)=34⋅134⋅1+14⋅14=3434+116=341316=1213=0.92.

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