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Question

# In answering a question on a multiple choice test, a student either knows the answer or guesses. Let be the probability that he knows the answer and be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability What is the probability that the student knows the answer given that he answered it correctly?

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Solution

## Let, P( A ), P( B ) and P( C ) be the probabilities that are defined below, The probability of student answer is P( A ). The probability of student guesses is P( B ). The probability of student answers correctly is P( C ). The probability that the student knows the answer, if he answered it correctly that is P( A C ), P( A C )= P( A )P( C A ) P( B )⋅P( C B )+P( A )⋅P( C A ) (1) The probability that student knows the answer is, P( A )= 3 4 The probability that the student answer correctly, given that he knows the answer, P( C A )=1 The probability that student guess the answer is, P( B )= 1 4 The probability that the student answer correctly, given that he guesses, P( C B )= 1 4 Put these values in equation (1), P( A C )= ( 3 4 ×1 ) ( 1 4 × 1 4 )+( 3 4 ×1 ) = 3 4 1 16 + 3 4 = 3 4 13 16 = 12 13 Thus, the required probability is 12 13 .

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