In an entrance test there are multiple choice questions. There are four options for each question, of which only one is correct. The probability that a student knows the answer to a question is 90\%. If he gets the correct answer to a question, then what is the probability that he was guessing ?
C. 1 37
Let E_1 be the event that the student knows the answer and E_2 be the event that the student guesses the answer. Let A be the event that the student gets the correct answer. From the given information: P(E_1) = 90 100 = 9 10 P(E_2) = 1 - P(E_1) = 1 10 If the student knows the answer, the probability of getting it correct is 1, so P(A|E_1) = 1. If the student guesses, since there are 4 options with only one correct, P(A|E_2) = 1 4 . The probability that the student was guessing given that the answer is correct is P(E_2|A). Using Bayes' theorem: P(E_2|A) = P(E_2) P(A|E_2) P(E_1) P(A|E_1) + P(E_2) P(A|E_2) P(E_2|A) = 1 10 1 4 9 10 1 + 1 10 1 4 P(E_2|A) = 1 40 9 10 + 1 40 = 1 40 37 40 = 1 37 Answer: 1 37
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