Question: A man has a 10 question multiple-choice test. Each question has three wrong choices and one correct choice. Assume 60% of the time he knows

A man has a 10 question multiple-choice test. Each question has three wrong choices and one correct choice. Assume 60% of the time he knows the answer and selects the correct one and the rest of the time he guesses and selects a choice uniformly at random. He can pass the exam if he gets more than 65% of the questions correct.

(a) If he answered a question correctly, what is the probability that he really knew the correct answer? And What is the probability that he can pass the test? (assume independence between the questions)

(b) His teacher allowed him to continue retaking the exam until he passes. On average, how many times will he need to take it until he passes? (assume independence between test attempts)

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