Question

# A student is taking a multiple-choice quiz in which each question has five possible answers, exactly...

A student is taking a multiple-choice quiz in which each question has five possible answers, exactly one of which is correct. He knows 85% of the material being tested. Assume the teacher has chosen the questions independently and at random from among all the questions she could have chosen.

When the student knows the correct answer to a given question, he has a 98% chance of marking it correctly. When he does not know the answer, he answers by marking at random one of the five possible answers. (Recall that “at random” means that each of the five possible answers is equally likely.)

For any question selected at random from the quiz, let K be the event that the student knows the correct answer and let M be the event that he marks the correct answer.

1) Which number is P(K)

2) Suppose a question is selected at random from the quiz.

(a) Calculate P(K ∩ M), the probability that the student knows the correct answer and marks it correctly.

(b) Calculate P(M), the probability of marking the correct answer.

(c) Calculate P(K I M), the probability that the student knows the answer to the question, given that he marked it correctly.

(d) Calculate P (K ∪ M )

3) Now suppose 3 questions are selected at random from the quiz. Calculate the probability that the student marks all three correctly. Assume that the student’s responses to different questions are independent of each other.

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