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A student takes a 20-question exam and guesses on each question. The lowest passing grade is 14 out of 20....

7. A student takes a 20-question exam and guesses on each question. The lowest passing grade is 14 out of 20.
(a) Assuming that the entire exam consists of true-false questions, find the probability that the student passes and find the expected number of correct answers.
(b) Assuming that the entire exam consists of multiple-choice questions, each with five choices, find the probability that the student passes and find the variance of the number of correct answers.

Answer

  1. (a) For true-false questions, the probability of guessing correctly is 0.5. The expected number of correct answers is given by the formula for the expected value of a binomial distribution: E(X) = n * p, where n is the number of questions and p is the probability of a correct answer. Thus, E(X) = 20 * 0.5 = 10.
    The probability of passing (getting at least 14 correct) can be calculated using the binomial probability formula or a cumulative binomial probability table/calculator.

  2. (b) For multiple-choice questions with five choices, the probability of guessing correctly is 0.2. The variance of the number of correct answers in a binomial distribution is given by Var(X) = n * p * (1 - p). Thus, Var(X) = 20 * 0.2 * 0.8 = 3.2.
    The probability of passing (getting at least 14 correct) can be calculated using the binomial probability formula or a cumulative binomial probability table/calculator.

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