# Richard has just been given a 8-question multiple-choice quiz in his history class. Each question has four answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all eight questions, find the indicated probabilities. (Round your answers to three decimal places.)(a) What is the probability that he will answer all questions correctly?(b) What is the probability that he will answer all questions incorrectly?(c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table.Then use the fact that P(r ≥ 1) = 1 − P(r = 0).Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference?They should be equal, but may not be due to table error.They should be equal, but differ substantially.     They should not be equal, but are equal.They should be equal, but may differ slightly due to rounding error.(d) What is the probability that Richard will answer at least half the questions correctly?

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Richard has just been given a 8-question multiple-choice quiz in his history class. Each question has four answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all eight questions, find the indicated probabilities. (Round your answers to three decimal places.)

(a) What is the probability that he will answer all questions correctly?

(b) What is the probability that he will answer all questions incorrectly?

(c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table.

Then use the fact that P(r ≥ 1) = 1 − P(r = 0).

Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference?
They should be equal, but may not be due to table error.They should be equal, but differ substantially.     They should not be equal, but are equal.They should be equal, but may differ slightly due to rounding error.

(d) What is the probability that Richard will answer at least half the questions correctly?

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Step 1

Note:

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Step 2

Introduction:

If X ~ Bin (n, p), then the probability mass function of X is:

p (x) = (nCx) px (1 – p)nx; for x = 0, 1, …, n; 0 < p <1; p (x) is 0 otherwise.

Here, number of trials is the number of questions in the quiz, n = 8.

Step 3

Part a:

Consider the event of getting a question correct as a success.

In the quiz, one out of 4 options in a question is correct. Since Richard randomly guesses the questions, the probability of giving the correct answer is 1/4 = 0.25.

Denote X as the number of correct answers, that is, the number of successes in n = 8 trials. Thus, the probability of success is, p = 0.25.

Assuming that ...

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