Richard has just been given a 4-question multiple-choice quiz in his history cla

Question

Richard has just been given a 4-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 four 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).

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

Explanation / Answer

Richard has just been given a 4-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 four questions, find the indicated probabilities. (Round your answers to three decimal places.)

n=4

p=1/4= 0.25

P(X=x) = (nCx) px (1-p)n-x

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

P( x=4) = 0.004

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

P( x=0) = 0.316

(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).

P= 1-P( x=0) = 1-0.316 =0.684

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

P= P( x=2)+P( x=3)+P( x=4)

= 0.2109+ 0.0469+ 0.0039

=0.2617

P=0.262 ( 3 decimals)

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