Question

Richard has just been given a 4-question multiple-choice quiz in his history class. Each question has five 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). Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference? They should not be equal, but are equal. They should be equal, but may differ slightly due to rounding error. They should be equal, but differ substantially. They should be equal, but may not be due to table error. (d) What is the probability that Richard will answer at least half the questions correctly?

Answer 1

it is a binomial probability distribution, because there is fixed number of trials,  
only two outcomes are there, success and failure  
trails are independent of each other  
and probability is given by  

P(X=x) = C(n,x)*px*(1-p)(n-x)

where  
Sample size , n =    4
Probability of an event of interest, p = 1/5 = 0.2

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a)

P ( X =    4   ) = C(   4   ,   4   )*   0.20   ^   4   *   0.800   ^   0   =   0.002

b)

P ( X =    0   ) = C(   4   ,   0   )*   0.20   ^   0   *   0.800   ^   4   =   0.410

c)

P(X≥1) = 1-P(X=0) = 1-0.410=0.590

other method

P(X≥1) = P(X=0) + P(X=2) + P(X=3) + P(X=4) = 0.590

yes, two results should be equal and they are equal

d)

P(X≥2) = P(X=2) + P(X=3) + P(X=4)=0.181