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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Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.