Richard has just been given a 8-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 guesseson all eight questions, find the indicated probabilities. (Round your answers to three decimal places.)USE SALT(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)(d)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 2 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?O They should be equal, but differ substantially.O They should not be equal, but are equal.O They should be equal, but may not be due to table error.O They should be equal, but may differ slightly due to rounding error.What is the probability that Richard will answer at least half the questions correctly?

Consider a random variable x that denotes the number of answers correctly answered by Richard.It is required to compute the probabilities that are used to quantify the chances of different outcomes based on Richrds guessing strategy.a) There are 5 choices for each question and only one option is correct, the probability of guessing the correct answer for one question is 15=0.2.Since each question is independent, the probability of guessing all 8 questions correctly is given as P(all correct)=(15)8=0.00000256Therefore the probability of guessing all 8 questions correctly is 0.000 b) The number of incorrect options in each question is 4.Hence the probability of guessing an incorrect option is 45=0.8Since there are 8 questions and the outcomes of each question are independent of one another, the probability of answering all questions incorrectly is given as P(all incorrect)=(45)8=0.167778Therefore the probability of guessing all 8 questions correctly is 0.168c)By the Mutually Exclusive Events:Using the PMF of the binomial distribution, the probability of answering at least one question correctly can be computed as shown in the following calculationsThus the probability of answering at least one question correctly by the use of the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table is 0.832By the Complement Rule method:Comparing the two results:Since the probability obtained from the above two methods are same, the correct option isThey should be equal but may differ slightly due to rounding errors.d)To find this probability, Richard should answer 82=4 questions correctly. Using the PMF of binomial destruction x=4,5,6,7,8 the required probability can be obtained.Therefore the probability that Richard will answer at least half the questions correctly is 0.056a) The probability of guessing all 8 questions correctly is 0.000b) The probability of guessing all 8 questions correctly is 0.168c) The probability of guessing all 8 questions correctly,By the Mutually Exclusive Events is 0.832By The Complement Rule method is 0.832On comparing the results: They should be equal but may differ slightly due to rounding errors.d) The probability that Richard will answer at least half the questions correctly is 0.056