Find the required probability for binomial distribution. 

Binomial Probabilities: Multiple-Choice Quiz Richard has just been given a 10-problem multiple-choice quiz in his history class. Each problem has 5 answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assume that Richard guesses on all 10 problem, find the indicated probabilities.

a) What is the probability that he would answer all problems correctly?

b) What is the probability that he would answer all problems incorrectly?

c) What is the probability that he would answer at least one of the problems correctly? Work out this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in Table 3 of appendix II. Then use the fact that P(r > 1) = 1 - P(r = 0). Evaluate the two results. Should they be equal? Are they equal? If not, how do you account for the difference?

d) What is the possibility that Richard will answer at least half the problems correctly?