Question

Consider a binomial probability distribution, it is unusual for the number of successes to be less than__________ or greater than____________.

a. Fill in the blanks above.

b. For a binomial experiment with 100 trials for which the probability of success on a single trial is 0.2, is it unusual to have more than five successes? Show all work and explain.

c. If you were simply guessing on a multiple-choice exam consisting of 6 questions with 3 possible responses for each question, would you be likely to get more than half of the questions correct? Show work and explain.

Answer 1

Answer)

Consider a binomial probability distribution, it is unusual for the number of successes to be less than mean - 2.5*s.d or greater than mean + 2.5*s.d

b. For a binomial experiment with 100 trials for which the probability of success on a single trial is 0.2, is it unusual to have more than five successes? Show all work and explain.

Answer)

Mean = n*p = 20

S.d = √{n*p*(1+p)) = 4

Mean + 2.5*4 = 30

Mean - 2.5*4 = 10

So yes it is unusual to have more than 5 success

c. If you were simply guessing on a multiple-choice exam consisting of 6 questions with 3 possible responses for each question, would you be likely to get more than half of the questions correct? Show work and explain.

Answer)

As there are fixed number of trials and probability of each and every trial is same and independent of each other

Here we need to use the binomial formula

P(r) = ncr*(p^r)*(1-p)^n-r

Ncr = n!/(r!*(n-r)!)

N! = N*n-1*n-2*n-3*n-4*n-5........till 1

For example 5! = 5*4*3*2*1

Special case is 0! = 1

P = probability of single trial = 1/3

N = number of trials = 6

R = desired success = more than half = p(4)+p(5)+p(6) = 0.09687788649

In probability if probability is greater than 0.05 it is not unusual

So answer is yes it is likely to get more than half