Mean and Variance of Binomial Distribution

Let X be the number of successes in n independent trials, each with probability of success P. Then X follows a binomial distribution with mean

Example: An insurance broker believes that for a particular contract the probability of making a sale is 0.4. Suppose that the broker has five contracts.

a.    Find the probability that she makes at most one sale.
b.    Find the probability that she makes between two and four sales (inclusive).
c.    Graph the probability distribution function.

Comments:
•    This shape is typical for binomial probability when P is neither very large nor very small.
•    At least extremes (o or 5 sales), the probabilities are quite small.

Example: Early in August an undergraduate  college discovers that it can accommodate a few extra students, Enrolling those additional students would provide a substantial increase in revenue without increasing the operating costs of the college; that is, no new classes would have to be added. From past experience the college knows that 40% of those students admitted will actually enroll.
a.    What is the probability that at most 6 students will enroll if the college offers admission to 10 more students?
b.    What is the probability that more than 12 will actually enroll if admission offered to 20 students?
c.    If 70% of those students admitted actually enroll, what is the probability that at least 12 out of 15 students will actually enroll?

Solution:
a.    This probability can be obtained using the cumulative binomial probability distribution from Table 3 in the Appendix. The probability of at most 6 students enrolling if n = 10 and P = 0.40 is

             
     
b.    The probability that at least 12 out of 15 students enroll is the same as the probability that at most 3 out of 15 students do not enroll (the probability of a student not enrolling is 1 – 0.70 = 0.30).