In a typical month, an insurance agent presents life insurance plans to 40 potential customers. Historically, one in four such customers chooses to buy life insurance from this agent. Based on the relevant binomial distribution, answer the following questions:
a. What is the probability that exactly five customers will buy life insurance from this agent in the coming month?
b. What is the probability that no more than 10 customers will buy life insurance from this agent in the coming month?
c. What is the probability that at least 20 customers will buy life insurance from this agent in the coming month?
d. Determine the mean and standard deviation of the number of customers who will buy life insurance from this agent in the coming month.
e. What is the probability that the number of customers who buy life insurance from this agent in the coming month will lie within two standard deviations of the mean?
f. What is the probability that the number of customers who buy life insurance from this agent in the coming month will lie within three standard deviations of the mean?