Statistics: Probability Example Questions

1:
When a customer places an order with Rudy's On-Line Office Supplies, a computerized accounting information system (AIS) automatically checks to see if the customer has exceeded his or her credit limit. Past records indicate that the probability of customers exceeding their credit limit is 0.05. Suppose that, on a given day, 20 customers place orders. Assume that the number of customers that the AIS detects as having exceeded their credit limit is distributed as a binomial random variable.

A: What is the mean and standard deviation of the number of customers exceeding their credit limits?
B: What is the probability that 0 customers will exceed their limit?
C: What is the probability that 1 customer will exceed his or her limit?
D: What is the probability that 2 or more customers will exceed their limits?

2:
The mean cost of a phone call handled by an automated customer-service system is $0.45. The mean cost of a phone call passed on to a "live" operator is $5.50. However, as more and more companies have implemented automated systems, customer annoyance with such systems has grown. Many customers are quick to leave the automated system when given an option such as "Press zero to talk to a customer-service representative." According to the Center for Client Retention, 40% of all callers to automated customer-service systems automatically opt to go to a live operator when given the chance. If 10 independent callers contact an automated customer-service system, what is the probability that:

A: 0 will automatically opt to talk to a live operator?
B: exactly 1 will automatically opt to talk to a live operator?
C: 2 or less will automatically opt to talk to a live operator?
D: all 10 will automatically opt to talk to a live operator?
E: If all 10 automatically opt to talk to a live operator, do you think that the 40% value given in the article applies to this particular system? Explain.

3:
One of the retail industry's biggest frustrations is customers who abuse the return and exchange policies (S. Kang, "New Return Policy: Retailers Say 'No" to Serial Exchanges, "The Wall Street Journal, November 29, 2004, pp. B1, B3). In recent a year, returns were 13% of sales in department stores. Consider a sample of 20 customers who make a purchase at a department store. Use the binomial model to answer the following questions:

A: What is the expected value, or mean, of the binomial distribution?
B: What is the standard deviation of the binomial distribution?
C: What is the probability that none of the 20 customers will make a return?
D: What is the probability that no more than 2 of the customers will make a return?
E: What is the probability that 3 or more of the customers will make a return?

4:
Refer to question 3: In the same year, returns were 1% of sales in grocery stores.

A: What is the expected value, or mean, of the binomial distribution?
B: What is the standard deviation of the binomial distribution?
C: What is the probability that none of the 20 customers will make a return?
D: What is the probability that no more than 2 of the customers will make a return?
E: What is the probability that 3 or more of the customers will make a return?
F: Compare the results of (a) through (e) to those of Problem 5.40 (a) through (e).