# STATS Random Problems

Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up teller window occur at random, with an arrival rate of 24 customers per hour or 0.4 customers per minute. a. What is the mean or expected number of customers that will arrive in a five-minute period? b. Assume that the Poisson probability distribution can be used to describe the arrival process. Use the arrival rate in part (a) and compute the probabilities that exactly 0, 1, 2, and 3 customers will arrive during a five-minute period. c. Delays are expected if more than three customers arrive during any five-minute period. What is the probability that delays will occur? In the Willow Brook National Bank waiting line system (see Problem 1), assume that the service times for the drive-up teller follow an exponential probability distribution with a service rate of 36 customers per hour, or 0.6 customer per minute. Use the exponential probability distribution to answer the following questions: a. What is the probability the service time is one minute or less? b. What is the probability the service time is two minutes or less? c. What is the probability the service time is more than two minutes? For the Burger Dome single-channel waiting line in Section 11.2, assume that the arrival rate is increased to 1 customer per minute and that the service rate is increased to 1.25 customers per minute. Compute the following operating characteristics for the new system: P 0 , L q , L , W q , W , and P w . Does this system provide better or poorer service compared to the original system? Discuss any differences and the reason for these differences. Consider the PortaCom project discussed in Section 12.1. a. An engineer on the product development team believes that first-year sales for the new printer will be 20,000 units. Using estimates of $45 per unit for the direct labor cost and $90 per unit for the parts cost, what is the first-year profit using the engineers sales estimate? b. The financial analyst on the product development team is more conservative, indicating that parts cost may well be $100 per unit. In addition, the analyst suggests that a sales volume of 10,000 units is more realistic. Using the most likely value of $45 per unit for the direct labor cost, what is the first-year profit using the financial analysts estimates? c. Why is the simulation approach to risk analysis preferable to generating a variety of what-if scenarios such as those suggested by the engineer and the financial analyst? Baseballs World Series is a maximum of seven games, with the winner being the first team to win four games. Assume that the Atlanta Braves are in the World Series and that the first two games are to be played in Atlanta, the next three games at the opponents ballpark, and the last two games, if necessary, back in Atlanta. Taking into account the projected starting pitchers for each game and the home field advantage, the probabilities of Atlanta winning each game are as follows: Game 1 2 3 4 5 6 7Probability of Win 0.60 0.55 0.48 0.45 0.48 0.55 0.50 a. Set up random number intervals that can be used to determine the winner of each game. Let the smaller random numbers indicate that Atlanta wins the game. For example, the random number interval 0.00 but less than 0.60 corresponds to Atlanta winning game 1. b. Use the random numbers in column 6 of Table 12.2 beginning with 0.3813 to simulate the playing of the World Series. Do the Atlanta Braves win the series? How many games are played? c. Discuss how repeated simulation trials could be used to estimate the overall probability of Atlanta winning the series as well as the most likely number of games in the series.