On the basis of past data, the owner of an automobile dealership finds that, on average, 8.5 cars are sold per day on Saturdays and Sundays during the months of January and February, with the sales rate relatively stable throughout the day. Moreover, purchases appear to be independent of one another. The dealership is open for10 hours per day on each of these days. There is no reason to believe that sales for the upcoming year will be any different from sales in the past.
a. On the first Saturday in February, the dealership will open at9 a.m. Find the probability that the time until the first sale is more than two hours, PE (T ≥ 2 hours | m = 8.5 cars per 10 hours).
b. Find the probability that the number of sales before 11 a.m. is equal to zero, PP (X = 0 in 2 hours | m = 8.5 cars per 10 hours). Compare your answer with that from part a. Can you explain why the answers are the same?
c. The owner of the dealership gives her salespeople bonuses, depending on the total number of cars sold. She gives them $200whenever exactly 13 cars are sold on a given day, $300 whenever14 cars are sold, $500 whenever 15 cars are sold, and $700whenever 16 or more cars are sold. On any given Saturday or Sunday in January or February, what is the expected bonus that the dealer will have to pay?
d. Consider the bonus scheme presented in part c. February contains exactly four Saturdays and four Sundays. What is the probability that the owner will have to pay the $200 bonuses exactly twice in those days?