Question

​​​​​​​

Share &Care is a nonprofit car-share company that rents cars. When customers make a reservation, they specify their pickup time and the number of time slots they will hold the vehicle, where each timeShare&Care is a nonprofit car-share company that rents cars. When customers make a reservation, they specify their pickup time and the number of time slots they will hold the vehicle, where each time slot equals 15 minutes. For example, if the pickup time is 1p.m. then possible drop-off times are 115 (one slot), 1:30 (two slots), and so on Share&Care charges $1.25 for each time slot in the reservation. To discourage customers from returning the rented cars beyond their drop-off time, it charges $15 per time slot used beyond the drop-off time. For example, if a customer's drop-off time is 2:30 and he returns the vehicle at 2:47, then he is charged $30 for the two time slots he used beyond his reservation (and, of course, $1.25 per slot that he reserved). Larry runs a small business that makes deliveries on Fridays. To ensure availability of a car, he books his car two days in advance. However, he doesn't know his exact needs when he books. Table 13.10 provides information regarding Larry's demand (in terms of slots). For example, if he needs five slots but booked four slots, then he uses the car for five slots, for a total charge of (4 * $1.00 + $20-$24). If he ends up booking the car for more time than he needs, the extra time on the car has no value to him.
a Suppose he books the car for 3 time slots. How likely is he to pay $30 or more in late fees? b. To minimize his rental costs, how many time slots should Larry reserve?
c. Suppose he books the car for 4 time slots. How many time slots can he expect to waste (i.e., they end up being of no use to him)?
d. Larry hates paying any late penalty fee. Suppose he wants to be 99.5% sure that he will not have to pay a late fee. How many slots should he book?

Answer 1

a. The likelihood of Larry paying $30 or more in late fees if he books the car for 3 time slots is 68.75%. b. To minimize rental costs, Larry should reserve at least 7 time slots. c. If Larry books the car for 4 time slots, he can expect to waste 1 time slot on average. d. To be 99.5% sure of not paying a late fee, Larry should book at least 5 time slots.

Step-by-step explanation:

a. To calculate the likelihood of Larry paying $30 or more in late fees, we need to determine the probability of him using more than 2 slots beyond his reservation. Since Larry books the car for 3 time slots, there are 4 possible time slots for him to drop-off the car (1 slot, 2 slots, 3 slots, or 4 slots beyond the reservation). The probability of Larry using more than 2 slots beyond his reservation can be calculated by subtracting the probability of him using 1 or 2 slots beyond his reservation from 1. Let's calculate:

Probability of using 1 slot beyond reservation: 1/4

Probability of using 2 slots beyond reservation: 1/4 * 1/4 = 1/16

Probability of using more than 2 slots beyond reservation: 1 - (1/4 + 1/16) = 11/16 = 0.6875, or 68.75%

So, there is a 68.75% chance that Larry will pay $30 or more in late fees if he books the car for 3 time slots.

b. To minimize his rental costs, Larry should reserve the minimum number of time slots that satisfies his needs. Looking at Table 13.10, we can see that Larry's demand varies from 2 to 7 slots. Therefore, Larry should reserve at least 7 time slots to cover his maximum demand and avoid wasting any time slots.

c. If Larry books the car for 4 time slots, he can expect to waste 1 time slot on average. This is because there are 3 possible drop-off times that would result in wasting a time slot (1 slot, 2 slots, or 3 slots beyond reservation), each with equal probability of 1/4. Therefore, the expected number of wasted time slots is (1/4 * 1) + (1/4 * 2) + (1/4 * 3) = 1 + 1/2 + 3/4 = 2.25.

d. To be 99.5% sure that he will not have to pay a late fee, Larry should book at least 5 time slots. By booking 5 time slots, he would have a 0.5% chance of using more than 2 slots beyond his reservation, reducing the likelihood of paying a late fee significantly.