This is a sample assignment on probability and statistics made for one of our students. This mathematics assignment will help you to understand binomial distribution for car rental fleet management and also learn how to tackle problems on probability and statistics and provide useful solutions applicable in the practical world. The coursework covers probability analysis using binomial distribution for car rental fleet management.
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Question: Calculate Probability Analysis for Car Rental Fleet Management During Peak Season
Details of the question:
One of the largest costs of a car rental company is the cost of repairing the cars both due to damage and to normal wear and tear. Rent‐a‐Car plc is analyzing the cost and pattern of vehicle repairs to ensure it has enough cars available during peak time. Peak time for Rent‐a‐Car, which is located in a beach resort, is during the months of July and August, where the average rental is for one week. Rent‐a‐Car has a fleet of 300 cars and over time it has collected data that indicate the probability of a car needing repairs after each rental is completed is 10%. Considering the location and the information above, most cars will be returned, cleaned and rented out on the same day. If a car needs repairs and no other car is available, Rent‐a‐Car will either send the customer away or, if the customer has pre‐ booked, rent a car from another company and honor the quote the customer has irrespective of what the other company charges, normally resulting in a loss for Rent‐a‐Car. The so‐called binomial distribution can be used to describe uncertain situations of this kind where there are two possible outcomes; a car needs a repair or not. Fortunately, the binomial distribution can be approximated by the normal distribution such that the number of cars available for rent is normally distributed with a mean of n∙p and a standard deviation of image.png where n is the fleet size and p is the probability of a car not needing repair.
a) If Rent‐a‐Car commits to 260 bookings in any one week, what is the probability that there will not have enough cars to meet all bookings?
b) If Rent‐a‐Car commits to 275 bookings in any one week, what is the probability that there will be cars available to meet all bookings?
c) If Rent‐a‐Car wants to have no more than a 4% probability of having idle cars, what’s the minimum number of rentals it should agree to in any one week?
d) You have been hired as a consultant for Rent‐a‐Car. You have understood the probability structure of the car availability problem and the next step is to advise them on how many car rentals to allow during the months of July and August. Please make a list of the data you would need to get from the car rental company in order to make your recommendation.
Note: Normal Distribution and standard normal distribution values have been taken from the given link (Page 164-166): https://www.actuaries.org.uk/system/files/field/document/Formulae%20and%20Tables.pdf
Solution
Rent‐a‐Car has a fleet of 300 cars, so n=300
probability of a car needing repairs after each rental is completed is 10%
so, the probability of a car not needing repairs after each rental is completed,
p=90%
Let X be a random variable representing the number of cars that do not need repairs after a rental
X ⏤ N(n*p,n*p*(1-p))
X ⏤ N(300*0.9, 300*0.9*(0.1_)
X ⏤ N(270, 27)
Now, Solutions of a,b,c, and d:
a) 1-0.97257=2.74%
b) 1-0.83147=16.85%
c) c>279
d) Data required from Car-Rental company:
- Number of cars given on rent per day at least for the last six months
- Of the above cars, number of cars that needed repairs after each rental was completed.
- Statistical analysis for past Peak time that is, last year’s July and August.
- All the 300 cars sorted by the condition of the car, namely: Great, Moderate, Poor.
- Cost of parking for idle cars
- Expected loss for when Rent-a-car will rent from another company when it has pre-booked and no car is available.
- Past data of cases of theft.
- Average number of days a car is rented by a customer.