Calculating Poisson Probability for Car Rental Income

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In summary, the conversation discusses the expected daily income from renting out four cars at $50 per day and the additional cost of $20 per day to buy another car for rental. The expected number of cars rented out is about 1.73, resulting in an expected income of $86.6. However, this calculation was incorrect as probabilities were only calculated up to 4 instead of using P(X>=4). The correct answer for the expected income is 4 x $50 = $200.
  • #1
thereddevils
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Homework Statement



A car rental shop has four cars to be rented out on a daily basis at $ 50 per car. The average daily demand for cars is four.

(1) Calculate the expected daily income received from the rentals

(2) If the shop wishes to have one more car, the additional cost incurred is $ 20 per day DEtermine whether the shop should buy another car for rental.

Homework Equations





The Attempt at a Solution



(1) X-p(4)

P(X=0)=0.0183
P(X=1)=0.07326
P(X=2)=0.1465
P(X=3)=0.195
P(X=4)=0.195

The expected number of cars rented out is about 1.73 so the expected income is

1.73 x $50= $ 86.6

But my answer is wrong.
 
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  • #2
Why do you stop calculating probabilities at 4... they hardly add up to the total of 1, so you are using an incomplete distribution here.

I'm not really into Poisson distributions, but if the average is 4, isn't the answer simply 4 x $50 as it would with a binomial distribution?
 
  • #3
CompuChip said:
Why do you stop calculating probabilities at 4... they hardly add up to the total of 1, so you are using an incomplete distribution here.

I'm not really into Poisson distributions, but if the average is 4, isn't the answer simply 4 x $50 as it would with a binomial distribution?

thanks Compuchip, i figured that out. I calculated P(X=4) instead of P(X>=4), that's why the probabilities do not add up to 1.
 

What is Poisson Probability?

Poisson Probability is a mathematical concept used to calculate the likelihood of a certain number of events occurring within a specific time or space, given a known average rate of occurrence.

How is Poisson Probability used in calculating car rental income?

In this context, Poisson Probability can be used to estimate the number of car rentals that will occur within a certain time period based on the average rate of rentals per day or week. This can help car rental companies make informed decisions about pricing, inventory, and staffing.

What factors influence Poisson Probability for car rental income?

The main factors that influence Poisson Probability for car rental income include the average rate of rentals per day or week, the duration of the rental period, and any external factors that may affect the demand for car rentals, such as holidays or events.

How is Poisson Probability calculated for car rental income?

The formula for calculating Poisson Probability is: P(x) = (e^-λ * λ^x) / x!, where P(x) is the probability of x events occurring, e is the base of natural logarithms, λ is the average rate of rentals per time period, and x is the number of events. This formula can be used to determine the probability of a specific number of car rentals occurring within a given time period.

What are the limitations of using Poisson Probability for car rental income?

While Poisson Probability can provide useful estimates for car rental companies, it does have limitations. It assumes that the rate of rentals remains constant over time, which may not always be the case. It also does not take into account any external factors that may impact demand for car rentals. Additionally, it is based on past data and may not accurately predict future events or changes in the market.

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