People visiting video rental stores often rent more than one DVD at a time.

Question

People visiting video rental stores often rent more than one DVD at a time. The probability distribution for DVD rentals per customer at Video To Go is given in the following table. There is a five-video limit per customer at this store, so nobody ever rents more than five DVDs.

x P(x)

0	0.04
1	0.52
2	0.24
3
4	0.06
5	0.03

Part (a)

Describe the random variable X in words.

1. X is the number of DVDs a customer owns
2. X is the total number of DVDs rented at the store    
3. X is the number of DVDs a customer rents
4. X is the number of customers that come into the storeX is the number of times a single customer comes into the store

Part (b)

1. Find the probability that a customer rents three DVDs. (Enter your answer to two decimal places.)

Part (c)

1. Find the probability that a customer rents at least four DVDs. (Enter your answer to two decimal places.)

Part (d)

1. Find the probability that a customer rents at most two DVDs. (Enter your answer to two decimal places.)

Part (e)

1. Another shop, Entertainment Headquarters, rents DVDs and video games. The probability distribution for DVD rentals per customer at this shop is given as follows. They also have a five-DVD limit per customer.

xP(x)
0	0.35
1	0.25
2	0.20
3	0.10
4	0.05
5	0.05

At which store is the expected number of DVDs rented per customer higher?

1. Entertainment Headquarters
2. Video to Go    

Part (f)

1. If Video to Go estimates that they will have 300 customers next week, how many DVDs do they expect to rent next week? Answer in sentence form.
2. The expected number of videos rented to 300 Video to Go customers is  .

If Video to Go expects 300 customers next week, and Entertainment HQ projects that they will have 420 customers, for which store is the expected number of DVD rentals for next week higher? Explain.

The expected number of videos rented to 420 Entertainment Headquarters customers is  . 

Answer 1

This table is read like this:

The probability that a person who comes in the store will rent zero DVDs is 4%

The probability that a person who comes in the store will rent one DVD is 52%

The probability that a person who comes in the store will rent two DVDs is 24%

The probability that a person who comes in the store will rent three DVDs is ( )%

The probability that a person who comes in the store will rent four DVDs is 6%

The probability that a person who comes in the store will rent five DVDs is 3%

So, "x" is the number of DVDs a customer rents

Since the probability is 100% that a person who comes in the store will rent either zero, 1, 2, 3, 4, or 5 DVDs, then we just subtract the others from 100% to get the probability they will rent 3 DVDs. So 100% - 4% - 52% - 24% - 6% - 3% = 11%. Therefore, the probability that a person who comes in the store will rent three DVDs is 11% or 0.11.

The probability that a customer rents at least four DVDs is equal to the probability they will rent 4 OR 5 DVDs so 6% + 3% = 9% or 0.09.

The probability that a customer rents at most two DVDs is equal to te probability they will rent zero OR one, OR two DVDs so 4% + 52% + 24% = 80% or 0.80.

At the second store, the probabilities that a person will rent higher numbers of DVDs seems to be lower. Let's compare the averages.

For store 1, the average is: 0(0.04) + 1(0.52) + 2(0.24) + 3(0.11) + 4(0.06) + 5(0.03) = 1.72 So the average number of DVDs rented is 1.72.

For store 2, the average is: 0(0.35) + 1(0.25) + 2(0.20) + 3(0.10) + 4(0.05) + 5(0.05) = 1.4 DVDs So the expected number of DVDs rented per customer is higher at the first store (Video To Go).

We just calculated the average number of DVDs that are rented at Video To Go as 1.72. If 300 customers come in, then they would expect to rent 300•1.72 = 516 DVDs.

Since we calculated that the average number of DVDs that are rented at Entertainment Headquarters is 1.4, then 420 customers would rent on average 420•1.4 = 588 DVDs

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