The manager of Avalon at the Prudential Center, a large apartment complex in Boston, collects the rent from each tenant on the first day of every month. Past records indicate that the mean number of tenants who do not pay the rent on time in any given month is 4.7. Consider the rent collection for the next month.

 

(a) Find the probability that every tenant will pay the rent on time.

(b) Find the probability that at least seven tenants will be late with their rent.

(c) Suppose the number of delinquent rent payments in a month is independent of the number in every other month. What is the probability that at most three tenants will be late with their rent in two consecutive months?

(d) State and explain any assumption you make to answer the above questions.

(a) Find the probability that every tenant will pay the rent on time.

P(X = 0) = 0.0091
 

(b) Find the probability that at least seven tenants will be late with their rent.

P(X ≥ 7) = 0.1954

 

(c) Suppose the number of delinquent rent payments in a month is independent of the number in every other month. What is the probability that at most three tenants will be late with their rent in two consecutive months?
P(x ≤ 3) = 0.0160

 

(d) State and explain any assumption you make to answer the above questions.

We assumed that the given follows a Poisson distribution. Likewise, we used μ = 4.7 as λ for the distribution. For letter (c) we also assumed a λ = 9.4 (which is double the λ for one month since there are two months on this scenario)

Step-by-step explanation

Given:

μ = 4.7 tenants who do not pay rent on time in any given month

Let x = number of tenants who do not pay the rent on time

 

Since we are dealing with the probability of events on a fixed time interval, we can use Poisson distribution. We use the following formula:

P(X=x)=x!e−λ×λx?

where x = number of expected occurrences

λ = mean number of occurrences

 

(a) Find the probability that every tenant will pay the rent on time.

If every tenant will pay the rent on time for the month, x = 0 (no tenants late)

λ = μ = 4.7

P(X=0)=0!e−4.7×4.70?

P(X=0)=0.009095277

P(X = 0) = 0.0091
 

(b) Find the probability that at least seven tenants will be late with their rent.

P(X ≥ 7)

We can get this also by:

P(X ≥ 7) = 1 - P(x < 7)

*since at least 7 extends up to infinity

P(x < 7) = P(x =0) + P(x = 1)+ P(x = 2) +P(x = 3) +  P(x = 4) + P(x = 5) + P(x = 6)

We get the corresponding probabilities using the same formula, but changing X to the respective values. The table below is the summary of the probabilities using the Poisson formula:

X	P(X)
0	0.009095
1	0.042748
2	0.100457
3	0.157383
4	0.184925
5	0.17383
6	0.136167

P(x < 7) = P(x =0) + P(x = 1)+ P(x = 2) +P(x = 3) +  P(x = 4) + P(x = 5) + P(x = 6)

P(x < 7) = 0.009095 + 0.042748 + 0.100457 + 0.157383 + 0.184925 + 0.17383 + 0.136167

P(x < 7) = 0.804605083
 

P(X ≥ 7) = 1 - P(x < 7)

P(X ≥ 7) = 1 - 0.804605083

P(X ≥ 7) = 0.195394917

P(X ≥ 7) = 0.1954

 

(c) Suppose the number of delinquent rent payments in a month is independent of the number in every other month. What is the probability that at most three tenants will be late with their rent in two consecutive months?

We can get this by using another lambda for the new timeframe: two months

λ = μ = 4.7 per month

λ = 4.7 x 2

λ = 9.4 tenants who do not pay rent on time in two months

 

Now, we use the same Poisson distribution formula.

P(x ≤ 3)

P(x ≤ 3) = P(x =0) + P(x = 1)+ P(x = 2) +P(x = 3)

 

For P(x =0)

P(X=0)=0!e−9.4×9.40?

P(X=0)=8.27241×10−5

P(X = 0) = 0.00008

 

We use the same formula for P(x = 1)+ P(x = 2) +P(x = 3) summarized in the following table:

X	P(X)
0	0.0000827241
1	0.000777606
2	0.003654749
3	0.011451548

 

P(x ≤ 3) = 0.0000827241 + 0.000777606 + 0.003654749 + 0.011451548

P(x ≤ 3) = 0.015966627
P(x ≤ 3) = 0.0160

 

(d) State and explain any assumption you make to answer the above questions.

We assumed that the given follows a Poisson distribution. Likewise, we used μ = 4.7 as λ for the distribution. For letter (c) we also assumed a λ = 9.4 (which is double the λ for one month since there are two months on this scenario)