Rent-A-Jalopy rents cars in NJ, NY, and PA. Some times cars are picked up and re
ID: 3226584 • Letter: R
Question
Rent-A-Jalopy rents cars in NJ, NY, and PA. Some times cars are picked up and returned in the same state. Other times the cars are picked up in one state and returned in another. Of the cars rented in NJ, 50% are returned to NJ, 20% are returned to NY, and 30% are returned to locations in PA. Of the cars rented in NY, 20% are returned to NJ, 60% are returned to NY, and 20% are returned to locations in PA. Of the cars rented in PA, 10% are returned to NJ, 20% are returned to NY, and 70% are returned to locations in PA. Use the matrix tool to answer the following questions. (Give your answers correct to 3 decimal places.) (a) If a car is rented in NJ, what is the probability it will be in NY after it has been rented twice? (b) If a car is rented in NY, what is the probability it will be in PA. after it has been rented three times? (c) In the long run, what fraction of the time does a rental car spend in NJ? In the long run, what fraction of the time does a rental car spend in NY? In the long run, what fraction of the time does a rental car spend in PA?Explanation / Answer
Create a 3-by-3 (single-step) transition matrix, where the columns have the conditional probabilities of a car being returned in NJ, NY, and PA, respectively, and the rows correspond to NJ, NY, and PA, the respective states where a car is initially rented. Call this matrix T.
T =
0.50 0.20 0.30
0.20 0.60 0.20
0.10 0.20 0.70 .
Now define Tn = T^n as the n-step transition matrix, and TN as the limiting ("long run") transition matrix as n approaches infinity. Clearly, T1 = T^1 is identical to T. In the context of this problem, n is the number of times a car gets rented. Tn[r, c] refers to the element in row r and column c of Tn, and Tn[., c] refers to any element in column c of Tn.
(a) Since the car is rented twice, compute T2 = T^2.
T2 =
0.32 0.28 0.40
0.24 0.44 0.32
0.16 0.28 0.56 .
Since the car originated in NJ, look in the first row of T2 for the probabilities of each return location. The second column corresponds to NY, so
P(ends in NY | started in NJ & rented 2 times) = T2[1, 2]
= 0.28 .
(b) The reasoning is similar here.
T3 =
0.256 0.312 0.432
0.240 0.376 0.384
0.192 0.312 0.496 .
P(ends in PA | started in NY & rented 3 times) = T3[2, 3]
= 0.384 .
(c) For TN the numbers in each column converge, so the original location of a car loses its relevance.
TN =
2/9 1/3 4/9
2/9 1/3 4/9
2/9 1/3 4/9 .
The long-run fraction of time a rental car spends in NJ = TN[., 1]
= 2/9.
The long-run fraction of time a rental car spends in NY = TN[., 2]
= 1/3.
The long-run fraction of time a rental car spends in PA = TN[., 3]
= 4/9.