Deck 9: Markov Chains

Question 1

The transition matrix for a Markov process is:
Find the first state matrix if the initial state is

A) [0.3 0.7]
B) [0.468 0.532]
C)
D) [0.9 0.1]

Answer 1

Question 2

The transition matrix for a Markov process is: Find the second state matrix if the initial state is

A) [0.72 0.28]
B) [0.3 0.7]
C) [ 0.468 0.538]
D) [0.6192 0.3808]

Answer 2

[ 0.468 0.538]

Question 3

The transition matrix for a Markov process is:

Find .

A)
B)
C)
D)

Answer 3

Question 4

The transition matrix for a Markov process is:

Find .

A)
B)
C)
D)

Question 5

Given the transition matrix:

Find .

A)
B)
C)
D)

Question 6

Using a graphing utility to compute powers of ,find the smallest n such that the corresponding entries in and are, when rounded to 3 decimal places, equal.

A) 3
B) 6
C) 11
D) 2

Question 7

Construct the transition matrix that represents the data:

-If it snows today, there is a 70 percent chance of snow tomorrow; however if it does not snow today, there is a 40 percent chance that it will not snow tomorrow.

A)
B)
C)
D)

Question 8

10 percent of the people in one generation who have a certain physical characteristic will pass that characteristic on to the next generation. 40 percent of the people in one generation who do not have this characteristic will pass it on to the next generation.

A)
B)
C)
D)

Question 9

A new anti-gravity commuter train has been installed. It is expected that each week 90% of the riders who used the existing system will continue to do so. Of those who traveled by car, 5% will begin to use the new anti-gravity train. Use this information to write the transition matrix that describes this process.

A)
B)
C)
D)

Question 10

A small town has only two dry cleaners, Fast and Speedy. Fast hopes to increase its market share by conducting an extensive advertising campaign. The initial market share for Fast was 40% and 60% for Speedy. Solve the problem:

-Find the probability that a customer using Fast initially will use Fast for his second batch of clothes. Use the following transition matrix.

A) 0. 73
B) 0. 29
C) 0. 27
D) 0. 71

Question 11

A small town has only two dry cleaners, Fast and Speedy. Fast hopes to increase its market share by conducting an extensive advertising campaign. The initial market share for Fast was 40% and 60% for Speedy. Solve the problem:

-Find the probability that a customer using Fast initially will use Fast for his third batch of clothes. Use the following transition matrix. Round your answer to the nearest hundredth.

A) 0. 64
B) 0. 36
C) 0. 35
D) 0. 65

Question 12

Given the transition matrix:

Find the probability of going from state A to state D in four trials.

A) 0.0396
B) 0.0144
C) 0
D) 0.3518

Question 13

Given the transition matrix:

Find the probability of going from state D to state A in four trials.

A) 0
B) 0.3518
C) 0.0288
D) 0.0396

Question 14

Given the transition matrix:

Find the probability of going from state C to state B in four trials.

A) 0.0396
B) 0.756
C) 0
D) 0.0144

Question 15

A trailer rental company has rental and return facilities at both a north and south location in a city. Assume a trailer must be returned to one or the other of these locations. If a trailer is rented at the north location, the probability that it will be returned there is .6; if a trailer is rented at the south location, the probability it will be returned there is .65. Assume the company rents all of its trailers each day and each trailer is rented (and returned) only once a day. If the company starts with 50% of the trailers at each location, what is the expected distribution (in percentages) the next day?

A) 62.5% of the trailers at the north location; 37.5% of the trailers at the south location
B) 47.5% of the trailers at the north location; 52.5% of the trailers at the south location
C) 52.5% of the trailers at the north location; 47.5% of the trailers at the south location
D) 37.5% of the trailers at the north location; 62.5% of the trailers at the south location

Question 16

Dublin is experiencing a population movement out of the city to the suburbs. Currently 85% of the total population live in the city with the remaining 15% living in the suburbs. It has been shown that each year 7% of the city residents move to the suburbs, while only 1% of the suburb population move back to the city. Assuming population remains constant for both, what percent of the total will remain in the city after 2 years. Express your answer rounded to hundredths of a percent.

A) 31.05%
B) 26.14%
C) 73.86%
D) 79.2%

Question 17

Laurinburg is experiencing a population movement out of the city to the suburbs. Currently 85% of the total population live in the city with the remaining 15% living in the suburbs. It has been shown that each year 7% of the city residents move to the suburbs, while only 1% of the suburb population move back to the city. Assuming population remains constant for both, what percent of the total will remain in the city after 5 years. Express your answer rounded to hundredths of a percent.

A) 64.44%
B) 39.72%
C) 60.28%
D) 35.56%

Question 18

Fayetteville is experiencing a population movement out of the city to the suburbs. Currently 85% of the total population live in the city with the remaining 15% living in the suburbs. It has been shown that each year 7% of the city residents move to the suburbs, while only 1% of the suburb population move back to the city. Assuming population remains constant for both, what percent of the total will remain in the suburbs after 5 years. Express your answer rounded to hundredths of a percent.

A) 35.56%
B) 60.28%
C) 39.72%
D) 64.44

Question 19

Decide whether or not the transition matrix is regular.

Question 20

Decide whether or not the transition matrix is regular.

Question 21

Decide whether or not the transition matrix is regular.

Question 22

Decide whether or not the transition matrix is regular.

Question 23

Decide whether or not the transition matrix is regular.

Question 24

Find the stationary matrix for the transition matrix P = .

A)
B) [0.16 0.36]
C) [0.6 0.4]
D) [0.4 0.6]

Question 25

Find the stationary matrix for the transition matrix P = .

A)
B) [0.6 0.4]
C)
D) [0.636 0.364]

Question 26

Find the stationary matrix for the transition matrix P = .

A)
B)
C)
D) [0.25 0.45 0.35]

Question 27

Find the stationary matrix for the transition matrix P = .

A)
B) [5 7 11]
C)
D)

Question 28

Find the stationary matrix for the transition matrix P = Round the numbers in your answer to the nearest hundredth.

A) [0.23 0.17 0.60]
B) [0.29 0.29 0.42]
C) [0.26 0.44 0.30]
D) [0.17 0.29 0.54]

Question 29

Find the stationary matrix for the transition matrix P = Round the numbers in your answer to the nearest hundredth.

A) [0.44 0.48 0.08]
B) [0.33 0.40 0.27]
C) [1 0 0]
D) [0.33 0.67 0]

Question 30

Find the limiting matrix corresponding to the transition matrix P = .Round to the nearest thousandths.

A)
B) [0.5 0.5]
C)
D) [0.462 0.538]

Question 31

Find the limiting matrix corresponding to the transition matrix P = .Round to the nearest hundredths.

A)
B)
C)
D)

Question 32

For the transition matrix P = find exactly by converting to fraction form.

A)
B)
C)
D) [0.213 0.698]

Question 33

The probability that an assembly line works correctly depends on whether the line worked correctly the last time. Find the probability that the line will work in the long run. Round your answer as appropriate.

A) 0. 143
B) 0. 900
C) 0. 857
D) 0. 855

Question 34

Weather is classified as sunny or cloudy in a certain place. What are the long-term predictions for sunny and cloudy days? Round numbers to the nearest thousandths.

A)
B)
C)
D)

Question 35

Rats are kept in a cage with two compartments (A and B). Rats in A move to B with probability 0.7. Rats in B move to A with probability 0.2. Find the long-term trend for rats in each compartment. Round numbers to the nearest thousandth.

A)
B)
C)
D)

Question 36

From statistics gathered over many seasons, it was determined that the probability a basketball player will make a basket after having made a basket on his previous attempt is .55, while the probability he will make a basket if he missed on his previous attempt is .48. In a current game a player has made 45% of his attempted shots. If the player shoots many more times in the game, what would be the overall percentage of baskets that he makes in this game?

A) 51%
B) 49%
C) 52%
D) 48%

Question 37

The probability that an assembly line operation works correctly depends on whether it worked correctly the last time it was used. There is a 0.91 chance that the line will work correctly if it worked correctly the time before and a 0.68 chance that it will work correctly if it did not work correctly the time before. After setting up a transition matrix with this information, find the long-run probability that the line will work correctly.

A) [0.117 0.883]
B) [0.883 0.883]
C) [0.802 0.198]
D) [0.883 0.117]

Question 38

The probability that a car owner will become a car renter in five years is 0.03. The probability that a renter will become an owner in five years is 0.1. Suppose the proportions in the population are 64% owners (O), 35.5% renters (R) and .5% neither (N) with the following transition matrix.

A) [0.666 0.333 0.001]
B) [0.94 0.879 0.001]
C) [0.64 0.355 0.005]
D) [0.94 0.06 0]

Question 39

According to data collected during one year in a large metropolitan community, 30% of commuters used public transportation to get to work, and this rose by 4% the following year. This is modeled by the transition matrix where P represents the percentage of people that use public transportation and the percentage of people that do not. Let Find .

A) [0.9 0.1]
B) [0.3 0.7]
C) [0.66 0.34]
D) [0.34 0.66]

Question 40

According to data collected during one year in a large metropolitan community, 30% of commuters used public transportation to get to work, and this rose by 4% the following year. This is modeled by the transition matrix where P represents the percentage of people that use public transportation and the percentage of people that do not.
Let . Find .Round your answer to the nearest thousandths.

A) [0.372 0.628]
B) [0.628 0.372]
C) [0.66 0.34]
D) [0.34 0.66]

Question 41

According to data collected during one year in a large metropolitan community, 30% of commuters used public transportation to get to work, and this rose by 4% the following year. This is modeled by the transition matrix where P represents the percentage of people that use public transportation and the percentage of people that do not.
What percentage of commuters in this community will use public transportation in the long run?

A) 50%
B) 34%
C) 30%
D) 37.2%

Question 42

Identify the absorbing state(s) in the transition matrix

A) A and B
B) A and D
C) C and D
D) B and D

Question 43

Find all absorbing states for the transition matrix, and indicate whether or not the matrix is that of an absorbing Markov chain.

A) State 3 is absorbing; matrix is an absorbing Markov chain.
B) State 2 is absorbing; matrix is not an absorbing Markov chain.
C) State 3 is absorbing; matrix is not an absorbing Markov chain.
D) State 2 is absorbing; matrix is an absorbing Markov chain.

Question 44

Find all absorbing states for the transition matrix, and indicate whether or not the matrix is that of an absorbing Markov chain.

A) State 2 and 3 are absorbing; the matrix is an absorbing Markov chain.
B) State 2 is an absorbing; the matrix is an absorbing Markov chain.
C) State 2 is an absorbing; the matrix is not an absorbing Markov chain.
D) State 2 and 3 are absorbing; the matrix is not an absorbing Markov chain.

Question 45

Find a standard form for the absorbing Markov chain with the transition matrix

A)
B)
C)
D)

Question 46

Find a standard form for the absorbing Markov chain with the transition matrix

A)
B)
C)
D)

Question 47

Find a standard form for the absorbing Markov chain with the transition matrix

A)
B)
C)
D)

Question 48

Find a standard form for the absorbing Markov chain with the following transition matrix

A)
B)
C)
D)

Question 49

For the transition matrix ,find the limiting matrix. Use fractional entries.

A)
B)
C)
D)

Question 50

Find the fundamental matrix F for the absorbing Markov chain with the given matrix. Express your answer in fraction form.

A) F =
B) F =
C) F =
D) F =

Question 51

Find the fundamental matrix F for the absorbing Markov chain with the given matrix. Express your answer in fraction form.

A) F =
B) F =
C) F = [3]
D) F =

Question 52

Find the fundamental matrix F for the absorbing Markov chain with the given matrix. Express your answer in fraction form.

A)
B)
C)
D)

Question 53

Find the limiting matrix corresponding to the transition matrix P = .Round to the nearest thousandths.

A)
B)
C) [0.429 0.571]
D)

Question 54

Find the limiting matrix corresponding to the transition matrix P = .Round to the nearest hundredths.

A) [0.875 0.125]
B)
C)
D)

Question 55

will end up in state A over the long run.

A)
B)
C) 1
D)

Question 56

A red urn contains 4 red marbles, 2 blue marbles, and 4 green marbles. A blue urn contains 2 red marbles, 2 blue marbles, and 1 green marble. A green urn contains 3 green marbles. A marble is selected from an urn, the color is noted, and the marble is returned to the urn from which it was drawn. The next marble is drawn from the urn whose color is the same as the marble just drawn. Thus, this is a Markov process with three states: draw from the red urn, draw from the blue urn, or draw from the green urn.Write the transition matrix P.

A)
B)
C)
D)

Question 57

A red urn contains 4 red marbles, 2 blue marbles, and 4 green marbles. A blue urn contains 2 red marbles, 2 blue marbles, and 1 green marble. A green urn contains 3 green marbles. A marble is selected from an urn, the color is noted, and the marble is returned to the urn from which it was drawn. The next marble is drawn from the urn whose color is the same as the marble just drawn. Thus, this is a Markov process with three states: draw from the red urn, draw from the blue urn, or draw from the green urn.Find the limiting matrix , if it exists, and describe the long-run behavior of this process.

A)
B)
C)
D)

Question 58

Suppose that for a certain absorbing Markov chain the fundamental matrix is found to be What is the expected number of times a person will have $3, given that she started with $1?

A) 0.5
B) 2.0
C) 1.5
D) 1.0

Question 59

Suppose that for a certain absorbing Markov chain the fundamental matrix is found to be What is the expected number of times a person will have $3, given that he started with $2?

A) 1.0
B) 1.5
C) 0.5
D) 2.0

Question 60

Among patients who were improving on a certain day (in the critical care unit of a certain hospital), the following was determined to be true on the next day: 58% were still improving; 10% were stable; 7% were deteriorating; and 25% had been discharged. Among the patients who were stable on a certain day, the following was determined to be true on the next day: 34% were improving; 44% were still stable; 18% were deteriorating; 3% had been discharged; and 1% had died. Among the patients who were deteriorating on a certain day, the following was determined to be true on the next day: 13% were improving; 42% were stable; 41% were still deteriorating; none had been discharged; and 4% had died. What is the expected number of additional days that a patient, who is improving on that certain day, will spend in the critical care unit? Round your answer to the nearest hundredth.

A) 5.76 days
B) 4.65 days
C) 7.53 days
D) 3.22 days