35)
Decide whether or not the transition matrix is regular. Answer Yes or No.
1 0 0
0 0 1
0 1 0
35)
A)
Yes
B)
No
36)
The transition matrix for a Markov process is:
State
A B
State A
B
0.3 0.7
0.9 0.1
= P
Find the first state matrix if the initial state is S0=.3 .7 .
36)
A)
[0.3 0.7]
B)
[0.468 0.532]
C)
0.72 0.28
D)
[0.9 0.1]
37)
Find the stationary matrix for the transition matrix P =
0.80 0.10 0.10
0.15 0.80 0.05
0.20 0.70 0.10
Round the numbers in your answer to the nearest hundredth.
37)
A)
[1 0 0]
B)
[0.33 0.40 0.27]
C)
[0.33 0.67 0]
D)
[0.44 0.48 0.08]
38)
Find all absorbing states for the transition matrix, and indicate whether or not the matrix is that of
an absorbing Markov chain.
1 2 3 4
1
2
3
4
0.1 0 0.1 0.8
0 0.1 0 0
0.1 0.2 0.3 0.4
0 0 0 1
38)
A)
State 2 is an absorbing; the matrix is not an absorbing Markov chain.
B)
State 2 and 3 are absorbing; the matrix is not an absorbing Markov chain.
C)
State 2 is an absorbing; the matrix is an absorbing Markov chain.
D)
State 2 and 3 are absorbing; the matrix is an absorbing Markov chain.
Solve the problem.
39)
Rats are kept in a cage with two compartments (A and B). Rats in A move to B with probability 0.5.
Rats in B move to A with probability 0.1. Find the long–term trend for rats in each compartment.
Round numbers to the nearest thousandth.
39)
A)
0.167 0.833
B)
0.475 0.525
C)
0.833 0.167
D)
0.500 0.500
Provide an appropriate response.
40)
The transition matrix for a Markov process is:
State
A B
State A
B
0.3 0.7
0.9 0.1
= P
Find the second state matrix if the initial state is S0=0.3 0.7 .
40)
A)
[0.3 0.7]
B)
[0.72 0.28]
C)
[0.6192 0.3808]
D)
[ 0.468 0.538]
41)
Find the stationary matrix for the transition matrix P =0.8 0.2
0.35 0.65 .
41)
A)
1 0
0 1
B)
[0.6 0.4]
C)
[0.636 0.364]
D)
0.8 0.2
0.35 0.65
42)
Find the limiting matrix P corresponding to the transition matrix P =
1
2
1
2
3
7
4
7
.
Round to the nearest thousandths.
42)
A)
0.462 0.538
0.462 0.538
B)
[0.5 0.5]
C)
[0.462 0.538]
D)
3
7
4
7
43)
Find all absorbing states for the transition matrix, and indicate whether or not the matrix is that of
an absorbing Markov chain.
1 2 3
1
2
3
0.9 0 0.1
0 1 0
0.6 0 0.4
43)
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 2 is absorbing; matrix is an absorbing Markov chain.
D)
State 3 is absorbing; matrix is not an absorbing Markov chain.
44)
Find a standard form for the absorbing Markov chain with the transition matrix
A B C
A
B
C
1 0 0
1
3
1
3
1
3
0 0 1
44)
A)
A C B
A
C
B
1 0 0
0 1 0
1
3
1
3
1
3
B)
A C B
A
C
B
1 0 0
0 1 0
3 3 3
C)
A B C
A
B
C
1 0 0
0 1 0
1
3
1
3
1
3
D)
A B C
A
B
C
1 0 0
3 3 3
0 0 1
45)
Find the fundamental matrix F for the absorbing Markov chain with the given matrix. Express your
answer in fraction form.
1 0 0 0
1
20 0 1
2
0 0 1 0
1
4
1
201
4
45)
A)
2 3
2 1
B)
3
2 1
1 2
C)
3 2
1
2 1
D)
3 1
2 2
46)
Decide whether or not the transition matrix is regular. Answer Yes or No.
0.7 0.3 0
0 0.4 0.6
1 0 0
46)
A)
Yes
B)
No
47)
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?
47)
A)
48%
B)
51%
C)
52%
D)
49%
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.
48)
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.
Fast Speedy
Fast
Speedy
0.16 0.84
0.31 0.69
48)
A)
0.74
B)
0.71
C)
0.28
D)
0.26
49)
Find the probability that a customer using Fast initially will use Fast for his second batch of clothes.
Use the following transition matrix.
Fast Speedy
Fast
Speedy
0.61 0.39
0.27 0.73
49)
A)
0.73
B)
0.27
C)
0.39
D)
0.61
Provide an appropriate response.
50)
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.
50)
A)
G B R
P =
G
B
R
0.4 0.2 0.4
0.2 0.4 0.4
0.4 0.2 0.4
B)
G B R
P =
G
B
R
1 0 0
0.2 0.4 0.4
0.4 0.2 0.4
C)
G B R
P =
G
B
R
3 0 0
2 2 1
4 2 4
D)
G B R
P =
G
B
R
1 0 0
0 1 0
0 0 1
51)
Find the limiting matrix P corresponding to the transition matrix P =
0.3 0.3 0.4
0.4 0.4 0.2
.
Round to the nearest hundredths.
51)
A)
0.1 0.1 0.8
0.3 0.3 0.4
0.4 0.4 0.2
B)
1 0 0
0 1 0
0 0 1
C)
0.36 0.36 0.28
0.28 0.28 0.44
0.24 0.24 0.52
D)
0.29 0.29 0.29
0.29 0.29 0.29
0.29 0.29 0.29
Solve the problem.
52)
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.
Sunny Cloudy
Sunny
Cloudy
0.9 0.1
0.2 0.8
52)
A)
0.855 0.145
B)
0.900 0.100
C)
0.333 0.667
D)
0.667 0.333
Provide an appropriate response.
53)
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.
53)
A)
79.2%
B)
73.86%
C)
26.14%
D)
31.05%
54)
Given the transition matrix:
A B C D A B C D
P =
A
B
C
D
0.4 0.3 0.2 0.1
0 1 0 0
0 0 1 0
0.2 0.5 0.1 0.2
P4=
A
B
C
D
0.0396 0.5942 0.3518 0.0144
0 1 0 0
0 0 1 0
0.0288 0.756 0.2044 0.0108
Find the probability of going from state A to state D in four trials.
54)
A)
0
B)
0.0396
C)
0.3518
D)
0.0144
Construct the transition matrix that represents the data.
55)
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.
T C
T
C
55)
A)
T C
T
C
0.95 0.10
0.05 0.95
B)
T C
T
C
0.90 0.05
0.10 0.95
C)
T C
T
C
0.95 0.10
0.05 0.90
D)
T C
T
C
0.90 0.10
0.05 0.95
Provide an appropriate response.
56)
Find a standard form for the absorbing Markov chain with the transition matrix
A B C D
A
B
C
D
1/4 1/4 1/4 1/4
0 1 0 0
1/2 0 1/2 0
0 0 0 1
56)
A)
A B C D
B
D
A
C
1 0 0 0
0 1 0 0
1/4 1/4 1/4 1/4
0 0 1/2 1/2
B)
B D A C
B
D
A
C
1 0 0 0
0 1 0 0
1/4 1/4 1/4 1/4
0 0 1/2 1/2
C)
A B C D
B
D
A
C
1 0 0 0
0 1 0 0
1/4 1/4 1/4 1/4
0 0 1/2 1/2
D)
D B C A
B
D
A
C
1 0 0 0
0 1 0 0
1/4 1/4 1/4 1/4
0 0 1/2 1/2
57)
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.
O R N
O
R
N
0.94 0.06 0
0.12 0.879 0.001
032 0.68
Find the long–range probabilities for the three categories.
57)
A)
[0.94 0.06 0]
B)
[0.666 0.333 0.001]
C)
[0.64 0.355 0.005]
D)
[0.94 0.879 0.001]
58)
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?
58)
A)
52.5% of the trailers at the north location; 47.5% of the trailers at the south location
B)
62.5% of the trailers at the north location; 37.5% of the trailers at the south location
C)
37.5% of the trailers at the north location; 62.5% of the trailers at the south location
D)
47.5% of the trailers at the north location; 52.5% of the trailers at the south location
59)
Given the transition matrix:
A B C D A B C D
P =
A
B
C
D
0.4 0.3 0.2 0.1
0 1 0 0
0 0 1 0
0.2 0.5 0.1 0.2
P4=
A
B
C
D
0.0396 0.5942 0.3518 0.0144
0 1 0 0
0 0 1 0
0.0288 0.756 0.2044 0.0108
Find the probability of going from state D to state A in four trials.
59)
A)
0.0396
B)
0
C)
0.0288
D)
0.3518
60)
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.
60)
A)
39.72%
B)
35.56%
C)
60.28%
D)
64.44%
Answer Key
Testname: C9
Answer Key
Testname: C9