You set out on a walk with three possible locations, denoted locations 1, 2 and 3. The probabilities of travelling from a specified location to the next are given by A = 0.5 0.2 0.4 0.4 0.6 0.1 0.1 0.2 0.5 If you know that you started in location 3, what is the probability that you end up at location 1 after two time periods?
You set out on a walk with three possible locations, denoted locations 1, 2 and 3. The probabilities of travelling from a specified location to the next are given by A = 0.5 0.2 0.4 0.4 0.6 0.1 0.1 0.2 0.5 If you know that you started in location 3, what is the probability that you end up at location 1 after two time periods?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Example 7.40:
You set out on a walk with three possible locations, denoted locations 1, 2 and 3. The
probabilities of travelling from a specified location to the next are given by
Solution.
If you know that you started in location 3, what is the probability that you end up at location 1
after two time periods?
Using the above equation, we get
To solve this we use the equation Xn+1 = AXn. Given that you began in location 3, the initial
state vector is
and subsequently
Exercise
X₁ = AX0
Probability =
A =
=
X₂ = AX₁
0.5 0.2 0.4
0.4 0.6 0.1
0.1 0.2 0.5
=
Xo
0
0.5 0.2 0.4
0.4 0.6 0.1
0.2 0.5
0.1
][
Therefore the probability of ending up in location 1 after two time periods is 0.42.
0.5 0.2 0.4
0.4 0.6 0.1
0.1 0.2 0.5
1
0.4
0.1
=
0.5
0.4
0.1
0.5
=
0.42
0.27
0.31
Refer to the example above. If you know that you started in location 2, what is the
probability that you will end up at location 1 after one time period?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7204a7a9-9779-428a-ab98-f09122257236%2Ff9c12527-b346-4633-b6f8-bb489d97770a%2Fhinf6nn_processed.png&w=3840&q=75)
Transcribed Image Text:Example 7.40:
You set out on a walk with three possible locations, denoted locations 1, 2 and 3. The
probabilities of travelling from a specified location to the next are given by
Solution.
If you know that you started in location 3, what is the probability that you end up at location 1
after two time periods?
Using the above equation, we get
To solve this we use the equation Xn+1 = AXn. Given that you began in location 3, the initial
state vector is
and subsequently
Exercise
X₁ = AX0
Probability =
A =
=
X₂ = AX₁
0.5 0.2 0.4
0.4 0.6 0.1
0.1 0.2 0.5
=
Xo
0
0.5 0.2 0.4
0.4 0.6 0.1
0.2 0.5
0.1
][
Therefore the probability of ending up in location 1 after two time periods is 0.42.
0.5 0.2 0.4
0.4 0.6 0.1
0.1 0.2 0.5
1
0.4
0.1
=
0.5
0.4
0.1
0.5
=
0.42
0.27
0.31
Refer to the example above. If you know that you started in location 2, what is the
probability that you will end up at location 1 after one time period?
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