Given the Markov Chain (S, ro, P) where S = {$1, 82, 83}, To , and [o 3 .21 Г.3 .14 P = |1 0 .4 p² = |0 .58 .36 P3 = |.58 .252 .376 .7 .28 .44 .2 [.14 .23 .196] 0 .7 .4 28 .518 .428 .23 .1792 .1984] 252 4372 3692 p5 [.1792 .20788 .19704] .4372 33264 37218 p6 [.20788 .191688 .197808] 33264 391672 36936
Given the Markov Chain (S, ro, P) where S = {$1, 82, 83}, To , and [o 3 .21 Г.3 .14 P = |1 0 .4 p² = |0 .58 .36 P3 = |.58 .252 .376 .7 .28 .44 .2 [.14 .23 .196] 0 .7 .4 28 .518 .428 .23 .1792 .1984] 252 4372 3692 p5 [.1792 .20788 .19704] .4372 33264 37218 p6 [.20788 .191688 .197808] 33264 391672 36936
A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Markov Chain Example**
Given the Markov Chain \((S, x_0, P)\) where \(S = \{s_1, s_2, s_3\}\), \(x_0 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\), and
\[ P = \begin{bmatrix}
0 & .3 & .2 \\
1 & 0 & .4 \\
0 & .7 & .4
\end{bmatrix} \]
\[P^2 = \begin{bmatrix}
.3 & .14 & .2 \\
0 & .58 & .36 \\
.7 & .28 & .44
\end{bmatrix} \]
\[P^3 = \begin{bmatrix}
.14 & .23 & .196 \\
.58 & .252 & .376 \\
.28 & .518 & .428
\end{bmatrix} \]
\[P^4 = \begin{bmatrix}
.23 & .1792 & .1984 \\
.252 & .4372 & .3692 \\
.518 & .3836 & .4344
\end{bmatrix} \]
\[P^5 = \begin{bmatrix}
.1792 & .20788 & .19704 \\
.4372 & .33264 & .37218 \\
.3836 & .45948 & .4308
\end{bmatrix} \]
\[P^6 = \begin{bmatrix}
.20788 & .191688 & .197808 \\
.33264 & .391672 & .36936 \\
.45948 & .41664 & .432832
\end{bmatrix} \]
Compute the probability that the Markov Chain is in state \(s_1\) at time 6.
---
**Interpretation of Matrices:**
In this case, each matrix \(P^n\) represents the state transition probabilities after \(n\) steps.
For example:
- \(P^2\) indicates the state probabilities after 2 steps.
- \(P^3\) indicates the state probabilities after 3 steps.
- and so on...
As \(n\) increases, \(P^n\) gives the probabilities of being in each state after \(n\) transitions, starting](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00a48d79-f805-418c-a679-1c91878d1d75%2F590b5f5e-52ef-4ed1-a14e-06092fbb209f%2Fjwnjk4w_processed.png&w=3840&q=75)
Transcribed Image Text:**Markov Chain Example**
Given the Markov Chain \((S, x_0, P)\) where \(S = \{s_1, s_2, s_3\}\), \(x_0 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\), and
\[ P = \begin{bmatrix}
0 & .3 & .2 \\
1 & 0 & .4 \\
0 & .7 & .4
\end{bmatrix} \]
\[P^2 = \begin{bmatrix}
.3 & .14 & .2 \\
0 & .58 & .36 \\
.7 & .28 & .44
\end{bmatrix} \]
\[P^3 = \begin{bmatrix}
.14 & .23 & .196 \\
.58 & .252 & .376 \\
.28 & .518 & .428
\end{bmatrix} \]
\[P^4 = \begin{bmatrix}
.23 & .1792 & .1984 \\
.252 & .4372 & .3692 \\
.518 & .3836 & .4344
\end{bmatrix} \]
\[P^5 = \begin{bmatrix}
.1792 & .20788 & .19704 \\
.4372 & .33264 & .37218 \\
.3836 & .45948 & .4308
\end{bmatrix} \]
\[P^6 = \begin{bmatrix}
.20788 & .191688 & .197808 \\
.33264 & .391672 & .36936 \\
.45948 & .41664 & .432832
\end{bmatrix} \]
Compute the probability that the Markov Chain is in state \(s_1\) at time 6.
---
**Interpretation of Matrices:**
In this case, each matrix \(P^n\) represents the state transition probabilities after \(n\) steps.
For example:
- \(P^2\) indicates the state probabilities after 2 steps.
- \(P^3\) indicates the state probabilities after 3 steps.
- and so on...
As \(n\) increases, \(P^n\) gives the probabilities of being in each state after \(n\) transitions, starting
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