Steady state and long-term probabilities Problem set A Answers

\(\begin{pmatrix} 0.315 \\ 0.526 \\ 0.159 \end{pmatrix}\)

\(−0.48 s_1 + 0.15 s_2 + 0.22 s_3 = 0\); \(0.29 s_1 − 0.43 s_2 + 0.31 s_3 = 0\); \(s_1 + s_2 + s_3 = 1\)

1. (a) \((\frac{34}{55}, \frac{21}{55})\)
2. (b) \((\frac{6}{17}, \frac{4}{17}, \frac{7}{17})\)

\((\frac{7}{22}, \frac{1}{11}, \frac{7}{22}, \frac{3}{11})\)

1. (a) \(\begin{pmatrix} 0.228 \\ 0.509 \\ 0.263 \end{pmatrix}\)
2. (b) \(\begin{pmatrix} 0.167 \\ 0.194 \\ 0.389 \\ 0.250 \end{pmatrix}\)

\(\begin{pmatrix} 0.132 \\ 0.343 \\ 0.184 \\ 0.342 \end{pmatrix}\)

0.19

1. (a) \(\begin{pmatrix} 0.307 & 0.307 & 0.307 \\ 0.414 & 0.414 & 0.414 \\ 0.279 & 0.279 & 0.279 \end{pmatrix}\)
2. (b) 41.4%

1. (a) \(\begin{pmatrix} 0.68 & 0.37 \\ 0.32 & 0.63 \end{pmatrix}\)
2. (b) 0.520
3. (c) \((\frac{37}{69}, \frac{32}{69})\). Long-term probability \(\frac{37}{69}\)

1. (a) \(0.74 \times 0.6 + p \times 0.4 = 0.6\)
2. (b) p = 0.39