Steady state and long-term probabilities Problem set B Answers

\(\begin{pmatrix} 0.225 \\ 0.360 \\ 0.416 \end{pmatrix}\)

\(−0.57 s_1 + 0.24 s_2 + 0.33 s_3 = 0\); \(0.37 s_1 − 0.54 s_2 + 0.09 s_3 = 0\); \(s_1 + s_2 + s_3 = 1\)

1. (a) \((\frac{38}{65}, \frac{27}{65})\)
2. (b) \((\frac{4}{19}, \frac{8}{19}, \frac{7}{19})\)

\((\frac{3}{13}, \frac{5}{26}, \frac{5}{13}, \frac{5}{26})\)

1. (a) \(\begin{pmatrix} 0.332 \\ 0.342 \\ 0.326 \end{pmatrix}\)
2. (b) \(\begin{pmatrix} 0.310 \\ 0.293 \\ 0.121 \\ 0.276 \end{pmatrix}\)

\(\begin{pmatrix} 0.193 \\ 0.295 \\ 0.230 \\ 0.283 \end{pmatrix}\)

0.36

1. (a) \(\begin{pmatrix} 0.282 & 0.282 & 0.282 \\ 0.527 & 0.527 & 0.527 \\ 0.191 & 0.191 & 0.191 \end{pmatrix}\)
2. (b) 19.1%

1. (a) \(\begin{pmatrix} 0.92 & 0.53 \\ 0.08 & 0.47 \end{pmatrix}\)
2. (b) 0.849
3. (c) \((\frac{53}{61}, \frac{8}{61})\). Long-term probability \(\frac{53}{61}\)

1. (a) \(p \times 0.4 + 0.28 \times 0.6 = 0.4\)
2. (b) p = 0.58