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We consider a two-state Markov chain with states A and B. Its transition matrix is \(\mathbf{M} = \begin{pmatrix} 0.4 & 0.6 \\ 0.7 & 0.3 \end{pmatrix}\) and the distribution at step \(n=1\) is \(\pi_1 = \begin{pmatrix} 0.5 & 0.5 \end{pmatrix}\).
Calculate the probability vectors \(\pi_2\) and \(\pi_3\).
Deduce the probabilities \(P(X_2 = B)\) and \(P(X_3 = A)\).
Express \(\pi_n\) in terms of \(\pi_1\) and \(\mathbf{M}\) for \(n \geq 1\).
Using a calculator, find \(\pi_{10}\) and \(\pi_{20}\). How does the distribution evolve?
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