\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
We consider a Markov chain \((X_n)\) represented by the probabilistic graph below, with an initial distribution \(\pi_0 = \begin{pmatrix} 1 & 0 \end{pmatrix}\).
  1. Determine the transition matrix \(\mathbf{M}\) associated with \((X_n)\).
  2. Using calculator, calculate \(\pi_5\) and \(\pi_{10}\). What can you conjecture?
  3. Determine the invariant distribution \(\pi\).

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