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A student eats lunch every day either at the Canteen (C) or at the Cafeteria (F).
We observe their behavior over time and note the following:
  • If they eat at the Canteen one day, they have a \(30\pourcent\) chance of eating at the Cafeteria the next day.
  • If they eat at the Cafeteria one day, they have a \(40\pourcent\) chance of returning to the Canteen the next day.
Let \(X_n\) be the random variable representing the student's lunch location on day \(n\).
  1. Justify that the sequence \((X_n)\) forms a time-homogeneous Markov chain.
  2. Identify the state space \(E\) and determine the transition probabilities \(p_{ij}\) for \(i, j \in E\).

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