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A customer shops every week at either Supermarket A or Supermarket B.
The following patterns are observed:
  • If the customer shops at A one week, there is a \(25\pourcent\) chance they will switch to B the following week.
  • If the customer shops at B one week, there is a \(15\pourcent\) chance they will return to A the following week.
Let \(X_n\) be the random variable representing the supermarket chosen in week \(n\).
  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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