Comme un jeu

In a certain country, two operators share the mobile telecommunications market. A study reveals that each year:

Of the customers of operator EfficaceRéseau (E), \(70\pourcent\) renew their contract with this operator, while the others switch to GénialPhone (G).
Of the customers of operator GénialPhone (G), \(45\pourcent\) switch to operator EfficaceRéseau (E), while the others renew with GénialPhone.

As of January 1st, 2020, it is assumed that \(10\pourcent\) of customers have a contract with EfficaceRéseau.
For \(n \in \mathbb{N}\), let \((X_n)\) be the sequence representing the operator (E or G) to which a randomly chosen customer is subscribed in the year \(2020 + n\). Let \(\pi_n = \begin{pmatrix} e_n & g_n \end{pmatrix}\) be the probability distribution.

1. Justify that \((X_n)\) is a Markov chain and draw the associated probabilistic graph.
2. Justify that the initial distribution (in order E, G) is \(\pi_0 = \begin{pmatrix} 0.1 & 0.9 \end{pmatrix}\).
3. Give the transition matrix \(\mathbf{M}\) associated with \((X_n)\).
Verify that on January 1st, 2022, approximately \(57\pourcent\) of customers have a contract with EfficaceRéseau.
1. What is the relationship between \(e_n\) and \(g_n\)?
2. Express \(e_{n+1}\) in terms of \(e_n\) and \(g_n\).
3. Deduce that \(e_{n+1} = 0.25 e_n + 0.45\) for all \(n \in \mathbb{N}\).

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