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In a certain city, two meal options share the student market: the
Canteen (C)
and the
Food Court (F)
. A survey indicates that each day:
Among students who ate at the
Canteen (C)
on day \(n\), \(80\pourcent\) eat again at the canteen on day \(n+1\), while the others go to the food court.
Among students who ate at the
Food Court (F)
on day \(n\), \(30\pourcent\) go to the canteen on day \(n+1\), while the others stay at the food court.
On day \(0\), it is assumed that \(20\pourcent\) of students eat at the canteen.
For \(n \in \mathbb{N}\), let \((X_n)\) be the sequence representing the place (C or F) where a randomly chosen student eats on day \(n\). Let \(\pi_n=\begin{pmatrix} c_n & f_n \end{pmatrix}\) be the probability distribution.
Justify that \((X_n)\) is a time-homogeneous Markov chain and draw the associated probabilistic graph.
Justify that the initial distribution (in order C, F) is \(\pi_0=\begin{pmatrix} 0.2 & 0.8 \end{pmatrix}\).
Give the transition matrix \(M\) associated with \((X_n)\).
Compute \(\pi_2\) and deduce the proportion of students who eat at the canteen on day \(2\).
What is the relationship between \(c_n\) and \(f_n\)?
Express \(c_{n+1}\) in terms of \(c_n\) and \(f_n\).
Deduce that \(c_{n+1} = 0.5\,c_n + 0.3\) for all \(n \in \mathbb{N}\).
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