\section*{Exercise 4 -- 5 points -- For candidates who have not followed the specialization course}
We study a model of virus propagation in a population, week after week. Each individual in the population can be:
\begin{itemize}
  \item either susceptible to being affected by the virus (``of type S'');
  \item either sick (affected by the virus);
  \item either immunized (cannot be affected by the virus).
\end{itemize}

For any natural integer $n$, the model of virus propagation is defined by the following rules:
\begin{itemize}
  \item Among individuals of type S in week $n$, in week $n+1$: $85\%$ remain of type S, $5\%$ become sick and $10\%$ become immunized;
  \item Among sick individuals in week $n$, in week $n+1$: $65\%$ remain sick, and $35\%$ are cured and become immunized.
  \item Any individual immunized in week $n$ remains immunized in week $n+1$.
\end{itemize}

We randomly choose an individual from the population. We consider the following events:\\
$S_{n}$: ``the individual is of type S in week $n$'';\\
$M_{n}$: ``the individual is sick in week $n$'';\\
$I_{n}$: ``the individual is immunized in week $n$''.\\
In week 0, all individuals are considered ``of type S'', so:
$$P(S_{0}) = 1 ; \quad P(M_{0}) = 0 \quad \text{and} \quad P(I_{0}) = 0.$$

\section*{Part A}
We study the evolution of the epidemic during weeks 1 and 2.
\begin{enumerate}
  \item Reproduce and complete the probability tree.
  \item Show that $P(I_{2}) = 0.2025$.
  \item Given that an individual is immunized in week 2, what is the probability, rounded to the nearest thousandth, that he was sick in week 1?
\end{enumerate}

\section*{Part B}
We study the long-term evolution of the disease.\\
For any natural integer $n$, we have: $u_{n} = P(S_{n})$, $v_{n} = P(M_{n})$ and $w_{n} = P(I_{n})$.