Data published on March 1st, 2023 by the Ministry of Ecological Transition on the registration of private vehicles in France in 2022 contain the following information:
\begin{itemize}
  \item $22.86\%$ of vehicles were new vehicles;
  \item $8.08\%$ of new vehicles were rechargeable hybrids;
  \item $1.27\%$ of used vehicles (that is, those that are not new) were rechargeable hybrids.
\end{itemize}

Throughout the exercise, probabilities will be rounded to the ten-thousandth.

\textbf{Part I}

In this part, we consider a private vehicle registered in France in 2022.\\
We denote:
\begin{itemize}
  \item $N$ the event ``the vehicle is new'';
  \item $R$ the event ``the vehicle is a rechargeable hybrid'';
  \item $\bar{N}$ and $\bar{R}$ the complementary events of $N$ and $R$.
\end{itemize}

\begin{enumerate}
  \item Represent the situation with a probability tree.
  \item Calculate the probability that this vehicle is new and a rechargeable hybrid.
  \item Prove that the value rounded to the ten-thousandth of the probability that this vehicle is a rechargeable hybrid is 0.0283.
  \item Calculate the probability that this vehicle is new given that it is a rechargeable hybrid.
\end{enumerate}

\textbf{Part II}

In this part, we choose 500 private rechargeable hybrid vehicles registered in France in 2022.\\
In what follows, we will assume that the probability that such a vehicle is new is equal to 0.65. We treat the choice of these 500 vehicles as a random draw with replacement.\\
We call $X$ the random variable representing the number of new vehicles among the 500 vehicles chosen.

\begin{enumerate}
  \item We assume that the random variable $X$ follows a binomial distribution. Specify the values of its parameters.
  \item Determine the probability that exactly 325 of these vehicles are new.
  \item Determine the probability $p(X \geq 325)$ then interpret the result in the context of the exercise.
\end{enumerate}

\textbf{Part III}

We now choose $n$ private rechargeable hybrid vehicles registered in France in 2022, where $n$ denotes a strictly positive natural number.\\
We recall that the probability that such a vehicle is new is equal to 0.65.\\
We treat the choice of these $n$ vehicles as a random draw with replacement.

\begin{enumerate}
  \item Give the expression as a function of $n$ of the probability $p_n$ that all these vehicles are used.
  \item We denote $q_n$ the probability that at least one of these vehicles is new. By solving an inequality, determine the smallest value of $n$ such that $q_n \geqslant 0.9999$.
\end{enumerate}