In France there are two formulas for obtaining a driving license:
\begin{itemize}
  \item Follow supervised driving training from age 15 for 2 years;
  \item Follow classical training (without supervised driving) from age 17.
\end{itemize}
In France currently, among young people who follow driving license training, $16\%$ choose supervised driving training, and among them, $74.7\%$ pass the driving test on their first attempt. Following classical training, the success rate on the first attempt is only $56.8\%$.

A young French person who has already taken the driving test is chosen at random, and we consider the following events $A$ and $R$:
\begin{itemize}
  \item $A$: ``the young person followed supervised driving training'';
  \item $R$: ``the young person obtained their license on their first attempt''.
\end{itemize}

Results should be rounded to $10^{-3}$ if necessary.

\textbf{Part A}
\begin{enumerate}
  \item Draw a probability tree modeling this situation.
  \item a. Prove that $P(R) = 0.59664$.

  In the following, we will keep the value 0.597 rounded to $10^{-3}$.\\
  b. Give this result as a percentage and interpret it in the context of the exercise.
  \item A young person who obtained their license on their first attempt is chosen. What is the probability that they followed supervised driving training?
  \item What should be the proportion of young people following supervised driving training if we wanted the overall success rate (regardless of the training chosen) on the first attempt at the driving test to exceed $70\%$?
\end{enumerate}

\textbf{Part B}

A driving school presents 10 candidates for the driving test for the first time, all of whom have followed supervised driving training. The fact of taking driving tests is modeled by independent random trials.

Let $X$ be the random variable giving the number of these 10 candidates who will obtain their license on their first attempt.
\begin{enumerate}
  \item Justify that $X$ follows a binomial distribution with parameters $n = 10$ and $p = 0.747$.
  \item Calculate $P(X \geqslant 6)$. Interpret this result.
  \item Determine $E(X)$ and $V(X)$.
  \item There are also 40 candidates who have not followed supervised driving training and who are taking the driving test for the first time. In the same way, let $Y$ be the random variable giving the number of these candidates who will obtain their license on their first attempt. We admit that $Y$ is independent of the variable $X$ and that in fact $E(Y) = 22.53$ and $V(Y) = 9.81$.

  Let $Z$ be the random variable counting the total number of candidates (among the 50) who will obtain their driving license on their first attempt at this driving school.\\
  a. Express $Z$ as a function of $X$ and $Y$. Deduce $E(Z)$ and $V(Z)$.\\
  b. Using Bienaymé-Chebyshev's inequality, show that the probability that there are fewer than 20 or more than 40 candidates who obtain their license on their first attempt is less than 0.12.
\end{enumerate}