According to a study, regular users of public transport represent $17\%$ of the French population. Among these regular users, $32\%$ are young people aged 18 to 24 years old.

A person is randomly interviewed and we note:
\begin{itemize}
  \item $R$ the event: ``The person interviewed regularly uses public transport''.
  \item $J$ the event: ``The person interviewed is aged 18 to 24 years old''.
\end{itemize}

\textbf{Part A:}
\begin{enumerate}
  \item Represent the situation using a probability tree, reporting the data from the problem statement.
  \item Calculate the probability $P(R \cap J)$.
  \item According to this same study, young people aged 18 to 24 represent $11\%$ of the French population. Show that the probability that the person interviewed is a young person aged 18 to 24 who does not regularly use public transport is 0.056 to $10^{-3}$ precision.
  \item Deduce the proportion of young people aged 18 to 24 among non-regular users of public transport.
\end{enumerate}

\textbf{Part B:}\\
During a census of the French population, a census taker randomly interviews 50 people in one day about their use of public transport. The French population is large enough to assimilate this census to sampling with replacement. Let $X$ be the random variable counting the number of people regularly using public transport among the 50 people interviewed.
\begin{enumerate}
  \item Determine, by justifying, the distribution of $X$ and specify its parameters.
  \item Calculate $P(X = 5)$ and interpret the result.
  \item The census taker indicates that there is more than a $95\%$ chance that, among the 50 people interviewed, fewer than 13 of them regularly use public transport. Is this statement true? Justify your answer.
  \item What is the average number of people regularly using public transport among the 50 people interviewed?
\end{enumerate}