\section*{Exercise 1 — 6 points}
\section*{Main topics covered: Probability}
At a ski resort, there are two types of passes depending on the skier's age:
\begin{itemize}
  \item a JUNIOR pass for people under 25 years old;
  \item a SENIOR pass for others.
\end{itemize}
Furthermore, a user can choose, in addition to the pass corresponding to their age, the skip-the-line option which allows them to reduce waiting time at the ski lifts.\\
We assume that:
\begin{itemize}
  \item $20 \%$ of skiers have a JUNIOR pass;
  \item $80 \%$ of skiers have a SENIOR pass;
  \item among skiers with a JUNIOR pass, $6 \%$ choose the skip-the-line option;
  \item among skiers with a SENIOR pass, $12.5 \%$ choose the skip-the-line option.
\end{itemize}
We interview a skier at random and consider the events:
\begin{itemize}
  \item $J$ : ``the skier has a JUNIOR pass'';
  \item $C$ : ``the skier chooses the skip-the-line option''.
\end{itemize}
The two parts can be worked on independently

\section*{Part A}
\begin{enumerate}
  \item Represent the situation with a probability tree.
  \item Calculate the probability $P ( J \cap C )$.
  \item Prove that the probability that the skier chooses the skip-the-line option is equal to 0.112.
  \item The skier has chosen the skip-the-line option. What is the probability that this is a skier with a SENIOR pass? Round the result to $10 ^ { - 3 }$.
  \item Is it true that people under twenty-five years old represent less than $15 \%$ of skiers who chose the skip-the-line option? Explain.
\end{enumerate}

\section*{Part B}
We recall that the probability that a skier chooses the skip-the-line option is equal to 0.112. We consider a sample of 30 skiers chosen at random.\\
Let $X$ be the random variable that counts the number of skiers in the sample who chose the skip-the-line option.
\begin{enumerate}
  \item We assume that the random variable $X$ follows a binomial distribution. Give the parameters of this distribution.
  \item Calculate the probability that at least one of the 30 skiers chose the skip-the-line option. Round the result to $10 ^ { - 3 }$.
  \item Calculate the probability that at most one of the 30 skiers chose the skip-the-line option. Round the result to $10 ^ { - 3 }$.
  \item Calculate the expected value of the random variable $X$.
\end{enumerate}