Exercise 1 — 4 points\\
Theme: probability\\
Parts A and B can be treated independently\\
Bicycle users in a city are classified into two disjoint categories:
\begin{itemize}
  \item those who use bicycles for professional travel;
  \item those who use bicycles only for leisure.
\end{itemize}
A survey gives the following results:
\begin{itemize}
  \item $21\%$ of users are under 35 years old. Among them, $68\%$ use their bicycle only for leisure while the others use it for professional travel;
  \item among those 35 years or older, only $20\%$ use their bicycle for professional travel, the others use it only for leisure.
\end{itemize}
A bicycle user from this city is randomly interviewed.\\
Throughout the exercise, the following events are considered:
\begin{itemize}
  \item $J$: ``the person interviewed is under 35 years old'';
  \item $T$: ``the person interviewed uses the bicycle for professional travel'';
  \item $\bar{J}$ and $\bar{T}$ are the complementary events of $J$ and $T$.
\end{itemize}

\textbf{Part A}
\begin{enumerate}
  \item Calculate the probability that the person interviewed is under 35 years old and uses their bicycle for professional travel.\\
You may use a probability tree.
  \item Calculate the exact value of the probability of $T$.
  \item Now consider a resident who uses their bicycle for professional travel. Prove that the probability that they are under 35 years old is 0.30 to within $10^{-2}$.
\end{enumerate}

\textbf{Part B}\\
In this part, we are interested only in people using their bicycle for professional travel.\\
We assume that $30\%$ of them are under 35 years old.\\
A sample of 120 people is randomly selected from among them to complete an additional questionnaire.\\
The selection of this sample is treated as random sampling with replacement.\\
Each individual in this sample is asked their age.\\
$X$ represents the number of people in the sample who are under 35 years old.\\
In this part, results should be rounded to $10^{-3}$.
\begin{enumerate}
  \item Determine the nature and parameters of the probability distribution followed by $X$.
  \item Calculate the probability that at least 50 bicycle users among the 120 are under 35 years old.
\end{enumerate}