A survey conducted in France provides the following information:
\begin{itemize}
  \item $60\%$ of people over 15 years old intend to watch the Paris 2024 Olympic and Paralympic Games (OPG) on television;
  \item among those who intend to watch the OPG, 8 out of 9 people declare that they regularly practice a sport.
\end{itemize}
A person over 15 years old is chosen at random. The following events are considered:
\begin{itemize}
  \item $J$: ``the person intends to watch the Paris 2024 OPG on television'';
  \item $S$: ``the chosen person declares that they regularly practice a sport''.
\end{itemize}
We denote by $\bar{J}$ and $\bar{S}$ their complementary events.

In questions 1. and 2., probabilities will be given in the form of an irreducible fraction.

\begin{enumerate}
  \item Demonstrate that the probability that the chosen person intends to watch the Paris 2024 OPG on television and declares that they regularly practice a sport is $\frac{8}{15}$. A weighted tree diagram may be used.

  According to this survey, two out of three people over 15 years old declare that they regularly practice a sport.

  \item[2.] a. Calculate the probability that the chosen person does not intend to watch the Paris 2024 OPG on television and declares that they regularly practice a sport.\\
  b. Deduce the probability of $S$ given $\bar{J}$ denoted $P_{\bar{J}}(S)$.

  In the rest of the exercise, results will be rounded to the nearest thousandth.

  \item[3.] As part of a promotional operation, 30 people over 15 years old are chosen at random. This choice is treated as sampling with replacement. Let $X$ be the random variable that gives the number of people declaring that they regularly practice a sport among the 30 people.\\
  a. Determine the nature and parameters of the probability distribution followed by $X$.\\
  b. Calculate the probability that exactly 16 people declare that they regularly practice a sport among the 30 people.\\
  c. The French judo federation wishes to offer a ticket for the final of the mixed team judo event at the Arena Champ-de-Mars for each person declaring that they regularly practice a sport among these 30 people. The price of a ticket is $380\,€$ and a budget of 10000 euros is available for this operation. What is the probability that this budget is insufficient?
\end{enumerate}