Exercise 1 — Part A

The centre offers people coming for a weekend an introductory roller skating formula consisting of two training sessions. We randomly choose a person among those who have subscribed to this formula. We denote by $A$ and $B$ the following events:
\begin{itemize}
  \item A: ``The person falls during the first session'';
  \item B: ``The person falls during the second session''.
\end{itemize}
For any event $E$, we denote $P(E)$ its probability and $\bar{E}$ its complementary event. Observations allow us to assume that $P(A) = 0{,}6$. Furthermore, we observe that:
\begin{itemize}
  \item If the person falls during the first session, the probability that they fall during the second is 0.3;
  \item If the person does not fall during the first session, the probability that they fall during the second is 0.4.
\end{itemize}

\begin{enumerate}
  \item Represent the situation with a probability tree.
  \item Calculate the probability $P(\bar{A} \cap \bar{B})$ and interpret the result.
  \item Show that $P(B) = 0{,}34$.
  \item The person does not fall during the second training session. Calculate the probability that they did not fall during the first session.
  \item We call $X$ the random variable which, for each sample of 100 people who have subscribed to the formula, associates the number of them who did not fall during either the first or the second session. We assimilate the choice of a sample of 100 people to a draw with replacement. We admit that the probability that a person does not fall during either the first or the second session is 0.24.
  \begin{enumerate}
    \item[a.] Show that the random variable $X$ follows a binomial distribution whose parameters you will specify.
    \item[b.] What is the probability of having, in a sample of 100 people who have subscribed to the formula, at least 20 people who do not fall during either the first or the second session?
    \item[c.] Calculate the expectation $E(X)$ and interpret the result in the context of the exercise.
  \end{enumerate}
\end{enumerate}