In a statistics school, after reviewing candidate files, recruitment is done in two ways:
\begin{itemize}
  \item $10\%$ of candidates are selected based on their file. These candidates must then take an oral examination after which $60\%$ of them are finally admitted to the school.
  \item Candidates who were not selected based on their file take a written examination after which $20\%$ of them are admitted to the school.
\end{itemize}

\textbf{Part 1}

A candidate for this recruitment competition is chosen at random. We denote:
\begin{itemize}
  \item $D$ the event ``the candidate was selected based on their file'';
  \item $A$ the event ``the candidate was admitted to the school'';
  \item $\bar{D}$ and $\bar{A}$ the complementary events of events $D$ and $A$ respectively.
\end{itemize}

\begin{enumerate}
  \item Represent the situation with a probability tree.
  \item Calculate the probability that the candidate is selected based on their file and admitted to the school.
  \item Show that the probability of event $A$ is equal to 0.24.
  \item A candidate admitted to the school is chosen at random. What is the probability that their file was not selected?
\end{enumerate}

\textbf{Part 2}

\begin{enumerate}
  \item We assume that the probability for a candidate to be admitted to the school is equal to 0.24.

We consider a sample of seven candidates chosen at random, treating this choice as a random draw with replacement. We denote by $X$ the random variable counting the candidates admitted to the school among the seven drawn.\\
a. We assume that the random variable $X$ follows a binomial distribution. What are the parameters of this distribution?\\
b. Calculate the probability that only one of the seven candidates drawn is admitted to the school. Give an answer rounded to the nearest hundredth.\\
c. Calculate the probability that at least two of the seven candidates drawn are admitted to this school. Give an answer rounded to the nearest hundredth.
  \item A secondary school presents $n$ candidates for recruitment in this school, where $n$ is a non-zero natural number.\\
We assume that the probability for any candidate from the secondary school to be admitted to the school is equal to 0.24 and that the results of the candidates are independent of each other.\\
a. Give the expression, as a function of $n$, of the probability that no candidate from this secondary school is admitted to the school.\\
b. From what value of the integer $n$ is the probability that at least one student from this secondary school is admitted to the school greater than or equal to 0.99?
\end{enumerate}