Exercise 3 (7 points)\\
The director of a large company proposed a training course to all its employees on the use of new software. This course was followed by $25\%$ of employees.

\begin{enumerate}
  \item In this company, $52\%$ of employees are women, of whom $40\%$ followed the course.
\end{enumerate}

A random employee of the company is questioned and we consider the events:
\begin{itemize}
  \item $F$: ``the employee questioned is a woman'',
  \item $S$: ``the employee questioned followed the course''.\\
$\bar{F}$ and $\bar{S}$ denote respectively the complementary events of events $F$ and $S$.\\
a. Give the probability of event $S$.\\
b. Copy and complete the blanks of the probability tree below on the four indicated branches.\\
c. Demonstrate that the probability that the person questioned is a woman who followed the course is equal to 0.208.\\
d. Given that the person questioned followed the course, what is the probability that it is a woman?\\
e. The director claims that, among the male employees of the company, fewer than $10\%$ followed the course. Justify the director's claim.
\end{itemize}

\begin{enumerate}
  \setcounter{enumi}{1}
  \item We denote by $X$ the random variable that associates to a sample of 20 employees of this company chosen at random the number of employees in this sample who followed the course. We assume that the number of employees in the company is sufficiently large to assimilate this choice to sampling with replacement.\\
a. Determine, by justifying, the probability distribution followed by the random variable $X$.\\
b. Determine, to $10^{-3}$ near, the probability that 5 employees in a sample of 20 followed the course.\\
c. The program below, written in Python language, uses the function binomial$(i, n, p)$ created for this purpose which returns the value of the probability $P(X = i)$ in the case where the random variable $X$ follows a binomial distribution with parameters $n$ and $p$.
\begin{verbatim}
def proba(k) :
    P=0
    for i in range(0,k+1) :
        P=P+binomiale(i,20,0.25)
    return P
\end{verbatim}
Determine, to $10^{-3}$ near, the value returned by this program when proba(5) is entered in the Python console. Interpret this value in the context of the exercise.\\
d. Determine, to $10^{-3}$ near, the probability that at least 6 employees in a sample of 20 followed the course.
  \item This question is independent of questions 1 and 2.\\
To encourage employees to follow the course, the company had decided to increase the salaries of employees who followed the course by $5\%$, compared to $2\%$ increase for employees who did not follow the course.\\
What is the average percentage increase in salaries for this company under these conditions?
\end{enumerate}