\textbf{Exercise 2} (4 points)

The flu virus affects each year, during the winter period, part of the population of a city. Vaccination against the flu is possible; it must be renewed each year.

\textbf{Part A}

A study conducted in the city's population at the end of the winter period found that:
\begin{itemize}
  \item $40 \%$ of the population is vaccinated;
  \item $8 \%$ of vaccinated people contracted the flu;
  \item $20 \%$ of the population contracted the flu.
\end{itemize}

A person is chosen at random from the city's population and we consider the events:\\
$V$: ``the person is vaccinated against the flu'';\\
$G$: ``the person contracted the flu''.

\begin{enumerate}
  \item a. Give the probability of event $G$.\\
b. Reproduce the probability tree below and complete the blanks indicated on four of its branches.
  \item Determine the probability that the chosen person contracted the flu and is vaccinated.
  \item The chosen person is not vaccinated. Show that the probability that they contracted the flu is equal to 0.28.
\end{enumerate}

\textbf{Part B}

In this part, the probabilities requested will be given to $10 ^ { - 3 }$ near.\\
A pharmaceutical laboratory conducts a study on vaccination against the flu in this city. After the winter period, $n$ inhabitants of the city are randomly interviewed, assuming that this choice amounts to $n$ successive independent draws with replacement. We assume that the probability that a person chosen at random in the city is vaccinated against the flu is equal to 0.4.\\
Let $X$ be the random variable equal to the number of vaccinated people among the $n$ interviewed.

\begin{enumerate}
  \item What is the probability distribution followed by the random variable $X$?
  \item In this question, we assume that $n = 40$.\\
a. Determine the probability that exactly 15 of the 40 people interviewed are vaccinated.\\
b. Determine the probability that at least half of the people interviewed are vaccinated.
  \item A sample of 3750 inhabitants of the city is interviewed, that is, we assume here that $n = 3750$.\\
Let $Z$ be the random variable defined by: $Z = \frac { X - 1500 } { 30 }$.\\
We admit that the probability distribution of the random variable $Z$ can be approximated by the standard normal distribution.\\
Using this approximation, determine the probability that there are between 1450 and 1550 vaccinated individuals in the sample interviewed.
\end{enumerate}