A car dealership sells two types of vehicles:
\begin{itemize}
  \item $60\%$ are fully electric vehicles;
  \item $40\%$ are rechargeable hybrid vehicles.
\end{itemize}
$75\%$ of buyers of fully electric vehicles and $52\%$ of buyers of rechargeable hybrid vehicles have the material possibility of installing a charging station at home.\\
A buyer is chosen at random and the following events are considered:
\begin{itemize}
  \item $E$: ``the buyer chooses a fully electric vehicle'';
  \item $B$: ``the buyer has the possibility of installing a charging station at home''.
\end{itemize}
Throughout the exercise, probabilities should be rounded to the nearest thousandth if necessary.
\begin{enumerate}
  \item Calculate the probability that the buyer chooses a fully electric vehicle and has the possibility of installing a charging station at home.\\
A weighted tree diagram may be used.
  \item Prove that $P(B) = 0.658$.
  \item A buyer has the possibility of installing a charging station at home. What is the probability that he chooses a fully electric vehicle?
  \item A sample of 20 buyers is chosen. This sampling is treated as drawing with replacement.\\
Let $X$ be the random variable that gives the total number of buyers able to install a charging station at home among the sample of 20 buyers.\\
a. Determine the nature and parameters of the probability distribution followed by $X$.\\
b. Calculate $P(X = 8)$.\\
c. Calculate the probability that at least 10 buyers can install a charging station.\\
d. Calculate the expected value of $X$.\\
e. The dealership manager decides to offer the installation of the charging station to buyers who have the possibility of installing one at home. This installation costs $1200$~\euro.\\
On average, what amount should she plan to spend on this offer when selling 20 vehicles?
\end{enumerate}