A car dealership sells vehicles with electric motors and vehicles with thermal engines. Some customers, before visiting the dealership website, consulted the dealership's digital platform. It was observed that:
\begin{itemize}
  \item $20\%$ of customers are interested in vehicles with electric motors and $80\%$ prefer to purchase a vehicle with a thermal engine;
  \item when a customer wishes to buy a vehicle with an electric motor, the probability that the customer consulted the digital platform is 0.5;
  \item when a customer wishes to buy a vehicle with a thermal engine, the probability that the customer consulted the digital platform is 0.375.
\end{itemize}
Consider the following events:
\begin{itemize}
  \item $C$: ``a customer consulted the digital platform'';
  \item $E$: ``a customer wishes to acquire a vehicle with an electric motor'';
  \item $T$: ``a customer wishes to acquire a vehicle with a thermal engine''.
\end{itemize}
Customers make choices independently of one another.
\begin{enumerate}
  \item a. Calculate the probability that a randomly chosen customer wishes to acquire a vehicle with an electric motor and consulted the digital platform.\\
  A weighted tree diagram may be used.\\
  b. Prove that $P(C) = 0.4$.\\
  c. Suppose that a customer consulted the digital platform. Calculate the probability that the customer wishes to buy a vehicle with an electric motor.
  \item The dealership welcomes an average of 17 clients daily. Let $X$ be the random variable giving the number of clients wishing to acquire a vehicle with an electric motor.\\
  a. Specify the nature and parameters of the probability distribution followed by $X$.\\
  b. Calculate the probability that at least three of the clients wish to buy a vehicle with an electric motor during a day. Give the result rounded to $10^{-2}$.
\end{enumerate}