\section*{Exercise 1 — 7 points}
\section*{Topics: Probability}
In Hugo's shop, customers can rent two types of bicycles: road bikes or mountain bikes.\\
Each type of bicycle can be rented in an electric version or not.\\
A customer is chosen at random from the shop, and we assume that:
\begin{itemize}
  \item If the customer rents a road bike, the probability that it is an electric bike is 0.4;
  \item If the customer rents a mountain bike, the probability that it is an electric bike is 0.7;
  \item The probability that the customer rents an electric bike is 0.58.
\end{itemize}
We denote by $\alpha$ the probability that the customer rents a road bike, with $0 \leqslant \alpha \leqslant 1$.\\
We consider the following events:
\begin{itemize}
  \item R: ``the customer rents a road bike'';
  \item $E$ : ``the customer rents an electric bike'';
  \item $\bar { R }$ and $\bar { E }$, complementary events of $R$ and $E$.
\end{itemize}
We model this random situation using the tree shown below.\\
If $F$ denotes any event, we denote by $p ( F )$ the probability of $F$.
\begin{enumerate}
  \item Copy this tree onto your answer sheet and complete it.
  \item a. Show that $p ( E ) = 0.7 - 0.3 \alpha$.\\
b. Deduce that: $\alpha = 0.4$.
  \item We know that the customer rented an electric bike. Determine the probability that they rented a mountain bike. Give the result rounded to the nearest hundredth.
  \item What is the probability that the customer rents an electric mountain bike?
  \item The daily rental price of a non-electric road bike is 25 euros, that of a non-electric mountain bike is 35 euros.\\
For each type of bike, choosing the electric version increases the daily rental price by 15 euros.\\
We denote by $X$ the random variable modeling the daily rental price of a bike.\\
a. Give the probability distribution of $X$. Present the results in the form of a table.\\
b. Calculate the expected value of $X$ and interpret this result.
  \item When 30 of Hugo's customers are chosen at random, we treat this choice as sampling with replacement. We denote by $Y$ the random variable associating to a sample of 30 randomly chosen customers the number of customers who rent an electric bike.\\
We recall that the probability of event $E$ is: $p ( E ) = 0.58$.\\
a. Justify that $Y$ follows a binomial distribution and specify its parameters.\\
b. Determine the probability that a sample contains exactly 20 customers who rent an electric bike. Give the result rounded to the nearest thousandth.\\
c. Determine the probability that a sample contains at least 15 customers who rent an electric bike. Give the result rounded to the nearest thousandth.
\end{enumerate}