\section*{Exercise 1}
4 points

\section*{Common to all candidates}
Parts $\mathrm { A } , \mathrm { B }$ and C can be treated independently.\\
Throughout the exercise, results should be rounded, if necessary, to the nearest thousandth.

\section*{Part A}
A merchant receives the results of a market study on consumer habits in France.\\
According to this study:

\begin{itemize}
  \item $54 \%$ of consumers prefer products of French manufacture;
  \item $65 \%$ of consumers regularly buy products from organic agriculture, and among them $72 \%$ prefer products of French manufacture.
\end{itemize}

A consumer is chosen at random. The following events are considered:

\begin{itemize}
  \item B: ``a consumer regularly buys products from organic agriculture'';
  \item $F$ : ``a consumer prefers products of French manufacture''.
\end{itemize}

We denote $P ( A )$ the probability of event $A$ and $P _ { C } ( A )$ the probability of $A$ given $C$.

\begin{enumerate}
  \item Justify that $P ( \bar { B } \cap F ) = 0.072$.
  \item Calculate $P _ { F } ( \bar { B } )$.
  \item A consumer is chosen who does not regularly buy products from organic agriculture.\\
What is the probability that he prefers products of French manufacture?
\end{enumerate}

\section*{Part B}
The merchant is interested in the quantity in kilograms of organic flour sold each month at retail in his store. This quantity is modeled by a random variable $X$ which follows the normal distribution with mean $\mu = 90$ and standard deviation $\sigma = 2$.

\begin{enumerate}
  \item At the beginning of each month, the merchant ensures he has 95 kg in stock.
\end{enumerate}

What is the probability that he cannot meet customer demand during the month?\\
2. Determine an approximate value to the nearest hundredth of the real number $a$ such that $P ( X < a ) = 0.02$.

Interpret the result in the context of the exercise.

\section*{Part C}
In this market study, it is specified that $46.8 \%$ of consumers in France prefer local products. The merchant observes that among his 2500 customers, 1025 regularly buy local products.\\
Is his customer base representative of consumers in France?