\textbf{Exercise 1 (5 points) -- Common to all candidates}

In this exercise, probabilities should be rounded to the nearest hundredth.

\textbf{Part A}

A wholesaler buys boxes of green tea from two suppliers. He buys 80\% of his boxes from supplier A and 20\% from supplier B.\\
10\% of the boxes from supplier A show traces of pesticides and 20\% of those from supplier B also show traces of pesticides.\\
A box is randomly selected from the wholesaler's stock and the following events are considered:\\
--- event $A$: ``the box comes from supplier A'';\\
--- event $B$: ``the box comes from supplier B'';\\
--- event $S$: ``the box shows traces of pesticides''.

\begin{enumerate}
  \item Translate the statement in the form of a weighted tree diagram.
  \item a. What is the probability of event $B \cap \bar{S}$?\\
b. Justify that the probability that the selected box shows no traces of pesticides is equal to 0.88.
  \item It is observed that the selected box shows traces of pesticides. What is the probability that this box comes from supplier B?
\end{enumerate}

\textbf{Part B}

The manager of a tea salon buys 10 boxes from the above wholesaler. It is assumed that the latter's stock is sufficiently large to model this situation by random selection of 10 boxes with replacement.\\
Consider the random variable $X$ which associates with this sample of 10 boxes the number of boxes without traces of pesticides.

\begin{enumerate}
  \item Justify that the random variable $X$ follows a binomial distribution and specify its parameters.
  \item Calculate the probability that all 10 boxes are free of pesticide traces.
  \item Calculate the probability that at least 8 boxes show no traces of pesticides.
\end{enumerate}

\textbf{Part C}

For advertising purposes, the wholesaler displays on his leaflets: ``88\% of our tea is guaranteed free of pesticide traces''.

An inspector from the fraud prevention unit wishes to study the validity of this claim. To this end, he randomly selects 50 boxes from the wholesaler's stock and finds 12 with traces of pesticides.

It is assumed that in the wholesaler's stock, the proportion of boxes without traces of pesticides is indeed equal to 0.88.\\
Let $F$ be the random variable which, for any sample of 50 boxes, associates the frequency of boxes containing no traces of pesticides.

\begin{enumerate}
  \item Give the asymptotic confidence interval for the random variable $F$ at the 95\% confidence level.
  \item Can the fraud prevention inspector decide, at the 95\% confidence level, that the advertisement is misleading?
\end{enumerate}