\section*{Exercise 4 (4 points)}
\section*{The two parts are independent.}
A laboratory manufactures a medicine packaged in the form of tablets.

\section*{Part A}
A quality control, concerning the mass of tablets, showed that $2 \%$ of tablets have non-conforming mass. These tablets are packaged in boxes of 100 chosen at random from the production line. We admit that the conformity of a tablet is independent of that of the others.

We denote by $N$ the random variable that associates to each box of 100 tablets the number of non-conforming tablets in this box.

\begin{enumerate}
  \item Justify that the random variable $N$ follows a binomial distribution whose parameters you will specify.
  \item Calculate the expectation of $N$ and give an interpretation in the context of the exercise.
  \item Results will be rounded to $10 ^ { - 3 }$ near.\\
a. Calculate the probability that a box contains exactly three non-conforming tablets.\\
b. Calculate the probability that a box contains at least 95 conforming tablets.
  \item The laboratory director wants to modify the number of tablets per box to be able to state: ``The probability that a box contains only conforming tablets is greater than 0.5''.\\
What is the maximum number of tablets a box should contain to meet this criterion? Justify.
\end{enumerate}

\section*{Part B}
We admit that the masses of tablets are independent of one another. We take a sample of 100 tablets and we denote $M _ { i }$, for $i$ natural integer between 1 and 100, the random variable that gives the mass in grams of the $i$-th tablet sampled.\\
We consider the random variable $S$ defined by:
$$S = M _ { 1 } + M _ { 2 } + \ldots + M _ { 100 } .$$
We admit that the random variables $M _ { 1 } , M _ { 2 } , \ldots , M _ { 100 }$ follow the same probability distribution with expectation $\mu = 2$ and standard deviation $\sigma$.

\begin{enumerate}
  \item Determine $E ( S )$ and interpret the result in the context of the exercise.
  \item We denote by $s$ the standard deviation of the random variable $S$.\\
Show that: $s = 10 \sigma$.
  \item We wish that the total mass, in grams, of the tablets contained in a box be strictly between 199 and 201 with a probability at least equal to 0.9.\\
a. Show that this condition is equivalent to:
$$P ( | S - 200 | \geqslant 1 ) \leqslant 0.1 .$$
b. Deduce the maximum value of $\sigma$ which allows, using the Bienaymé--Chebyshev inequality, to ensure this condition.
\end{enumerate}