At the output of a tennis ball manufacturing factory, a ball is judged to be compliant in $85\%$ of cases.

\begin{enumerate}
  \item We test successively 20 balls. We consider that the number of balls is large enough to assimilate these tests to sampling with replacement. We denote $X$ the random variable that counts the number of compliant balls among the 20 tested.\\
a. What is the distribution followed by $X$ and what are its parameters? Justify.\\
b. Calculate $P(X \leqslant 18)$.\\
c. What is the probability that at least two balls are not compliant among the 20 balls tested?\\
d. Determine the expectation of $X$.
  \item We now test $n$ balls successively. We consider the $n$ tests as a sample of $n$ independent random variables $X$ following the Bernoulli distribution with parameter 0.85.\\
We consider the random variable
$$M_n = \sum_{i=1}^{n} \frac{X_i}{n} = \frac{X_1}{n} + \frac{X_2}{n} + \frac{X_3}{n} + \ldots + \frac{X_n}{n}$$
a. Determine the expectation and variance of $M_n$.\\
b. After recalling the Bienaymé-Chebyshev inequality, show that, for every natural integer $n$, $P\left(0.75 < M_n < 0.95\right) \geqslant 1 - \frac{12.75}{n}$.\\
c. Deduce an integer $n$ such that the average number of compliant balls for a sample of size $n$ belongs to the interval $]0.75 ; 0.95[$ with a probability greater than 0.9.
\end{enumerate}