Let $X$ be the random variable giving the number of points scored by Victor during a match.

We admit that the expected value $E ( X ) = 22$ and the variance $V ( X ) = 65$.
Victor plays $n$ matches, where $n$ is a strictly positive integer.
Let $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ be the random variables giving the number of points scored during the $1 ^ { \text {st} } , 2 ^ { \mathrm { nd } } , \ldots , n$-th matches. We admit that the random variables $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are independent and follow the same distribution as $X$.
We set $\quad M _ { n } = \frac { X _ { 1 } + X _ { 2 } + \ldots + X _ { n } } { n }$.

\begin{enumerate}
  \item In this question, we take $n = 50$.\\
a. What does the random variable $M _ { 50 }$ represent?\\
b. Determine the expected value and variance of $M _ { 50 }$.\\
c. Prove that $P \left( \left| M _ { 50 } - 22 \right| \geqslant 3 \right) \leqslant \frac { 13 } { 90 }$.\\
d. Deduce that the probability of the event ``$19 < M _ { 50 } < 25$'' is strictly greater than 0.85.
  \item Indicate, by justifying, whether the following statement is true or false:\\
``There is no natural number $n$ such that $P \left( \left| M _ { n } - 22 \right| \geqslant 3 \right) < 0.01$ ''.
\end{enumerate}