There are four blood groups in the human species: $\mathrm { A } , \mathrm { B } , \mathrm { AB }$ and O. Each blood group can present a rhesus factor. When it is present, we say that the rhesus is positive, otherwise we say that it is negative.

Within the French population, we know that:
\begin{itemize}
  \item $45 \%$ of individuals belong to group A, and among them $85 \%$ are rhesus positive;
  \item $10 \%$ of individuals belong to group B, and among them $84 \%$ are rhesus positive;
  \item $3 \%$ of individuals belong to group AB, and among them $82 \%$ are rhesus positive.
\end{itemize}

We randomly choose a person from the French population. We denote by:
\begin{itemize}
  \item A the event ``The chosen person is of blood group A'';
  \item B the event ``The chosen person is of blood group B'';
  \item $AB$ the event ``The chosen person is of blood group AB'';
  \item O the event ``The chosen person is of blood group O'';
  \item $R$ the event ``The chosen person has a positive rhesus factor''.
\end{itemize}
For any event $E$, we denote by $\bar { E }$ the complementary event of $E$ and $p ( E )$ the probability of $E$.

\begin{enumerate}
  \item Copy the tree opposite and complete the ten blanks.
  \item Show that $p ( B \cap R ) = 0{,}084$. Interpret this result in the context of the exercise.
  \item We specify that $p ( R ) = 0{,}8397$. Show that $p _ { O } ( R ) = 0{,}83$.
  \item We say that an individual is a ``universal donor'' when their blood can be transfused to any person without risk of incompatibility. Blood group O with negative rhesus is the only one satisfying this characteristic. Show that the probability that an individual randomly chosen from the French population is a universal donor is 0,0714.
  \item During a blood collection, a sample of 100 people is chosen from the population of a French city. This population is large enough to assimilate this choice to sampling with replacement. We denote by $X$ the random variable that associates to each sample of 100 people the number of universal donors in that sample.\\
  a. Justify that $X$ follows a binomial distribution and specify its parameters.\\
  b. Determine to $10 ^ { - 3 }$ near the probability that there are at most 7 universal donors in this sample.\\
  c. Show that the expectation $E ( X )$ of the random variable $X$ is equal to 7,14 and that its variance $V ( X )$ is equal to 6,63 to $10 ^ { - 2 }$ near.
  \item During the national blood donation week, a blood collection is organized in $N$ randomly chosen French cities numbered $1,2,3 , \ldots , N$ where $N$ is a non-zero natural integer. We consider the random variable $X _ { 1 }$ which associates to each sample of 100 people from city 1 the number of universal donors in that sample. We define in the same way the random variables $X _ { 2 }$ for city $2 , \ldots , X _ { N }$ for city $N$. We assume that these random variables are independent and that they have the same expectation equal to 7,14 and the same variance equal to 6,63. We consider the random variable $M _ { N } = \frac { X _ { 1 } + X _ { 2 } + \ldots + X _ { N } } { N }$.\\
  a. What does the random variable $M _ { N }$ represent in the context of the exercise?\\
  b. Calculate the expectation $E \left( M _ { N } \right)$.\\
  c. We denote by $V \left( M _ { N } \right)$ the variance of the random variable $M _ { N }$. Show that $V \left( M _ { N } \right) = \frac { 6{,}63 } { N }$.\\
  d. Determine the smallest value of $N$ for which the Bienaymé-Chebyshev inequality allows us to assert that:
  $$P \left( 7 < M _ { N } < 7{,}28 \right) \geqslant 0{,}95 .$$
\end{enumerate}