In the journal Lancet Public Health, researchers claim that on May 11, 2020, 5.7\% of French adults had already been infected with COVID 19.

\begin{enumerate}
  \item An individual is drawn from the adult French population on May 11, 2020. Let $I$ be the event: ``the adult has already been infected with COVID 19''. What is the probability that this individual drawn has already been infected with COVID 19?
  \item A sample of 100 people from the population is drawn, assumed to be chosen independently of each other. This sampling is assimilated to a draw with replacement. Let $X$ be the random variable that counts the number of people who have already been infected.\\
a. Justify that $X$ follows a binomial distribution and give its parameters.\\
b. Calculate its mathematical expectation. Interpret this result in the context of the exercise.\\
c. What is the probability that there is no infected person in the sample? Give an approximate value to $10^{-4}$ near of the result.\\
d. What is the probability that there are at least 2 infected people in the sample? Give an approximate value to $10^{-4}$ near of the result.\\
e. Determine the smallest integer $n$ such that $P(X \leq n) > 0.9$. Interpret this result in the context of the exercise.
\end{enumerate}