4 points\\
Parts $\boldsymbol{A}$ and $\boldsymbol{B}$ are independent.\\
The requested probabilities will be given to $10^{-3}$ near.\\
To help detect certain allergies, a blood test can be performed whose result is either positive or negative. In a population, this test gives the following results:
\begin{itemize}
  \item If an individual is allergic, the test is positive in $97\%$ of cases;
  \item If an individual is not allergic, the test is negative in $95{,}7\%$ of cases.
\end{itemize}
Furthermore, $20\%$ of individuals in the concerned population have a positive test.\\
We randomly choose an individual from the population, and we denote:
\begin{itemize}
  \item $A$ the event ``the individual is allergic'';
  \item $T$ the event ``the individual has a positive test''.
\end{itemize}
We denote by $\bar{A}$ and $\bar{T}$ the complementary events of $A$ and $T$. We also call $x$ the probability of event $A$: $x = p(A)$.

\textbf{Part A}
\begin{enumerate}
  \item Reproduce and complete the tree describing the situation, indicating on each branch the corresponding probability.
  \item a. Prove the equality: $p(T) = 0{,}927x + 0{,}043$.\\
  b. Deduce the probability that the chosen individual is allergic.
  \item Justify by a calculation the following statement: ``If the test of an individual chosen at random is positive, there is more than $80\%$ chance that this individual is allergic''.
\end{enumerate}

\textbf{Part B}\\
A survey on allergies is conducted in a city by interviewing 150 randomly chosen residents, and we assume that this choice amounts to successive independent draws with replacement.\\
We know that the probability that a randomly chosen resident in this city is allergic is equal to $0{,}08$.\\
We denote by $X$ the random variable that associates to a sample of 150 randomly chosen residents the number of allergic people in this sample.
\begin{enumerate}
  \item What is the probability distribution followed by the random variable $X$? Specify its parameters.
  \item Determine the probability that exactly 20 people among the 150 interviewed are allergic.
  \item Determine the probability that at least $10\%$ of the people among the 150 interviewed are allergic.
\end{enumerate}