A pharmaceutical laboratory has just developed a new anti-doping test.

\section*{Part A}
A study on this new test gives the following results:
\begin{itemize}
  \item if an athlete is doped, the probability that the test result is positive is 0.98 (test sensitivity);
  \item if an athlete is not doped, the probability that the test result is negative is 0.995 (test specificity).
\end{itemize}
The test is administered to an athlete selected at random from among the participants in an athletics competition.\\
We denote by $D$ the event ``the athlete is doped'' and $T$ the event ``the test is positive''.\\
We assume that the probability of event $D$ is equal to 0.08.

\begin{enumerate}
  \item Represent the situation in the form of a probability tree.
  \item Prove that $P ( T ) = 0.083$.
  \item a. Given that an athlete presents a positive test, what is the probability that he is doped?\\
b. The laboratory decides to market the test if the probability of the event ``an athlete presenting a positive test is doped'' is greater than or equal to 0.95.\\
Will the test proposed by the laboratory be marketed? Justify.
\end{enumerate}

\section*{Part B}
In a sporting competition, we assume that the probability that a tested athlete presents a positive test is 0.103.

\begin{enumerate}
  \item In this question 1., we assume that the organizers decide to test 5 athletes selected at random from among the athletes in this competition.\\
We denote by $X$ the random variable equal to the number of athletes presenting a positive test among the 5 tested athletes.\\
a. Give the distribution followed by the random variable $X$. Specify its parameters.\\
b. Calculate the expectation $E ( X )$ and interpret the result in the context of the exercise.\\
c. What is the probability that at least one of the 5 tested athletes presents a positive test?
  \item How many athletes must be tested at minimum so that the probability of the event ``at least one tested athlete presents a positive test'' is greater than or equal to 0.75? Justify.
\end{enumerate}