Chikungunya is a viral disease transmitted from one human to another by bites of infected female mosquitoes. A test has been developed for the screening of this virus. The laboratory manufacturing this test provides the following characteristics:
\begin{itemize}
  \item the probability that a person affected by the virus has a positive test is 0.98;
  \item the probability that a person not affected by the virus has a positive test is 0.01.
\end{itemize}
An individual is chosen at random from a target population. We call:
\begin{itemize}
  \item $M$ the event: "The chosen individual is affected by chikungunya"
  \item $T$ the event: "The test of the chosen individual is positive"
\end{itemize}
We denote by $\bar{M}$ (respectively $\bar{T}$) the opposite event of the event $M$ (respectively $T$). We denote by $p$ $(0 \leqslant p \leqslant 1)$ the proportion of people affected by the disease in the target population.

\begin{enumerate}
  \item a. Copy and complete the probability tree.\\
  b. Express $P(M \cap T)$, $P(\bar{M} \cap T)$ then $P(T)$ as a function of $p$.
  \item a. Prove that the probability of $M$ given $T$ is given by the function $f$ defined on $[0;1]$ by:
$$f(p) = \frac{98p}{97p + 1}$$
  b. Study the variations of the function $f$.
  \item We consider that the test is reliable when the probability that a person with a positive test is actually affected by chikungunya is greater than 0.95. Using the results of question 2., from what proportion $p$ of sick people in the population is the test reliable?
\end{enumerate}