All probabilities, unless otherwise stated, will be rounded to $10 ^ { - 3 }$ in this exercise.
``Chikungunya virus, transmitted to humans by the bite of the tiger mosquito, causes patients acute joint pain that can be persistent. In 2005, a major chikungunya epidemic affected the islands of the Indian Ocean and notably the island of Réunion, with several hundred thousand reported cases. In 2007, the disease appeared in Europe, then at the end of 2013, in the Caribbean and reached the American continent in 2014''.
(https ://www.pasteur.fr/fr/centre-medical/fiches-maladies/chikungunya)
A test has been developed for the detection of this virus.
The laboratory manufacturing this test provides the following characteristics:

\begin{itemize}
  \item the probability that an individual affected by the virus has a positive test is 0.999;
  \item the probability that an individual not affected by the virus has a positive test is 0.005.
\end{itemize}

A systematic screening test is carried out in a target population.
An individual is chosen at random from this 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}

The test is considered reliable when the probability that an individual with a positive test is affected by the virus is greater than 0.95.

\section*{Part A: Study of an example}
\begin{enumerate}
  \item Give the probabilities $P _ { M } ( T )$ and $P _ { \bar { M } } ( T )$.
``In March 2005, the epidemic spread rapidly on the island of Réunion, with a major surge between late April and early June followed by persistence of viral transmission during the austral winter. In total, 270,000 people were infected out of a total population of 750,000 individuals''.
(https ://www.pasteur.fr/fr/centre-medical/fiches-maladies/chikungunya)
At the end of 2005, the laboratory conducted a mass screening test of the population of the island of Réunion. In this part, the target population is therefore the population of the island of Réunion.
  \item Give the exact value of $P ( M )$.
  \item Copy and complete the weighted tree given below.
\includegraphics[max width=\textwidth, alt={}, center]{57113a15-a80d-4e49-a2e2-b5c752aabe7c-3_542_615_324_276}
  \item Calculate the probability that an individual is affected by the virus and has a positive test.
  \item Calculate the probability that an individual has a positive test.
  \item Calculate the probability that an individual with a positive test is affected by the virus.
  \item Can we estimate that this test is reliable? Argue.
\end{enumerate}

\section*{Part B: Screening on a target population}
In this part, we denote by $p$ the proportion of people affected by chikungunya virus in a target population.
We seek here to test the reliability of this laboratory's test as a function of $p$.

\begin{enumerate}
  \item Copy, adapting it, the weighted tree from question A3 taking into account the new data.
  \item Express the probability $P ( T )$ as a function of $p$.
  \item Show that $P _ { T } ( M ) = \frac { 999 p } { 994 p + 5 }$.
  \item For which values of $p$ can we consider that this test is reliable?
\end{enumerate}

\section*{Part C: Study on a sample}
During the epidemic, it is admitted that the probability of being affected by chikungunya on the island of Réunion is 0.36.
We consider a sample of $n$ individuals chosen at random, assimilating this choice to a random draw with replacement. We denote by $X$ the random variable counting the number of infected individuals in this sample among the $n$ drawn.
It is admitted that $X$ follows a binomial distribution with parameters $n$ and $p = 0.36$.
Determine from how many individuals $n$ the probability of the event ``at least one of the $n$ inhabitants of this sample is affected by the virus'' is greater than 0.99. Explain the approach.