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.

A test has been implemented: this allows to determine (even long after infection), whether or not a person has already been infected with COVID 19. If the test is positive, this means that the person has already been infected with COVID 19.

The sensitivity of a test is the probability that it is positive given that the person has been infected with the disease. The specificity of a test is the probability that the test is negative given that the person has not been infected with the disease.

The test manufacturer provides the following characteristics:
\begin{itemize}
  \item Its sensitivity is 0.8.
  \item Its specificity is 0.99.
\end{itemize}

An individual is drawn and subjected to the test from the adult French population on May 11, 2020. Let $T$ be the event ``the test performed is positive''.

\begin{enumerate}
  \item Complete the probability tree with the data from the statement.
  \item Show that $p(T) = 0.05503$.
  \item What is the probability that an individual has been infected given that their test is positive? Give an approximate value to $10^{-4}$ near of the result.
\end{enumerate}