A test is developed to detect a disease in a country.\\
According to the health authorities of this country, $7\%$ of the inhabitants are infected with this disease.\\
Among infected individuals, $20\%$ test negative.\\
Among healthy individuals, $1\%$ test positive.\\
A person is chosen at random from the population.\\
We denote:
\begin{itemize}
  \item $M$ the event: ``the person is infected with the disease'';
  \item $T$ the event: ``the test is positive''.
\end{itemize}

\begin{enumerate}
  \item Construct a probability tree modelling the proposed situation.
  \item a. What is the probability that the person is infected with the disease and that their test is positive?\\
b. Show that the probability that their test is positive is 0.0653.
  \item It is known that the test of the chosen person is positive.\\
What is the probability that they are infected?\\
Give the result as an approximation to $10 ^ { - 2 }$ near.
  \item Ten people are chosen at random from the population. The size of the population of this country allows us to treat this sample as a draw with replacement.\\
Let $X$ be the random variable that counts the number of individuals with a positive test among the ten people.\\
a. What is the probability distribution followed by $X$? Specify its parameters.\\
b. Determine the probability that exactly two people have a positive test.\\
Give the result as an approximation to $10 ^ { - 2 }$ near.
  \item Determine the minimum number of people to test in this country so that the probability that at least one of these people has a positive test is greater than $99\%$.
\end{enumerate}