Exercise 1 (7 points)\\
Theme: probability\\
The coyote is a wild animal close to the wolf, which lives in North America.\\
In the state of Oklahoma, in the United States, $70\%$ of coyotes are affected by a disease called ehrlichiosis.\\
There is a test that helps detect this disease. When this test is applied to a coyote, its result is either positive or negative, and we know that:
\begin{itemize}
  \item If the coyote is sick, the test is positive in $97\%$ of cases.
  \item If the coyote is not sick, the test is negative in $95\%$ of cases.
\end{itemize}

\textbf{Part A}\\
Veterinarians capture a coyote from Oklahoma at random and perform a test for ehrlichiosis on it. We consider the following events:
\begin{itemize}
  \item $M$: ``the coyote is sick'';
  \item $T$: ``the coyote's test is positive''.
\end{itemize}
We denote $\bar{M}$ and $\bar{T}$ respectively the complementary events of $M$ and $T$.

\begin{enumerate}
  \item Copy and complete the probability tree below that models the situation.
  \item Determine the probability that the coyote is sick and that its test is positive.
  \item Prove that the probability of $T$ is equal to 0.694.
  \item The ``positive predictive value of the test'' is called the probability that the coyote is actually sick given that its test is positive. Calculate the positive predictive value of the test. Round the result to the nearest thousandth.
  \item a. By analogy with the previous question, propose a definition of the ``negative predictive value of the test'' and calculate this value rounding to the nearest thousandth.\\
  b. Compare the positive and negative predictive values of the test, and interpret.
\end{enumerate}

\textbf{Part B}\\
Recall that the probability that a randomly captured coyote has a positive test is 0.694.

\begin{enumerate}
  \item When five coyotes are captured at random, this choice is treated as sampling with replacement. We denote $X$ the random variable that associates to a sample of five randomly captured coyotes the number of coyotes in this sample having a positive test.\\
  a. What is the probability distribution followed by $X$? Justify and specify its parameters.\\
  b. Calculate the probability that in a sample of five randomly captured coyotes, only one has a positive test. Round the result to the nearest hundredth.\\
  c. A veterinarian claims that there is more than a one in two chance that at least four out of five coyotes have a positive test: is this claim true? Justify your answer.
  \item To test medications, veterinarians need to have a coyote with a positive test. How many coyotes must they capture so that the probability that at least one of them has a positive test is greater than 0.99?
\end{enumerate}