In this exercise, the results of the probabilities requested will be, if necessary, rounded to the nearest thousandth.

Feline leukaemia is a disease affecting cats; it is caused by a virus. In a large veterinary centre, the proportion of cats carrying the disease is estimated at $40\%$. A screening test for the disease is carried out among the cats present in this veterinary centre. This test has the following characteristics.
\begin{itemize}
  \item When the cat carries the disease, its test is positive in $90\%$ of cases.
  \item When the cat does not carry the disease, its test is negative in $85\%$ of cases.
\end{itemize}

A cat is chosen at random from the veterinary centre and the following events are considered:
\begin{itemize}
  \item $M$: ``The cat carries the disease'';
  \item $T$: ``The cat's test is positive'';
  \item $\bar{M}$ and $\bar{T}$ denote the complementary events of events $M$ and $T$ respectively.
\end{itemize}

\begin{enumerate}
  \item a. Represent the situation with a probability tree.\\
b. Calculate the probability that the cat carries the disease and that its test is positive.\\
c. Show that the probability that the cat's test is positive is equal to 0.45.\\
d. A cat is chosen from among those whose test is positive. Calculate the probability that it carries the disease.
  \item A sample of 20 cats is chosen at random from the veterinary centre. It is assumed that this choice can be treated as sampling with replacement.
\end{enumerate}

Let $X$ be the random variable giving the number of cats presenting a positive test in the chosen sample.\\
a. Determine, by justifying, the distribution followed by the random variable $X$.\\
b. Calculate the probability that there are exactly 5 cats with a positive test in the sample.\\
c. Calculate the probability that there are at most 8 cats with a positive test in the sample.\\
d. Determine the expected value of the random variable $X$ and interpret the result in the context of the exercise.

3. In this question, a sample of $n$ cats is chosen from the centre, which is again treated as sampling with replacement. Let $p_n$ be the probability that there is at least one cat presenting a positive test in this sample.\\
a. Show that $p_n = 1 - 0{,}55^n$.\\
b. Describe the role of the program below written in Python language, in which the variable $n$ is a natural integer and the variable $P$ is a real number.
\begin{verbatim}
def seuil() :
    n = 0
    P = 0
    while P < 0,99 :
        n = n + 1
        P = 1 - 0,55**n
    return n
\end{verbatim}
c. Determine, by specifying the method used, the value returned by this program.