A company receives numerous emails daily.\\
Among these emails, $8\%$ are ``spam'', that is, emails with advertising or malicious intent, which it is desirable not to open.\\
An email received by the company is chosen at random.\\
The properties of the email software used in the company allow us to state that:

\begin{itemize}
  \item The probability that the chosen email is classified as ``undesirable'' given that it is spam is equal to 0.9.
  \item The probability that the chosen email is classified as ``undesirable'' given that it is not spam is equal to 0.01.
\end{itemize}

We denote:

\begin{itemize}
  \item S the event ``the chosen email is spam'';
  \item I the event ``the chosen email is classified as undesirable by the email software''.
  \item $\bar{S}$ and $\bar{I}$ the complementary events of $S$ and $I$ respectively.
\end{itemize}

\begin{enumerate}
  \item Model the situation studied using a probability tree, on which the probabilities associated with each branch should appear.
  \item a. Prove that the probability that the chosen email is a spam message and is classified as undesirable is equal to 0.072.\\
b. Calculate the probability that the chosen message is classified as undesirable.\\
c. The chosen message is classified as undesirable. What is the probability that it is actually a spam message? Give the answer rounded to the nearest hundredth.
  \item A random sample of 50 emails is chosen from those received by the company. It is assumed that this choice amounts to a random draw with replacement of 50 emails from the set of all emails received by the company.\\
Let $Z$ be the random variable counting the number of spam emails among the 50 chosen.\\
a. What is the probability distribution followed by the random variable $Z$, and what are its parameters?\\
b. What is the probability that, among the 50 chosen emails, at least two are spam? Give the answer rounded to the nearest hundredth.
\end{enumerate}