\textbf{Part B}

This same entrepreneur decides to install anti-spam software. This software detects unwanted messages called spam (malicious messages, advertisements, etc.) and moves them to a file called the ``spam folder''. The manufacturer claims that $95\%$ of spam messages are moved. For his part, the entrepreneur knows that $60\%$ of the messages he receives are spam. After installing the software, he observes that $58.6\%$ of messages are moved to the spam folder. For a message chosen at random, we consider the following events:

\begin{itemize}
  \item $D$: ``the message is moved'';
  \item $S$: ``the message is spam''.
\end{itemize}

\begin{enumerate}
  \item Calculate $P ( S \cap D )$.
  \item A message that is not spam is chosen at random. Show that the probability that it is moved equals 0.04.
  \item A message that is not moved is chosen at random. What is the probability that this message is spam?
  \item For the software chosen by the company, the manufacturer estimates that $2.7\%$ of messages moved to the spam folder are reliable messages. In order to test the software's effectiveness, the secretariat takes the trouble to count the number of reliable messages among the moved messages. It finds 13 reliable messages among the 231 messages moved during one week.\\
Do these results call into question the manufacturer's claim?
\end{enumerate}