Thomas owns an MP3 player on which he has stored several thousand musical pieces. The set of musical pieces he owns is divided into three distinct genres according to the following distribution: 30\% classical music, 45\% variety, the rest being jazz. Thomas used two encoding qualities to store his musical pieces: high quality encoding and standard encoding. We know that:
\begin{itemize}
  \item $\frac { 5 } { 6 }$ of the classical music pieces are encoded in high quality.
  \item $\frac { 5 } { 9 }$ of the variety pieces are encoded in standard quality.
\end{itemize}

We consider the following events:\\
$C$ : ``The piece heard is a classical music piece'';\\
$V :$ ``The piece heard is a variety piece'';\\
$J$ : ``The piece heard is a jazz piece'';\\
$H$ : ``The piece heard is encoded in high quality'';\\
$S$ : ``The piece heard is encoded in standard quality''.

\section*{Part 1}
Thomas decides to listen to a piece at random from all the pieces stored on his MP3 using the ``random play'' function. A probability tree may be helpful.

\begin{enumerate}
  \item What is the probability that it is a classical music piece encoded in high quality?
  \item We know that $P ( H ) = \frac { 13 } { 20 }$.\\
a. Are the events $C$ and $H$ independent?\\
b. Calculate $P ( J \cap H )$ and $P _ { J } ( H )$.
\end{enumerate}

\section*{Part 2}
During a long train journey, Thomas listens to, using the ``random play'' function of his MP3, 60 musical pieces.

\begin{enumerate}
  \item Determine the asymptotic fluctuation interval at the 95\% threshold of the proportion of classical music pieces in a sample of size 60.
  \item Thomas counted that he had listened to 12 classical music pieces during his journey. Can we think that the ``random play'' function of Thomas's MP3 player is defective?
\end{enumerate}

\section*{Part 3}
Consider the random variable $X$ which, to each song stored on the MP3 player, associates its duration expressed in seconds, and we establish that $X$ follows the normal distribution with mean 200 and standard deviation 20.

We listen to a musical piece at random.

\begin{enumerate}
  \item Give an approximate value to $10 ^ { - 3 }$ of $P ( 180 \leqslant X \leqslant 220 )$.
  \item Give an approximate value to $10 ^ { - 3 }$ of the probability that the piece heard lasts more than 4 minutes.
\end{enumerate}