A company specializes in the sale of tiles.

Parts A, B and C are independent.

\section*{Part A}
We assume in this part that the company sells batches of tiles containing $25\%$ of tiles with pattern and $75\%$ of white tiles.\\
During a quality control, it is observed that:
\begin{itemize}
  \item $2.25\%$ of the tiles are cracked;
  \item $6\%$ of the tiles with pattern are cracked.
\end{itemize}
A tile is randomly selected.\\
We denote by $M$ the event ``the tile has a pattern'' and $F$ the event ``the tile is cracked''.

\begin{enumerate}
  \item Translate the situation using a probability tree.
  \item We know that the selected tile is cracked. Prove that the probability that it is a tile with pattern is $\frac{2}{3}$.
  \item Calculate $P_{\bar{M}}(F)$, the probability of $F$ given $\bar{M}$.
\end{enumerate}

\section*{Part B}
We model the thickness in millimeters of a randomly selected tile by a random variable $X$ that follows a normal distribution with mean $\mu = 11$ and standard deviation $\sigma$.\\
A tile is marketable if its thickness measures between $10.1\text{ mm}$ and $11.9\text{ mm}$. We know that $99\%$ of the tiles are marketable.

\begin{enumerate}
  \item Prove that $P(X < 10.1) = 0.005$.
  \item We introduce the random variable $Z$ such that
$$Z = \frac{X - 11}{\sigma}.$$
a. Give the distribution followed by the random variable $Z$.\\
b. Prove that $P\left(Z \leqslant -\frac{0.9}{\sigma}\right) = 0.005$.\\
c. Deduce the value of $\sigma$ rounded to the nearest hundredth.
\end{enumerate}

\section*{Part C}
We consider the function $f$ defined on $[0; 2\pi]$ by
$$f(x) = -1.5\cos(x) + 1.5$$
We admit that the function $f$ is continuous on $[0; 2\pi]$.\\
We denote by $\mathscr{C}_1$ the representative curve of the function $f$ in an orthonormal coordinate system.

\begin{enumerate}
  \item Prove that the function $f$ is positive on $[0; 2\pi]$.
  \item In the figure above, the curve drawn in dashes, denoted $\mathscr{C}_2$, is the curve symmetric to $\mathscr{C}_1$ with respect to the $x$-axis.\\
The shape of a tile is that of the region bounded by the curves $\mathscr{C}_1$ and $\mathscr{C}_2$.\\
We denote by $\mathscr{A}$ its area, expressed in square units.\\
Calculate $\mathscr{A}$.
\end{enumerate}