A company specializes in the sale of tiles.
Parts A, B and C are independent.
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:
- $2.25\%$ of the tiles are cracked;
- $6\%$ of the tiles with pattern are cracked.
A tile is randomly selected. We denote by $M$ the event ``the tile has a pattern'' and $F$ the event ``the tile is cracked''.
- Translate the situation using a probability tree.
- We know that the selected tile is cracked. Prove that the probability that it is a tile with pattern is $\frac{2}{3}$.
- Calculate $P_{\bar{M}}(F)$, the probability of $F$ given $\bar{M}$.
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.
- Prove that $P(X < 10.1) = 0.005$.
- 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.
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.
- Prove that the function $f$ is positive on $[0; 2\pi]$.
- 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 $\mathscr{A}$.