A company packages white sugar from two farms $U$ and $V$ in 1 kg packets of different qualities. Extra fine sugar is packaged separately in packets bearing the label ``extra fine''. Throughout the exercise, results should be rounded, if necessary, to the nearest thousandth.

To calibrate the sugar according to the size of its crystals, it is passed through a series of three sieves. Sugar crystals with a size less than $0.2\,\mathrm{mm}$ are found in the sealed bottom container at the end of calibration. They will be packaged in packets bearing the label ``extra fine sugar''.

\begin{enumerate}
  \item A sugar crystal is randomly selected from farm U. The size of this crystal, expressed in millimeters, is modeled by the random variable $X_{\mathrm{U}}$ which follows the normal distribution with mean $\mu_{\mathrm{U}} = 0.58\,\mathrm{mm}$ and standard deviation $\sigma_{\mathrm{U}} = 0.21\,\mathrm{mm}$.\\
a. Calculate the probabilities of the following events: $X_{\mathrm{U}} < 0.2$ and $0.5 \leqslant X_{\mathrm{U}} < 0.8$.\\
b. 1800 grams of sugar from farm $U$ is passed through the series of sieves. Deduce from the previous question an estimate of the mass of sugar recovered in the sealed bottom container and an estimate of the mass of sugar recovered in sieve 2.
  \item A sugar crystal is randomly selected from farm V. The size of this crystal, expressed in millimeters, is modeled by the random variable $X_{\mathrm{V}}$ which follows the normal distribution with mean $\mu_{\mathrm{V}} = 0.65\,\mathrm{mm}$ and standard deviation $\sigma_{\mathrm{V}}$ to be determined. During the calibration of a large quantity of sugar crystals from farm V, it is observed that $40\%$ of these crystals end up in sieve 2. What is the value of the standard deviation $\sigma_{\mathrm{V}}$ of the random variable $X_{\mathrm{V}}$?
\end{enumerate}