A company manufactures chocolate tablets of 100 grams. The quality control department performs several types of control.

\section*{Part A Control before market release}
A chocolate tablet must weigh 100 grams with a tolerance of two grams more or less. It is therefore put on the market if its mass is between 98 and 102 grams. The mass (expressed in grams) of a chocolate tablet can be modelled by a random variable $X$ following the normal distribution with mean $\mu = 100$ and standard deviation $\sigma = 1$. The adjustment of the manufacturing chain machines allows us to modify the value of $\sigma$.
\begin{enumerate}
  \item Calculate the probability of the event $M$ : ``the tablet is put on the market''.
  \item We wish to modify the adjustment of the machines so that the probability of this event reaches 0.97. Determine the value of $\sigma$ so that the probability of the event ``the tablet is put on the market'' equals 0.97.
\end{enumerate}

\section*{Part B Control upon reception}
The department controls the quality of cocoa beans delivered by producers. One of the quality criteria is the moisture content which must be $7\%$. The bean is then said to be compliant. The company has three different suppliers: the first supplier provides half of the bean stock, the second $30\%$ and the last provides $20\%$ of the stock. For the first, $98\%$ of its production respects the moisture content; for the second, which is somewhat cheaper, $90\%$ of its production is compliant, and the third supplies $20\%$ of non-compliant beans. We randomly choose a bean from the received stock. We denote $F _ { i }$ the event ``the bean comes from supplier $i$'', for $i$ taking the values 1, 2 or 3, and $C$ the event ``the bean is compliant''.
\begin{enumerate}
  \item Determine the probability that the bean comes from supplier 1, given that it is compliant.
\end{enumerate}