A company manufactures spherical wooden balls using two production machines A and B. The company considers that a ball can be sold only when its diameter is between $0.9 \mathrm{~cm}$ and $1.1 \mathrm{~cm}$.

Parts A, B and C are independent.

\textbf{Part A}

A study of the operation of the machines made it possible to establish the following results:
\begin{itemize}
  \item $96\%$ of daily production is saleable.
  \item Machine A provides $60\%$ of daily production.
  \item The proportion of saleable balls among the production of machine A is $98\%$.
\end{itemize}

A ball is chosen at random from the production of a given day. The following events are defined:\\
$A$: ``the ball was manufactured by machine A'';\\
$B$: ``the ball was manufactured by machine B'';\\
$V$: ``the ball is saleable''.

\begin{enumerate}
  \item Determine the probability that the chosen ball is saleable and comes from machine A.
  \item Justify that $P(B \cap V) = 0.372$ and deduce the probability that the chosen ball is saleable given that it comes from machine B.
  \item A technician claims that $70\%$ of non-saleable balls come from machine B. Is he correct?
\end{enumerate}

\textbf{Part B}

\begin{enumerate}
  \item A statistical study leads to modelling the diameter of a ball randomly selected from the production of machine B by a random variable $X$ which follows a normal distribution with mean $\mu = 1$ and standard deviation $\sigma = 0.055$.\\
Verify that the probability that a ball produced by machine B is saleable is indeed that found in Part A, to the nearest hundredth.
  \item In the same way, the diameter of a ball randomly selected from the production of machine A is modelled using a random variable $Y$ which follows a normal distribution with mean $\mu = 1$ and standard deviation $\sigma'$, $\sigma'$ being a strictly positive real number.\\
Given that $P(0.9 \leqslant Y \leqslant 1.1) = 0.98$, determine an approximate value to the nearest thousandth of $\sigma'$.
\end{enumerate}

\textbf{Part C}

The saleable balls then pass through a machine that colours them randomly and with equal probability in white, black, blue, yellow or red. After being mixed, the balls are packaged in bags. The quantity produced is large enough that filling a bag can be treated as successive sampling with replacement of balls from daily production.

\begin{enumerate}
  \item In this question only, the bags are all composed of 40 balls.\\
a. A bag of balls is chosen at random. Determine the probability that the chosen bag contains exactly 10 black balls. Round the result to $10^{-3}$.\\
b. In a bag of 40 balls, 12 black balls were counted. Does this observation allow us to question the adjustment of the machine that colours the balls?
  \item If the company wishes the probability of obtaining at least one black ball in a bag to be greater than or equal to $99\%$, what is the minimum number of balls each bag must contain to achieve this objective?
\end{enumerate}