\textbf{Exercise 1: Probability}

A company manufactures components for the automotive industry. These components are designed on three assembly lines numbered 1 to 3.
\begin{itemize}
  \item Half of the components are designed on line $\mathrm{n}^{\circ}1$;
  \item $30\%$ of the components are designed on line $\mathrm{n}^{\circ}2$;
  \item the remaining components are designed on line $\mathrm{n}^{\circ}3$.
\end{itemize}
At the end of the manufacturing process, it appears that $1\%$ of the parts from line $\mathrm{n}^{\circ}1$ have a defect, as do $0.5\%$ of the parts from line $\mathrm{n}^{\circ}2$ and $4\%$ of the parts from line $\mathrm{n}^{\circ}3$.\\
One of these components is randomly selected. We denote:
\begin{itemize}
  \item $C_1$ the event ``the component comes from line $\mathrm{n}^{\circ}1$'';
  \item $C_2$ the event ``the component comes from line $\mathrm{n}^{\circ}2$'';
  \item $C_3$ the event ``the component comes from line $\mathrm{n}^{\circ}3$'';
  \item $D$ the event ``the component is defective'' and $\bar{D}$ its complementary event.
\end{itemize}
Throughout the exercise, probability calculations will be given as exact decimal values or rounded to $10^{-4}$ if necessary.

\textbf{PART A}
\begin{enumerate}
  \item Represent this situation with a probability tree.
  \item Calculate the probability that the selected component comes from line $\mathrm{n}^{\circ}3$ and is defective.
  \item Show that the probability of event $D$ is $P(D) = 0.0145$.
  \item Calculate the probability that a defective component comes from line $\mathrm{n}^{\circ}3$.
\end{enumerate}

\textbf{PART B}

The company decides to package the produced components by forming batches of $n$ units. We denote $X$ the random variable which, to each batch of $n$ units, associates the number of defective components in this batch.\\
Given the company's production and packaging methods, we can consider that $X$ follows the binomial distribution with parameters $n$ and $p = 0.0145$.
\begin{enumerate}
  \item In this question, the batches contain 20 units. We set $n = 20$.\\
    a. Calculate the probability that a batch contains exactly three defective components.\\
    b. Calculate the probability that a batch contains no defective components. Deduce the probability that a batch contains at least one defective component.
  \item The company director wishes the probability of having no defective components in a batch of $n$ components to be greater than 0.85.\\
    He proposes to form batches of at most 11 components. Is he correct? Justify your answer.
\end{enumerate}

\textbf{PART C}

The manufacturing costs of the components of this company are 15 euros if they come from assembly line $\mathrm{n}^{\circ}1$, 12 euros if they come from assembly line $\mathrm{n}^{\circ}2$ and 9 euros if they come from assembly line $\mathrm{n}^{\circ}3$.\\
Calculate the average manufacturing cost of a component for this company.