Customs authorities are interested in imports of headphones bearing the logo of a certain brand. Customs seizures allow them to estimate that:
\begin{itemize}
  \item $20 \%$ of headphones bearing this brand's logo are counterfeits;
  \item $2 \%$ of non-counterfeit headphones have a design defect;
  \item $10 \%$ of counterfeit headphones have a design defect.
\end{itemize}
The fraud agency randomly orders a headphone displaying the brand's logo from an internet site. Consider the following events:
\begin{itemize}
  \item C: ``the headphone is counterfeit'';
  \item $D$: ``the headphone has a design defect'';
  \item $\bar { C }$ and $\bar { D }$ denote respectively the complementary events of $C$ and $D$.
\end{itemize}
Throughout the exercise, probabilities will be rounded to $10 ^ { - 3 }$ if necessary.

\section*{Part 1}
\begin{enumerate}
  \item Calculate $P ( C \cap D )$. You may use a probability tree.
  \item Prove that $P ( D ) = 0,036$.
  \item The headphone has a defect. What is the probability that it is counterfeit?
\end{enumerate}

\section*{Part 2}
We order $n$ headphones bearing this brand's logo. We treat this experiment as a random draw with replacement. Let $X$ be the random variable giving the number of headphones with a design defect in this batch.
\begin{enumerate}
  \item In this question, $n = 35$.\\
a. Justify that $X$ follows a binomial distribution $\mathscr { B } ( n , p )$ where $n = 35$ and $p = 0,036$.\\
b. Calculate the probability that among the ordered headphones, exactly one has a design defect.\\
c. Calculate $P ( X \leqslant 1 )$.
  \item In this question, $n$ is not fixed. What is the minimum number of headphones to order so that the probability that at least one headphone has a defect is greater than 0.99?
\end{enumerate}