In an effort to improve its sustainable development policy, a company conducted a statistical survey on its waste production.

In this survey, waste is classified into three categories:
\begin{itemize}
  \item $69 \%$ of waste is mineral and non-hazardous;
  \item $28 \%$ of waste is non-mineral and non-hazardous;
  \item the remaining waste is hazardous waste.
\end{itemize}

This statistical survey also tells us that:
\begin{itemize}
  \item $73 \%$ of mineral and non-hazardous waste is recyclable;
  \item $49 \%$ of non-mineral and non-hazardous waste is recyclable;
  \item $35 \%$ of hazardous waste is recyclable.
\end{itemize}

In this company, a piece of waste is randomly selected. We consider the following events:
\begin{itemize}
  \item $M$ : ``The selected waste is mineral and non-hazardous'';
  \item N : ``The selected waste is non-mineral and non-hazardous'';
  \item $D$ : ``The selected waste is hazardous'';
  \item R: ``The selected waste is recyclable''.
\end{itemize}
We denote by $\bar{R}$ the complementary event of event $R$.

\textbf{Part A}
\begin{enumerate}
  \item Copy and complete the probability tree below representing the situation described in the problem.
  \item Justify that the probability that the waste is hazardous and recyclable is equal to 0.0105.
  \item Determine the probability $P(M \cap \bar{R})$ and interpret the answer obtained in the context of the exercise.
  \item Prove that the probability of event $R$ is $P(R) = 0.6514$.
  \item Suppose that the selected waste is recyclable. Determine the probability that this waste is non-mineral and non-hazardous. Give the answer rounded to the ten-thousandth.
\end{enumerate}

\textbf{Part B}

We recall that the probability that a randomly selected piece of waste is recyclable is equal to 0.6514.

\begin{enumerate}
  \item In order to control the quality of collection in the company, a sample of 20 pieces of waste is randomly selected from production. We assume that the stock is sufficiently large to treat the sampling of this sample as drawing with replacement.

We denote by $X$ the random variable equal to the number of recyclable pieces of waste in this sample.\\
a. We admit that the random variable $X$ follows a binomial distribution. Specify its parameters.\\
b. Give the probability that the sample contains exactly 14 recyclable pieces of waste. Give the answer rounded to the ten-thousandth.
  \item In this question, we now select $n$ pieces of waste, where $n$ denotes a strictly positive natural number.\\
a. Give the expression as a function of $n$ of the probability $p_n$ that no piece of waste in this sample is recyclable.\\
b. Determine the value of the natural number $n$ from which the probability that at least one piece of waste in the sample is recyclable is greater than or equal to 0.9999.
\end{enumerate}