A merchant sells two types of mattresses: SPRING mattresses and FOAM mattresses. We assume that each customer buys only one mattress.

We have the following information:
\begin{itemize}
  \item $20\%$ of customers buy a SPRING mattress. Among them, $90\%$ are satisfied with their purchase.
  \item $82\%$ of customers are satisfied with their purchase.
\end{itemize}

The two parts can be treated independently.

\textbf{Part A}

We randomly select a customer and note the events:
\begin{itemize}
  \item R: ``the customer buys a SPRING mattress'',
  \item S: ``the customer is satisfied with their purchase''.
\end{itemize}

We denote $x = P_{\bar{R}}(S)$, where $P_{\bar{R}}(S)$ denotes the probability of $S$ given that $R$ is not realized.

\begin{enumerate}
  \item Copy and complete the probability tree below describing the situation.
  \item Prove that $x = 0.8$.
  \item A customer satisfied with their purchase is selected. What is the probability that they bought a SPRING mattress? Round the result to $10^{-2}$.
\end{enumerate}

\textbf{Part B}

\begin{enumerate}
  \item We randomly select 5 customers. We consider the random variable $X$ which gives the number of customers satisfied with their purchase among these 5 customers.

  a. We admit that $X$ follows a binomial distribution. Give its parameters.

  b. Determine the probability that at most three customers are satisfied with their purchase. Round the result to $10^{-3}$.

  \item Let $n$ be a non-zero natural number. We now randomly select $n$ customers. This selection can be treated as a random draw with replacement.

  a. We denote $p_n$ the probability that all $n$ customers are satisfied with their purchase. Prove that $p_n = 0.82^n$.

  b. Determine the natural numbers $n$ such that $p_n < 0.01$. Interpret in the context of the exercise.
\end{enumerate}