A marketing agency studied customer satisfaction regarding customer service when purchasing a television. These purchases were made either online, in an appliance store chain, or in a large supermarket. Online purchases represent $60 \%$ of sales, appliance store purchases $30 \%$ of sales, and large supermarket purchases $10 \%$ of sales. A survey shows that the proportion of customers satisfied with customer service is:

\begin{itemize}
  \item $75 \%$ for online customers;
  \item $90 \%$ for appliance store customers;
  \item $80 \%$ for large supermarket customers.
\end{itemize}

A customer who purchased the television model in question is chosen at random. The following events are defined:

\begin{itemize}
  \item I: ``the customer made their purchase online'';
  \item $M$: ``the customer made their purchase in an appliance store'';
  \item $G$: ``the customer made their purchase in a large supermarket'';
  \item S: ``the customer is satisfied with customer service''.
\end{itemize}

If $A$ is any event, we denote by $\bar { A }$ its complementary event and $P ( A )$ its probability.

\begin{enumerate}
  \item Reproduce and complete the tree diagram opposite.
  \item Calculate the probability that the customer made their purchase online and is satisfied with customer service.
  \item Prove that $P ( S ) = 0.8$.
  \item A customer is satisfied with customer service. What is the probability that they made their purchase online? Give the result rounded to $10 ^ { - 3 }$.
  \item To conduct the study, the agency must contact 30 customers each day among the television buyers. We assume that the number of customers is large enough to treat the choice of 30 customers as sampling with replacement. Let $X$ be the random variable that, for each sample of 30 customers, associates the number of customers satisfied with customer service.\\
a. Justify that $X$ follows a binomial distribution and specify its parameters.\\
b. Determine the probability, rounded to $10 ^ { - 3 }$, that at least 25 customers are satisfied in a sample of 30 customers contacted on the same day.
  \item By solving an inequality, determine the minimum sample size of customers to contact so that the probability that at least one of them is not satisfied is greater than $0.99$.
  \item In the two questions a. and b. that follow, we are only interested in online purchases.\\
When a television order is placed by a customer, the delivery time of the television is modeled by a random variable $T$ equal to the sum of two random variables $T _ { 1 }$ and $T _ { 2 }$.
\end{enumerate}

The random variable $T _ { 1 }$ models the integer number of days for the television to be transported from a storage warehouse to a distribution platform. The random variable $T _ { 2 }$ models the integer number of days for the television to be transported from this platform to the customer's home.

We admit that the random variables $T _ { 1 }$ and $T _ { 2 }$ are independent, and we are given:

\begin{itemize}
  \item The expectation $E \left( T _ { 1 } \right) = 4$ and the variance $V \left( T _ { 1 } \right) = 2$;
  \item The expectation $E \left( T _ { 2 } \right) = 3$ and the variance $V \left( T _ { 2 } \right) = 1$.\\
a. Determine the expectation $E ( T )$ and the variance $V ( T )$ of the random variable $T$.\\
b. A customer places a television order online. Justify that the probability that they receive their television between 5 and 9 days after their order is greater than or equal to $\frac { 2 } { 3 }$.
\end{itemize}