An apricot producer buys his trees from three different providers : 35\% of the trees come from the tree nursery $T_1$, 25\% from the tree nursery $T_2$ and the rest from the tree nursery $T_3$.

Each provider has two qualities of apricot trees : high quality or medium quality. The delivery from the tree nursery $T_1$ has 80\% of high quality trees, the one from $T_2$ has 50\% of high quality trees and the one from $T_3$ only 30\%.

The producer chooses a tree randomly within all deliveries. We consider the following events :

\begin{itemize}
    \item $\event{T}_1$ : "the tree comes from the tree nursery $T_1$"
    \item $\event{T}_2$ : "the tree comes from the tree nursery $T_2$"
    \item $\event{T}_3$ : "the tree comes from the tree nursery $T_3$"
    \item $\event{H}$ : "it's a high quality tree"
    \item $\event{M}$ : "it's a medium quality tree"
\end{itemize}

\begin{enumerate}
\setcounter{enumi}{2}
    \item \begin{enumerate}
        \item \nbPointsEnumii{2}\textbf{Construct} a tree diagram that models the situation.

        \item \nbPointsEnumii{2}\textbf{Calculate} the probability that the tree is a high quality one coming from $T_3$.

        \item \nbPointsEnumii{2}\textbf{Show} that the probability of picking a high quality tree $P(\event{H})$ is $0.525$.

        \item \nbPointsEnumii{2}The tree is a high quality one, \textbf{calculate} the probability it comes from $T_1$. Give your answer correct to 3~d.p.
    \end{enumerate}
\end{enumerate}

A random sample of 10~apricots is selected within the stock of the apricot producer. We assume the stock is large enough so that this selection can be considered as drawing apricots with replacement.

We also assume that the probability for an apricot to be of high quality is the same as for a tree to be a high quality one.

Let $X$ be the random variable counting the number of high quality apricots.

\begin{enumerate}
\setcounter{enumi}{3}
    \item \begin{enumerate}
        \item \nbPointsEnumii{1}\textbf{Give} the parameters of the binomial distribution followed by $X$.
        
        \item \nbPointsEnumii{2}\textbf{Calculate} the probability that exactly 5~apricots are high quality ones. Give your answer correct to 3~d.p.
        
        \item \nbPointsEnumii{2}\textbf{Calculate} the probability that at least 2~apricots are high quality ones. Give your answer correct to 3~d.p.
    \end{enumerate}
\end{enumerate}
