A garden centre sells young tree saplings that come from three horticulturists: $35 \%$ of the plants come from horticulturist $\mathrm { H } _ { 1 } , 25 \%$ from horticulturist $\mathrm { H } _ { 2 }$ and the rest from horticulturist $\mathrm { H } _ { 3 }$. Each horticulturist supplies two categories of trees: conifers and deciduous trees.\\
The delivery from horticulturist $\mathrm { H } _ { 1 }$ contains $80 \%$ conifers while that from horticulturist $\mathrm { H } _ { 2 }$ contains only $50 \%$ and that from horticulturist $\mathrm { H } _ { 3 }$ only $30 \%$.

We consider the following events:\\
$H _ { 1 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 1 }$'',\\
$H _ { 2 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 2 }$'',\\
$H _ { 3 }$ : ``the tree chosen was purchased from horticulturist $\mathrm { H } _ { 3 }$'',\\
$C$ : ``the tree chosen is a conifer'',\\
$F$ : ``the tree chosen is a deciduous tree''.

\begin{enumerate}
  \item The garden centre manager chooses a tree at random from his stock.
  \begin{enumerate}
    \item[a.] Construct a probability tree representing the situation.
    \item[b.] Calculate the probability that the tree chosen is a conifer purchased from horticulturist $\mathrm { H } _ { 3 }$.
    \item[c.] Justify that the probability of event $C$ is equal to 0.525.
    \item[d.] The tree chosen is a conifer. What is the probability that it was purchased from horticulturist $\mathrm { H } _ { 1 }$? Round to $10 ^ { - 3 }$.
  \end{enumerate}
  \item A random sample of 10 trees is chosen from the stock of this garden centre. We assume that this stock is large enough that this choice can be treated as sampling with replacement of 10 trees from the stock.\\
  Let $X$ be the random variable that gives the number of conifers in the chosen sample.
  \begin{enumerate}
    \item[a.] Justify that $X$ follows a binomial distribution and specify its parameters.
    \item[b.] What is the probability that the sample contains exactly 5 conifers? Round to $10 ^ { - 3 }$.
    \item[c.] What is the probability that this sample contains at least two deciduous trees? Round to $10 ^ { - 3 }$.
  \end{enumerate}
\end{enumerate}