In tennis, the player who is serving can, in case of failure on the first serve, serve a second ball.\\
In match play, Abel succeeds with his first serve in $70\%$ of cases. When the first serve is successful, he wins the point in $80\%$ of cases.\\
On the other hand, after a failure on his first serve, Abel wins the point in $45\%$ of cases.\\
Abel is serving.\\
Consider the following events:
\begin{itemize}
  \item S: ``Abel succeeds with his first serve''
  \item G: ``Abel wins the point''.
\end{itemize}

\begin{enumerate}
  \item Describe the event $S$ then translate the situation with a probability tree.
  \item Calculate $P(S \cap G)$.
  \item Justify that the probability of event $G$ is equal to 0.695.
  \item Abel has won the point. What is the probability that he succeeded with his first serve?
  \item Are events $S$ and $G$ independent? Justify.
\end{enumerate}