In the orthonormal coordinate system (O; I, J), we have represented:
\begin{itemize}
  \item the line with equation $y = x$;
  \item the line with equation $y = 1$;
  \item the line with equation $x = 1$;
  \item the parabola with equation $y = x ^ { 2 }$.
\end{itemize}
We can thus divide the square OIKJ into three zones.

\textbf{Part A}\\
Prove the results shown in the table below.

\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
ZONE & ZONE 1 & ZONE 2 & ZONE 3 \\
\hline
AREA & $\frac { 1 } { 2 }$ & $\frac { 1 } { 3 }$ & $\frac { 1 } { 6 }$ \\
\hline
\end{tabular}
\end{center}

\textbf{Part B: a first game}\\
A player throws a dart at the square above. It is admitted that the probability that it lands on a zone is equal to the area of that zone. Thus, the probability that the dart lands on ZONE 3 is equal to $\frac { 1 } { 6 }$.
\begin{itemize}
  \item If the dart lands on ZONE 3, then the player tosses a fair coin. If the coin lands on HEADS, then the player wins, otherwise he loses.
  \item If the dart lands on a zone other than ZONE 3, then the player rolls a fair six-sided die. If the die lands on FACE 6, then the player wins, otherwise he loses.
\end{itemize}
We note the following events:\\
$T$: ``the dart lands on ZONE 3'';\\
$G$: ``the player wins''.
\begin{enumerate}
  \item Represent the situation with a weighted tree.
  \item Prove that the probability of event $G$ is equal to $\frac { 2 } { 9 }$.
  \item Given that the player has won, what is the probability that the dart landed on ZONE 3?
\end{enumerate}

\textbf{Part C: a second game}\\
A player, called player $n^{\mathrm{o}}1$, throws a dart at the previous square. As in Part B, it is admitted that the probability that the dart lands on each of the zones is equal to the area of that zone.\\
The player wins a sum equal, in euros, to the number of the zone. For example, if the dart lands on ZONE 3, the player wins 3 euros.\\
We denote $X _ { 1 }$ the random variable giving the winnings of player $n^{\circ}1$.\\
We denote respectively $E \left( X _ { 1 } \right)$ and $V \left( X _ { 1 } \right)$ the expectation and variance of the random variable $X _ { 1 }$.
\begin{enumerate}
  \item a. Calculate $E \left( X _ { 1 } \right)$.\\
b. Show that $V \left( X _ { 1 } \right) = \frac { 5 } { 9 }$.
  \item A player $n^{\circ}2$ and a player $n^{\circ}3$ play in turn, under the same conditions as player $n^{\circ}1$. It is admitted that the games of these three players are independent of each other.\\
We denote $X _ { 2 }$ and $X _ { 3 }$ the random variables giving the winnings of players $n^{\circ}2$ and $n^{\circ}3$. We denote $Y$ the random variable defined by $Y = X _ { 1 } + X _ { 2 } + X _ { 3 }$.\\
a. Determine the probability that $Y = 9$.\\
b. Calculate $E ( Y )$.\\
c. Justify that $V ( Y ) = \frac { 5 } { 3 }$.
\end{enumerate}