
\begin{enumerate}
\item On a :

\begin{center}
\begin{tikzpicture}%
[level 1/.style={sibling distance=40mm, distance=20mm},%
level 2/.style={sibling distance=10mm, distance=30 mm},%
level 3/.style={distance=10mm,->,>=stealth',draw},yscale=1.5
]
\node {$\Omega$} 
	child { node {$R_1$} 
				child { node {$R_2$} edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{4}{7}$}}
				child { node {$B_2$} edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{2}{7}$}}
				child { node {$N_2$} edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{1}{7}$}}	
			edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{k}{2k+3}$}	
		  }
	child { node {$B_1$} 
				child { node {$R_2$} edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{3}{7}$} }
				child { node {$B_2$}edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{3}{7}$} }
				child { node {$N_2$}edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{1}{7}$} }	
			edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{k+1}{2k+3}$}	
		  }
	child { node {$N_1$} 
				child { node {$R_2$} edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{3}{7}$} }
				child { node {$B_2$} edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{2}{7}$}}
				child { node {$N_2$}edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{2}{7}$} }
			edge from parent node[fill=pagecorrige!30,inner sep=2pt] {$\frac{2}{2k+3}$}		
		  };
\end{tikzpicture}
\end{center}

\item D'après la formule des probabilités totales :
\begin{align*}
\proba{D} & = \probacond{B_2}{R_1}+\probacond{N_2}{R_1}+\probacond{R_2}{B_1}+\probacond{N_2}{B_1}+\probacond{R_2}{N_1}+\probacond{B_2}{N_1}\\
& = 1- \left(\probacond{R_2}{R_1}+\probacond{B_2}{B_1}+\probacond{N_2}{N_1}\right)\\
& =1-\left(\frac{k}{2k+3}\times\frac{4}{7} + \frac{k+1}{2k+3}\times\frac{3}{7} + \frac{2}{2k+3}\times\frac{2}{7}\right)\\
& =1-\frac{4k+3k+3+4}{14k+21}\\
\Aboxed{\proba{D} & = \dfrac{k+2}{2k+3}}
\end{align*}
\end{enumerate}
