
Avant tout, construisons un arbre de probabilités schématisant notre problème :

\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[right] (a1) at (1.5,1) {A$_1$};
\node[right] (a1b) at (1.5,-1) {$\overline{\text{A}_1}$};
\draw (0,0) -- (1.5,1) node[midway,fill=pagecorrige!30] {\small 0,4};
\draw (0,0) -- (1.5,-1) node[midway,fill=pagecorrige!30] {\small 0,6};
\node[right] at (3.5,-0.5) {R$_1$};
\node[right] at (3.5,-1.5) {$\overline{\text{R}_1}$};
\draw (a1b.east) -- (3.5,-0.5) node[midway,fill=pagecorrige!30] {\small 0,2};
\draw (a1b.east) -- (3.5,-1.5) node[midway,fill=pagecorrige!30] {\small 0,8};
\node[right] (a2) at (3.5,1.5) {A$_2$};
\node[right] (a2b) at (3.5,0.5) {$\overline{\text{A}_2}$};
\draw (a1.east) -- (3.5,0.5) node[midway,fill=pagecorrige!30] {\small 0,7};
\draw (a1.east) -- (3.5,1.5) node[midway,fill=pagecorrige!30] {\small 0,3};
\node[right] at (5.5,1) {R$_2$};
\node[right] at (5.5,0) {$\overline{\text{R}_2}$};
\draw (a2b.east) -- (5.5,1) node[midway,fill=pagecorrige!30] {\small 0,2};
\draw (a2b.east) -- (5.5,0) node[midway,fill=pagecorrige!30] {\small 0,8};
\end{tikzpicture}
\end{center}

\begin{enumerate}
\item $\begin{aligned}[t]
\proba{R_1} & = \probacond{R_1}{\evcont{A_1}}
\times\proba{\evcont{A_1}} \\
& = 0,2\times 0,6 \\
\Aboxed{\proba{R_1} & = 0,12}
\end{aligned}$

\item D'après la formule des probabilités totales, 
\begin{align*}
\proba{R} & = \proba{R_1}+\proba{R_2}\\
& = 0,12+0,4\times0,7\times0,2\\
& = 0,12+0,056\\
\Aboxed{\proba{R} & = 0,176}
\end{align*}

\item Nous cherchons \probacond{R_1}{R}.
\begin{align*}
\probacond{R_1}{R} & = \dfrac{\proba{R_1\cap R}}{\proba{R}}
\intertext{Or, \proba{R_1\cap R}=\proba{R_1}.}
& = \dfrac{0,12}{0,176}\\
\Aboxed{\probacond{R_1}{R}&\approx0,682}
\end{align*}
\end{enumerate}
