With three identical valves $V_1, V_2$ and $V_3$, we manufacture a hydraulic circuit. The circuit is operational if $V_1$ is operational or if $V_2$ and $V_3$ are simultaneously operational.

We treat as a random experiment the fact that each valve is or is not operational after 6000 hours. We denote:
\begin{itemize}
  \item $F_1$ the event: ``valve $V_1$ is operational after 6000 hours''.
  \item $F_2$ the event: ``valve $V_2$ is operational after 6000 hours''.
  \item $F_3$ the event: ``valve $V_3$ is operational after 6000 hours''.
  \item $E$: the event: ``the circuit is operational after 6000 hours''.
\end{itemize}

We assume that the events $F_1, F_2$ and $F_3$ are pairwise independent and each have probability equal to 0.3.

\begin{enumerate}
  \item The probability tree shown represents part of the situation. Reproduce this tree and place the probabilities on the branches.
  \item Prove that $P(E) = 0.363$.
  \item Given that the circuit is operational after 6000 hours, calculate the probability that valve $V_1$ is operational at that time. Round to the nearest thousandth.
\end{enumerate}