A video game rewards players who have won a challenge with a randomly drawn object. The drawn object can be ``common'' or ``rare''. Two types of objects, common or rare, are available: swords and shields.

The video game designers have planned that:
\begin{itemize}
  \item the probability of drawing a rare object is $7\%$;
  \item if a rare object is drawn, the probability that it is a sword is $80\%$;
  \item if a common object is drawn, the probability that it is a sword is $40\%$.
\end{itemize}

\textbf{Part A}

A player has just won a challenge and draws an object at random. We denote:
\begin{itemize}
  \item R the event ``the player draws a rare object'';
  \item $E$ the event ``the player draws a sword'';
  \item $\bar{R}$ and $\bar{E}$ the complementary events of events $R$ and $E$.
\end{itemize}

\begin{enumerate}
  \item Draw a probability tree modelling the situation, then calculate $P(R \cap E)$.
  \item Calculate the probability of drawing a sword.
  \item The player has drawn a sword. Determine the probability that it is a rare object. Round the result to the nearest thousandth.
\end{enumerate}