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 B}

A player wins 30 challenges. We denote $X$ the random variable corresponding to the number of rare objects the player obtains after winning 30 challenges. The successive draws are considered independent.

\begin{enumerate}
  \item Determine, by justifying, the probability distribution followed by the random variable $X$. Specify its parameters, as well as its expected value.
  \item Determine $P(X < 6)$. Round the result to the nearest thousandth.
  \item Determine the largest value of $k$ such that $P(X \geqslant k) \geqslant 0.5$. Interpret the result in the context of the exercise.
  \item The video game developers want to offer players the option to buy a ``gold ticket'' which allows them to draw $N$ objects. The probability of drawing a rare object remains $7\%$. The developers would like that by buying a gold ticket, the probability that a player obtains at least one rare object in these $N$ draws is greater than or equal to $0.95$.\\
Determine the minimum number of objects to draw to achieve this objective. Care should be taken to detail the approach used.
\end{enumerate}