\textbf{Exercise 4 — 7 points}\\
Theme: Probability\\
An urn contains white and black tokens all indistinguishable to the touch.\\
A game consists of drawing at random successively and with replacement two tokens from this urn.\\
The following game rule is established:
\begin{itemize}
  \item a player loses 9 euros if the two tokens drawn are white;
  \item a player loses 1 euro if the two tokens drawn are black;
  \item a player wins 5 euros if the two tokens drawn are of different colors.
\end{itemize}

\begin{enumerate}
  \item We consider that the urn contains 2 black tokens and 3 white tokens.
  \begin{enumerate}
    \item[a.] Model the situation using a probability tree.
    \item[b.] Calculate the probability of losing $9\,\text{\euro}$ in one game.
  \end{enumerate}

  \item We now consider that the urn contains 3 white tokens and at least two black tokens but we do not know the exact number of black tokens. We will call $N$ the number of black tokens.
  \begin{enumerate}
    \item[a.] Let $X$ be the random variable giving the gain of the game for one game. Determine the probability distribution of this random variable.
    \item[b.] Solve the inequality for real $x$:
$$-x^2 + 30x - 81 > 0$$
    \item[c.] Using the result of the previous question, determine the number of black tokens the urn must contain so that this game is favorable to the player.
    \item[d.] How many black tokens should the player request in order to obtain a maximum average gain?
  \end{enumerate}

  \item We observe 10 players who try their luck by playing one game of this game, independently of each other. We assume that 7 black tokens have been placed in the urn (with 3 white tokens). What is the probability of having at least 1 player winning 5 euros?
\end{enumerate}