A bag contains the following eight letters: A B C D E F G H (2 vowels and 6 consonants). A game consists of drawing simultaneously at random two letters from this bag. You win if the draw consists of one vowel and one consonant.
A player draws simultaneously two letters from the bag. a. Determine the number of possible draws. b. Determine the probability that the player wins this game.
Questions 2 and 3 of this exercise are independent.
For the rest of the exercise, we admit that the probability that the player wins is equal to $\frac{3}{7}$.
To play, the player must pay $k$ euros, where $k$ is a non-zero natural integer. If the player wins, he receives 10 euros, otherwise he receives nothing. We denote $G$ the random variable equal to the algebraic gain of a player (that is, the sum received minus the sum paid). a. Determine the probability distribution of $G$. b. What must be the maximum value of the sum paid at the start for the game to remain favourable to the player?
Ten players each play one game. The letters drawn are returned to the bag after each game. We denote $X$ the random variable equal to the number of winning players. a. Justify that $X$ follows a binomial distribution and give its parameters. b. Calculate the probability, rounded to $10^{-3}$, that there are exactly four winning players. c. Calculate $P(X \geqslant 5)$ by rounding to $10^{-3}$. Give an interpretation of the result obtained. d. Determine the smallest natural integer $n$ such that $P(X \leqslant n) \geqslant 0.9$.
A bag contains the following eight letters: A B C D E F G H (2 vowels and 6 consonants).\\
A game consists of drawing simultaneously at random two letters from this bag.\\
You win if the draw consists of one vowel and one consonant.
\begin{enumerate}
\item A player draws simultaneously two letters from the bag.\\
a. Determine the number of possible draws.\\
b. Determine the probability that the player wins this game.
\end{enumerate}
\section*{Questions 2 and 3 of this exercise are independent.}
For the rest of the exercise, we admit that the probability that the player wins is equal to $\frac{3}{7}$.\\
\begin{enumerate}
\setcounter{enumi}{1}
\item To play, the player must pay $k$ euros, where $k$ is a non-zero natural integer. If the player wins, he receives 10 euros, otherwise he receives nothing.\\
We denote $G$ the random variable equal to the algebraic gain of a player (that is, the sum received minus the sum paid).\\
a. Determine the probability distribution of $G$.\\
b. What must be the maximum value of the sum paid at the start for the game to remain favourable to the player?
\item Ten players each play one game. The letters drawn are returned to the bag after each game.\\
We denote $X$ the random variable equal to the number of winning players.\\
a. Justify that $X$ follows a binomial distribution and give its parameters.\\
b. Calculate the probability, rounded to $10^{-3}$, that there are exactly four winning players.\\
c. Calculate $P(X \geqslant 5)$ by rounding to $10^{-3}$. Give an interpretation of the result obtained.\\
d. Determine the smallest natural integer $n$ such that $P(X \leqslant n) \geqslant 0.9$.
\end{enumerate}