During a fair, a game organizer has, on one hand, a wheel with four white squares and eight red squares and, on the other hand, a bag containing five tokens bearing the numbers $1, 2, 3, 4$ and 5. The game consists of spinning the wheel, each square having equal probability of being obtained, then extracting one or two tokens from the bag according to the following rule:
if the square obtained by the wheel is white, then the player extracts one token from the bag;
if the square obtained by the wheel is red, then the player extracts successively and without replacement two tokens from the bag.
The player wins if the token(s) drawn all bear an odd number.
A player plays one game and we denote by $B$ the event ``the square obtained is white'', $R$ the event ``the square obtained is red'' and $G$ the event ``the player wins the game''. a. Give the value of the conditional probability $P _ { B } ( G )$. b. It is admitted that the probability of drawing successively and without replacement two odd tokens is equal to 0.3. Copy and complete the following probability tree.
a. Show that $P ( G ) = 0.4$. b. A player wins the game. What is the probability that he obtained a white square by spinning the wheel?
Are the events $B$ and $G$ independent? Justify.
The same player plays ten games. The tokens drawn are returned to the bag after each game. We denote by $X$ the random variable equal to the number of games won. a. Explain why $X$ follows a binomial distribution and specify its parameters. b. Calculate the probability, rounded to $10 ^ { - 3 }$, that the player wins exactly three games out of the ten games played. c. Calculate $P ( X \geqslant 4 )$ rounded to $10 ^ { - 3 }$. Give an interpretation of the result obtained.
A player plays $n$ games and we denote by $p _ { n }$ the probability of the event ``the player wins at least one game''. a. Show that $p _ { n } = 1 - 0.6 ^ { n }$. b. Determine the smallest value of the integer $n$ for which the probability of winning at least one game is greater than or equal to 0.99.
During a fair, a game organizer has, on one hand, a wheel with four white squares and eight red squares and, on the other hand, a bag containing five tokens bearing the numbers $1, 2, 3, 4$ and 5.\\
The game consists of spinning the wheel, each square having equal probability of being obtained, then extracting one or two tokens from the bag according to the following rule:
\begin{itemize}
\item if the square obtained by the wheel is white, then the player extracts one token from the bag;
\item if the square obtained by the wheel is red, then the player extracts successively and without replacement two tokens from the bag.
\end{itemize}
The player wins if the token(s) drawn all bear an odd number.
\begin{enumerate}
\item A player plays one game and we denote by $B$ the event ``the square obtained is white'', $R$ the event ``the square obtained is red'' and $G$ the event ``the player wins the game''.\\
a. Give the value of the conditional probability $P _ { B } ( G )$.\\
b. It is admitted that the probability of drawing successively and without replacement two odd tokens is equal to 0.3.\\
Copy and complete the following probability tree.
\item a. Show that $P ( G ) = 0.4$.\\
b. A player wins the game. What is the probability that he obtained a white square by spinning the wheel?
\item Are the events $B$ and $G$ independent? Justify.
\item The same player plays ten games. The tokens drawn are returned to the bag after each game.\\
We denote by $X$ the random variable equal to the number of games won.\\
a. Explain why $X$ follows a binomial distribution and specify its parameters.\\
b. Calculate the probability, rounded to $10 ^ { - 3 }$, that the player wins exactly three games out of the ten games played.\\
c. Calculate $P ( X \geqslant 4 )$ rounded to $10 ^ { - 3 }$. Give an interpretation of the result obtained.
\item A player plays $n$ games and we denote by $p _ { n }$ the probability of the event ``the player wins at least one game''.\\
a. Show that $p _ { n } = 1 - 0.6 ^ { n }$.\\
b. Determine the smallest value of the integer $n$ for which the probability of winning at least one game is greater than or equal to 0.99.
\end{enumerate}