An online gaming company offers a new smartphone application called ``Heart Tickets!''. Each participant generates on their smartphone a ticket containing a $3 \times 3$ grid on which three hearts are placed randomly. The ticket is winning if the three hearts are positioned side by side on the same line, on the same column or on the same diagonal.
Justify that there are exactly 84 different ways to position the three hearts on a grid.
Show that the probability that a ticket is winning equals $\frac{2}{21}$.
When a player generates a ticket, the company deducts \euro{}1 from their bank account. If the ticket is winning, the company then gives the player \euro{}5. Is the game favorable to the player?
A player decides to generate 20 tickets on this application. We assume that the generations of tickets are independent of each other. a. Give the probability distribution of the random variable $X$ which counts the number of winning tickets among the 20 tickets generated. b. Calculate the probability, rounded to $10^{-3}$, of the event $(X = 5)$. c. Calculate the probability, rounded to $10^{-3}$, of the event $(X \geqslant 1)$ and interpret the result in the context of the exercise.
An online gaming company offers a new smartphone application called ``Heart Tickets!''. Each participant generates on their smartphone a ticket containing a $3 \times 3$ grid on which three hearts are placed randomly. The ticket is winning if the three hearts are positioned side by side on the same line, on the same column or on the same diagonal.
\begin{enumerate}
\item Justify that there are exactly 84 different ways to position the three hearts on a grid.
\item Show that the probability that a ticket is winning equals $\frac{2}{21}$.
\item When a player generates a ticket, the company deducts \euro{}1 from their bank account. If the ticket is winning, the company then gives the player \euro{}5. Is the game favorable to the player?
\item A player decides to generate 20 tickets on this application. We assume that the generations of tickets are independent of each other.\\
a. Give the probability distribution of the random variable $X$ which counts the number of winning tickets among the 20 tickets generated.\\
b. Calculate the probability, rounded to $10^{-3}$, of the event $(X = 5)$.\\
c. Calculate the probability, rounded to $10^{-3}$, of the event $(X \geqslant 1)$ and interpret the result in the context of the exercise.
\end{enumerate}