A game offered at a fairground consists of making three successive shots at a moving target.

It has been observed that:
\begin{itemize}
  \item If the player hits the target on one shot then they miss it on the next shot in $65\%$ of cases;
  \item If the player misses the target on one shot then they hit it on the next shot in $50\%$ of cases.
\end{itemize}

The probability that a player hits the target on their first shot is 0.6.\\
For any event $A$, we denote $p(A)$ its probability and $\bar{A}$ the complementary event of $A$.\\
We randomly choose a player for this shooting game.\\
We consider the following events:
\begin{itemize}
  \item $A_1$: ``The player hits the target on the $1^{\text{st}}$ shot''
  \item $A_2$: ``The player hits the target on the $2^{\mathrm{nd}}$ shot''
  \item $A_3$: ``The player hits the target on the $3^{\mathrm{rd}}$ shot''.
\end{itemize}

\textbf{Part A}

\begin{enumerate}
  \item Copy and complete, with the corresponding probabilities on each branch, the probability tree below modelling the situation.
\end{enumerate}

Let $X$ be the random variable that gives the number of times the player hits the target during the three shots.\\
2. Show that the probability that the player hits the target exactly twice during the three shots is equal to 0.4015.\\
3. The objective of this question is to calculate the expectation of the random variable $X$, denoted $E(X)$.\\
a. Copy and complete the table below giving the probability distribution of the random variable $X$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$X = x_i$ & 0 & 1 & 2 & 3 \\
\hline
$p\left(X = x_i\right)$ & 0.1 &  &  & 0.0735 \\
\hline
\end{tabular}
\end{center}
b. Calculate $E(X)$.\\
c. Interpret the previous result in the context of the exercise.

\textbf{Part B}

We consider $N$, a natural number greater than or equal to 1.\\
A group of $N$ people comes to this stand to play this game under identical and independent conditions.\\
A player is declared a winner when they hit the target three times.\\
We denote $Y$ the random variable that counts among the $N$ people the number of players declared winners.

\begin{enumerate}
  \item In this question, $N = 15$.\\
a. Justify that $Y$ follows a binomial distribution and determine its parameters.\\
b. Give the probability, rounded to $10^{-3}$, that exactly 5 players win this game.
  \item By the method of your choice, which you will explain, determine the minimum number of people who must come to this stand so that the probability that there is at least one winning player is greater than or equal to 0.98.
\end{enumerate}