\section*{Exercise 2 (5 points)}
An online platform offers two types of video games: a game of type $A$ and a game of type $B$.

\section*{Part A}
The durations of games of type $A$ and type $B$, expressed in minutes, can be modeled respectively by two random variables $X_A$ and $X_B$.\\
The random variable $X_A$ follows the uniform distribution on the interval $[9; 25]$.\\
The random variable $X_B$ follows the normal distribution with mean $\mu$ and standard deviation 3.

\begin{enumerate}
  \item a. Calculate the average duration of a game of type $A$.\\
  b. Specify using the graph the average duration of a game of type $B$.
  \item We choose at random, with equal probability, a game type. What is the probability that the duration of a game is less than 20 minutes? Give the result rounded to the nearest hundredth.
\end{enumerate}

\section*{Part B}
It is admitted that, as soon as the player completes a game, the platform proposes a new game according to the following model:
\begin{itemize}
  \item if the player completes a game of type $A$, the platform proposes to play again a game of type $A$ with probability 0.8;
  \item if the player completes a game of type $B$, the platform proposes to play again a game of type $B$ with probability 0.7.
\end{itemize}
For a natural number $n$ greater than or equal to 1, we denote $A_n$ and $B_n$ the events:\\
$A_n$: ``the $n$-th game is a game of type $A$.''\\
$B_n$: ``the $n$-th game is a game of type $B$.''\\
For any natural number $n$ greater than or equal to 1, we denote $a_n$ the probability of event $A_n$.

\begin{enumerate}
  \item a. Copy and complete the probability tree.\\
  b. Show that for any natural number $n \geqslant 1$, we have: $a_{n+1} = 0.5\,a_n + 0.3$.
\end{enumerate}

In the rest of the exercise, we denote $a$ the probability that the player plays game $A$ during his first game, where $a$ is a real number belonging to the interval $[0; 1]$. The sequence $(a_n)$ is therefore defined by: $a_1 = a$, and for any natural number $n \geqslant 1$, $a_{n+1} = 0.5\,a_n + 0.3$.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Study of a particular case. In this question, we assume that $a = 0.5$.\\
  a. Show by induction that for any natural number $n \geqslant 1$, we have: $0 \leqslant a_n \leqslant 0.6$.\\
  b. Show that the sequence $(a_n)$ is increasing.\\
  c. Show that the sequence $(a_n)$ is convergent and specify its limit.
  \item Study of the general case. In this question, the real number $a$ belongs to the interval $[0; 1]$.\\
  We consider the sequence $(u_n)$ defined for any natural number $n \geqslant 1$ by $u_n = a_n - 0.6$.\\
  a. Show that the sequence $(u_n)$ is a geometric sequence.\\
  b. Deduce that for any natural number $n \geqslant 1$, we have: $a_n = (a - 0.6) \times 0.5^{n-1} + 0.6$.\\
  c. Determine the limit of the sequence $(a_n)$. Does this limit depend on the value of $a$?\\
  d. The platform broadcasts an advertisement inserted at the beginning of games of type $A$ and another inserted at the beginning of games of type $B$. Which advertisement should be the most viewed by a player intensively playing video games?
\end{enumerate}