During a training session, a volleyball player practises serving. The probability that he succeeds on the first serve is equal to 0.85.

We further assume that the following two conditions are satisfied:
\begin{itemize}
  \item if the player succeeds on a serve, then the probability that he succeeds on the next one is equal to 0.6;
  \item if the player fails a serve, then the probability that he fails the next one is equal to 0.6.
\end{itemize}

For any non-zero natural number $n$, we denote by $R _ { n }$ the event ``the player succeeds on the $n$-th serve'' and $\overline { R _ { n } }$ the complementary event.

\textbf{Part A}\\
We are interested in the first two serves of the training session.

\begin{enumerate}
  \item Represent the situation with a probability tree.
  \item Prove that the probability of event $R _ { 2 }$ is equal to 0.57.
  \item Given that the player succeeded on the second serve, calculate the probability that he failed the first one.
  \item Let $Z$ be the random variable equal to the number of successful serves during the first two serves.\\
a. Determine the probability distribution of $Z$ (you may use the probability tree from question 1).\\
b. Calculate the mathematical expectation $\mathrm { E } ( Z )$ of the random variable $Z$.
\end{enumerate}
Interpret this result in the context of the exercise.

\textbf{Part B}\\
We now consider the general case.\\
For any non-zero natural number $n$, we denote by $x _ { n }$ the probability of event $R _ { n }$.

\begin{enumerate}
  \item a. Give the conditional probabilities $P _ { R _ { n } } \left( R _ { n + 1 } \right)$ and $P _ { \overline { R _ { n } } } \left( \overline { R _ { n + 1 } } \right)$.\\
b. Show that, for any non-zero natural number $n$, we have: $x _ { n + 1 } = 0.2 x _ { n } + 0.4$.
  \item Let the sequence $(u _ { n })$ be defined for any non-zero natural number $n$ by: $u _ { n } = x _ { n } - 0.5$.\\
a. Show that the sequence $(u _ { n })$ is a geometric sequence.\\
b. Determine the expression of $x _ { n }$ as a function of $n$. Deduce the limit of the sequence $\left( x _ { n } \right)$.\\
c. Interpret this limit in the context of the exercise.
\end{enumerate}