Léa spends a good part of her days playing a video game and is interested in the chances of winning her next games.

She estimates that if she has just won a game, she wins the next one in $70\%$ of cases. But if she has just suffered a defeat, according to her, the probability that she wins the next one is 0.2. Furthermore, she thinks she has an equal chance of winning the first game as of losing it.

For all non-zero natural integer $n$, we define the following events:
\begin{itemize}
  \item $G _ { n }$: ``Léa wins the $n$-th game of the day'';
  \item $D _ { n }$: ``Léa loses the $n$-th game of the day''.
\end{itemize}

For all non-zero natural integer $n$, we denote $g _ { n }$ the probability of event $G _ { n }$. We have therefore $g _ { 1 } = 0.5$.

\begin{enumerate}
  \item What is the value of the conditional probability $p _ { G _ { 1 } } \left( D _ { 2 } \right)$?
  \item Copy and complete the probability tree below which models the situation for the first two games of the day.
  \item Calculate $g _ { 2 }$.
  \item Let $n$ be a non-zero natural integer.\\
a. Copy and complete the probability tree below which models the situation for the $n$-th and $(n+1)$-th games of the day.\\
b. Justify that for all non-zero natural integer $n$,
$$g _ { n + 1 } = 0.5 g _ { n } + 0.2 .$$
  \item For all non-zero natural integer $n$, we set $v _ { n } = g _ { n } - 0.4$.\\
a. Show that the sequence $( v _ { n } )$ is geometric. We will specify its first term and its common ratio.\\
b. Show that, for all non-zero natural integer $n$:
$$g _ { n } = 0.1 \times 0.5 ^ { n - 1 } + 0.4 .$$
  \item Study the variations of the sequence $( g _ { n } )$.
  \item Give, by justifying, the limit of the sequence $( g _ { n } )$. Interpret the result in the context of the problem.
  \item Determine, by calculation, the smallest integer $n$ such that $g _ { n } - 0.4 \leqslant 0.001$.
  \item Copy and complete lines 4, 5 and 6 of the following function, written in Python language, so that it returns the smallest rank from which the terms of the sequence $\left( g _ { n } \right)$ are all less than or equal to $0.4 + e$, where $e$ is a strictly positive real number.
\begin{verbatim}
def seuil(e) :
    g = 0.5
    n = 1
    while...:
        g = 0.5 * g + 0.2
        n = ...
    return (n)
\end{verbatim}
\end{enumerate}