Let $n$ be a non-zero natural integer.
In the context of a random experiment, we consider a sequence of events $A _ { n }$ and we denote $p _ { n }$ the probability of the event $A _ { n }$.
For parts $\mathbf { A }$ and $\mathbf { B }$ of the exercise, we consider that:

\begin{itemize}
  \item If the event $A _ { n }$ is realized then the event $A _ { n + 1 }$ is realized with probability 0.3.
  \item If the event $A _ { n }$ is not realized then the event $A _ { n + 1 }$ is realized with probability 0.7.
\end{itemize}

We assume that $p _ { 1 } = 1$.

\section*{Part A:}
\begin{enumerate}
  \item Copy and complete the probabilities on the branches of the probability tree below.
  \item Show that $p _ { 3 } = 0.58$.
  \item Calculate the conditional probability $P _ { A _ { 3 } } \left( A _ { 2 } \right)$, round the result to $10 ^ { - 2 }$ near.
\end{enumerate}

\section*{Part B:}
In this part, we study the sequence $( p _ { n } )$ with $n \geqslant 1$.

\begin{enumerate}
  \item Copy and complete the probabilities on the branches of the probability tree below.
  \item a. Show that, for all non-zero natural integer $n$: $p _ { n + 1 } = - 0.4 p _ { n } + 0.7$.
\end{enumerate}

We consider the sequence $( u _ { n } )$, defined for all non-zero natural integer $n$ by:
$u _ { n } = p _ { n } - 0.5$.\\
b. Show that $( u _ { n } )$ is a geometric sequence for which we will specify the common ratio and the first term.\\
c. Deduce the expression of $u _ { n }$, then of $p _ { n }$ as a function of $n$.\\
d. Determine the limit of the sequence $\left( p _ { n } \right)$.

\section*{Part C:}
Let $x \in ] 0 ; 1 [$, we assume that $P _ { \overline { A _ { n } } } \left( A _ { n + 1 } \right) = P _ { A _ { n } } \left( \overline { A _ { n + 1 } } \right) = x$. We recall that $p _ { 1 } = 1$.

\begin{enumerate}
  \item Show that for all non-zero natural integer $n$: $p _ { n + 1 } = ( 1 - 2 x ) p _ { n } + x$.
  \item Prove by induction on $n$ that, for all non-zero natural integer $n$:
$$p _ { n } = \frac { 1 } { 2 } ( 1 - 2 x ) ^ { n - 1 } + \frac { 1 } { 2 }$$
  \item Show that the sequence $\left( p _ { n } \right)$ is convergent and give its limit.
\end{enumerate}