A robot is positioned on a horizontal axis and moves several times by one meter on this axis, randomly to the right or to the left.\\
During the first movement, the probability that the robot moves to the right is equal to $\frac{1}{3}$.\\
If it moves to the right, the probability that the robot moves to the right again during the next movement is equal to $\frac{3}{4}$.\\
If it moves to the left, the probability that the robot moves to the left again during the next movement is equal to $\frac{1}{2}$.\\
For every natural integer $n \geqslant 1$, we denote:
\begin{itemize}
  \item $D_n$ the event: ``the robot moves to the right during the $n$-th movement'';
  \item $\overline{D_n}$ the complementary event of $D_n$;
  \item $p_n$ the probability of event $D_n$.
\end{itemize}
We therefore have $p_1 = \frac{1}{3}$.

\textbf{Part A: study of the special case where $n = 2$}\\
In this part, the robot performs two successive movements.
\begin{enumerate}
  \item Reproduce and complete the following weighted tree.
  \item Determine the probability that the robot moves to the right twice.
  \item Show that $p_2 = \frac{7}{12}$.
  \item The robot moved to the left during the second movement. What is the probability that it moved to the right during the first movement?
\end{enumerate}

\textbf{Part B: study of the sequence $(p_n)$.}\\
We wish to estimate the movement of the robot after a large number of steps.
\begin{enumerate}
  \item Prove that for every natural integer $n \geqslant 1$, we have:
$$p_{n+1} = \frac{1}{4} p_n + \frac{1}{2}.$$
You may use a tree to help.
  \item a. Show by induction that for every natural integer $n \geqslant 1$, we have:
$$p_n \leqslant p_{n+1} < \frac{2}{3}.$$
b. Is the sequence $(p_n)$ convergent? Justify.
  \item We consider the sequence $(u_n)$ defined for every natural integer $n \geqslant 1$, by $u_n = p_n - \frac{2}{3}$.\\
a. Show that the sequence $(u_n)$ is geometric and specify its first term and its common ratio.\\
b. Determine the limit of the sequence $(p_n)$ and interpret the result in the context of the exercise.
\end{enumerate}

\textbf{Part C}\\
In this part, we consider another robot that performs ten movements of one meter independent of each other, each movement to the right having a fixed probability equal to $\frac{3}{4}$.\\
What is the probability that it returns to its starting point after the ten movements? Round the result to $10^{-3}$ near.