(Candidates who have not followed the specialisation course) A computer game of chance is set up as follows:
If the player wins a game, the probability that he wins the next game is $\frac{1}{4}$;
If the player loses a game, the probability that he loses the next game is $\frac{1}{2}$;
The probability of winning the first game is $\frac{1}{4}$.
For every non-zero natural number $n$, we denote by $G_{n}$ the event ``the $n^{\mathrm{th}}$ game is won'' and we denote by $p_{n}$ the probability of this event. We thus have $p_{1} = \frac{1}{4}$.
Show that $p_{2} = \frac{7}{16}$.
Show that, for every non-zero natural number $n$, $p_{n+1} = -\frac{1}{4} p_{n} + \frac{1}{2}$.
We thus obtain the first values of $p_{n}$:
$n$
1
2
3
4
5
6
7
$p_{n}$
0,25
0,4375
0,3906
0,4023
0,3994
0,4001
0,3999
What conjecture can be made?
We define, for every non-zero natural number $n$, the sequence $(u_{n})$ by $u_{n} = p_{n} - \frac{2}{5}$. a. Prove that the sequence $(u_{n})$ is a geometric sequence and specify its common ratio. b. Deduce that, for every non-zero natural number $n$, $p_{n} = \frac{2}{5} - \frac{3}{20}\left(-\frac{1}{4}\right)^{n-1}$. c. Does the sequence $(p_{n})$ converge? Interpret this result.
\textbf{(Candidates who have not followed the specialisation course)}
A computer game of chance is set up as follows:
\begin{itemize}
\item If the player wins a game, the probability that he wins the next game is $\frac{1}{4}$;
\item If the player loses a game, the probability that he loses the next game is $\frac{1}{2}$;
\item The probability of winning the first game is $\frac{1}{4}$.
\end{itemize}
For every non-zero natural number $n$, we denote by $G_{n}$ the event ``the $n^{\mathrm{th}}$ game is won'' and we denote by $p_{n}$ the probability of this event. We thus have $p_{1} = \frac{1}{4}$.
\begin{enumerate}
\item Show that $p_{2} = \frac{7}{16}$.
\item Show that, for every non-zero natural number $n$, $p_{n+1} = -\frac{1}{4} p_{n} + \frac{1}{2}$.
\item We thus obtain the first values of $p_{n}$:
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$n$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
$p_{n}$ & 0,25 & 0,4375 & 0,3906 & 0,4023 & 0,3994 & 0,4001 & 0,3999 \\
\hline
\end{tabular}
\end{center}
What conjecture can be made?
\item We define, for every non-zero natural number $n$, the sequence $(u_{n})$ by $u_{n} = p_{n} - \frac{2}{5}$.\\
a. Prove that the sequence $(u_{n})$ is a geometric sequence and specify its common ratio.\\
b. Deduce that, for every non-zero natural number $n$, $p_{n} = \frac{2}{5} - \frac{3}{20}\left(-\frac{1}{4}\right)^{n-1}$.\\
c. Does the sequence $(p_{n})$ converge? Interpret this result.
\end{enumerate}