Exercise 1 — 5 points Theme: probability, sequences Parts A and B can be treated independently Part A Each day, an athlete must jump over a hurdle at the end of training. Based on the previous season, his coach estimates that
if the athlete clears the hurdle one day, then he will clear it in $90\%$ of cases the next day;
if the athlete does not clear the hurdle one day, then in $70\%$ of cases he will not clear it the next day either.
For every natural integer $n$, we denote:
$R_{n}$ the event: ``The athlete successfully clears the hurdle during the $n$-th session'',
$p_{n}$ the probability of event $R_{n}$. We consider that $p_{0} = 0.6$.
Let $n$ be a natural integer, copy the weighted tree below and complete the blanks.
Justify using the tree that, for every natural integer $n$, we have: $$p_{n+1} = 0.6 p_{n} + 0.3 .$$
Consider the sequence $(u_{n})$ defined, for every natural integer $n$, by $u_{n} = p_{n} - 0.75$. a. Prove that the sequence $(u_{n})$ is a geometric sequence and specify its common ratio and first term. b. Prove that, for every natural integer $n$: $$p_{n} = 0.75 - 0.15 \times 0.6^{n} .$$ c. Deduce that the sequence $(p_{n})$ is convergent and determine its limit $\ell$. d. Interpret the value of $\ell$ in the context of the exercise.
Part B After many training sessions, the coach now estimates that the athlete clears each hurdle with a probability of 0.75 and this independently of whether or not he cleared the previous hurdles. We denote $X$ the random variable that gives the number of hurdles cleared by the athlete at the end of a 400 metres hurdles race which has 10 hurdles.
Specify the nature and parameters of the probability distribution followed by $X$.
Determine, to $10^{-3}$ near, the probability that the athlete clears all 10 hurdles.
Calculate $p(X \geqslant 9)$, to $10^{-3}$ near.
\textbf{Exercise 1 — 5 points}\\
Theme: probability, sequences
\textbf{Parts A and B can be treated independently}
\textbf{Part A}
Each day, an athlete must jump over a hurdle at the end of training. Based on the previous season, his coach estimates that
\begin{itemize}
\item if the athlete clears the hurdle one day, then he will clear it in $90\%$ of cases the next day;
\item if the athlete does not clear the hurdle one day, then in $70\%$ of cases he will not clear it the next day either.
\end{itemize}
For every natural integer $n$, we denote:
\begin{itemize}
\item $R_{n}$ the event: ``The athlete successfully clears the hurdle during the $n$-th session'',
\item $p_{n}$ the probability of event $R_{n}$. We consider that $p_{0} = 0.6$.
\end{itemize}
\begin{enumerate}
\item Let $n$ be a natural integer, copy the weighted tree below and complete the blanks.
\item Justify using the tree that, for every natural integer $n$, we have:
$$p_{n+1} = 0.6 p_{n} + 0.3 .$$
\item Consider the sequence $(u_{n})$ defined, for every natural integer $n$, by $u_{n} = p_{n} - 0.75$.\\
a. Prove that the sequence $(u_{n})$ is a geometric sequence and specify its common ratio and first term.\\
b. Prove that, for every natural integer $n$:
$$p_{n} = 0.75 - 0.15 \times 0.6^{n} .$$
c. Deduce that the sequence $(p_{n})$ is convergent and determine its limit $\ell$.\\
d. Interpret the value of $\ell$ in the context of the exercise.
\end{enumerate}
\textbf{Part B}
After many training sessions, the coach now estimates that the athlete clears each hurdle with a probability of 0.75 and this independently of whether or not he cleared the previous hurdles.\\
We denote $X$ the random variable that gives the number of hurdles cleared by the athlete at the end of a 400 metres hurdles race which has 10 hurdles.
\begin{enumerate}
\item Specify the nature and parameters of the probability distribution followed by $X$.
\item Determine, to $10^{-3}$ near, the probability that the athlete clears all 10 hurdles.
\item Calculate $p(X \geqslant 9)$, to $10^{-3}$ near.
\end{enumerate}