The director of a school wishes to conduct a study among students who took the final examination to analyze how they think they performed on this exam. For this study, students are asked at the end of the exam to answer individually the question: ``Do you think you passed the exam?''.

Only the answers ``yes'' or ``no'' are possible, and it is observed that $91.7\%$ of the students surveyed answered ``yes''.
Following the publication of exam results, it is discovered that:
\begin{itemize}
  \item $65\%$ of students who failed answered ``no'';
  \item $98\%$ of students who passed answered ``yes''.
\end{itemize}
A student who took the exam is randomly selected. We denote by $R$ the event ``the student passed the exam'' and $Q$ the event ``the student answered ``yes'' to the question''. For any event $A$, we denote by $P(A)$ its probability and $\bar{A}$ its complementary event.

Throughout the exercise, probabilities are, if necessary, rounded to $10^{-3}$ near.

\begin{enumerate}
  \item Specify the values of the probabilities $P(Q)$ and $P_{\bar{R}}(\bar{Q})$.
  \item Let $x$ be the probability that the randomly selected student passed the exam.\\
a. Copy and complete the weighted tree below.\\
b. Show that $x = 0.9$.
  \item The student selected answered ``yes'' to the question. What is the probability that he passed the exam?
  \item The grade obtained by a randomly selected student is an integer between 0 and 20. It is assumed to be modeled by a random variable $N$ that follows the binomial distribution with parameters $(20; 0.615)$.

The director wishes to award a prize to students with the best results.

Starting from which grade should she award prizes so that $65\%$ of students are rewarded?
  \item Ten students are randomly selected.

The random variables $N_1, N_2, \ldots, N_{10}$ model the grade out of 20 obtained on the exam by each of them. It is admitted that these variables are independent and follow the same binomial distribution with parameters $(20; 0.615)$.\\
Let $S$ be the variable defined by $S = N_1 + N_2 + \cdots + N_{10}$.\\
Calculate the expectation $E(S)$ and the variance $V(S)$ of the random variable $S$.
  \item Consider the random variable $M = \frac{S}{10}$.\\
a. What does this random variable $M$ model in the context of the exercise?\\
b. Justify that $E(M) = 12.3$ and $V(M) = 0.47355$.\\
c. Using the Bienaymé-Chebyshev inequality, justify the statement below.\\
``The probability that the average grade of ten randomly selected students is strictly between 10.3 and 14.3 is at least $80\%$''.
\end{enumerate}