We seek to study the number of students knowing the meaning of the acronym URSSAF. For this, we survey them by proposing a multiple choice questionnaire. Each student must choose from three possible answers, denoted $A$, $B$ and $C$, the correct answer being $A$. We denote by $r$ the probability that a student knows the correct answer. Any student knowing the correct answer responds $A$, otherwise they respond at random (equiprobably).

\begin{enumerate}
  \item We survey a student at random. We denote:\\
$A$ the event ``the student responds $A$'',\\
$B$ the event ``the student responds $B$'',\\
$C$ the event ``the student responds $C$'',\\
$R$ the event ``the student knows the answer'',\\
$\bar{R}$ the complementary event of $R$.\\
a. Translate this situation using a probability tree.\\
b. Show that the probability of event $A$ is $P(A) = \frac{1}{3}(1 + 2r)$.\\
c. Express as a function of $r$ the probability that a person who chose $A$ knows the correct answer.
  \item To estimate $r$, we survey 400 people and denote by $X$ the random variable counting the number of correct answers. We will assume that surveying 400 students at random is equivalent to performing sampling with replacement of 400 students from the set of all students.\\
a. Give the distribution of $X$ and its parameters $n$ and $p$ as a function of $r$.\\
b. In an initial survey, we observe that 240 students respond $A$, out of 400 surveyed.\\
Give a 95\% confidence interval for the estimate of $p$.\\
Deduce a 95\% confidence interval for $r$.\\
c. In what follows, we assume that $r = 0.4$. Given the large number of students, we will consider that $X$ follows a normal distribution.\\
i. Give the parameters of this normal distribution.\\
ii. Give an approximate value of $P(X \leqslant 250)$ to $10^{-2}$ precision.
\end{enumerate}