\textbf{Exercise 4 -- Candidates who have not followed the specialisation course}

\textbf{Part A}

During an evening, a television channel broadcast a match. This channel then offered a programme analysing this match. We have the following information:
\begin{itemize}
  \item $56\%$ of viewers watched the match;
  \item one quarter of viewers who watched the match also watched the programme;
  \item $16.2\%$ of viewers watched the programme.
\end{itemize}
We randomly interview a viewer. We denote the events:
\begin{itemize}
  \item $M$: ``the viewer watched the match'';
  \item $E$: ``the viewer watched the programme''.
\end{itemize}
We denote by $x$ the probability that a viewer watched the programme given that they did not watch the match.

\begin{enumerate}
  \item Construct a probability tree illustrating the situation.
  \item Determine the probability of $M \cap E$.
  \item \begin{enumerate}
    \item[a.] Verify that $p ( E ) = 0.44 x + 0.14$.
    \item[b.] Deduce the value of $x$.
  \end{enumerate}
  \item The interviewed viewer did not watch the programme. What is the probability, rounded to $10 ^ { - 2 }$, that they watched the match?
\end{enumerate}

\textbf{Part B}

This institute decides to model the time spent, in hours, by a viewer watching television on the evening of the match, by a random variable $T$ following the normal distribution with mean $\mu = 1.5$ and standard deviation $\sigma = 0.5$.

\begin{enumerate}
  \item What is the probability, rounded to $10 ^ { - 3 }$, that a viewer spent between one hour and two hours watching television on the evening of the match?
  \item Determine the approximation to $10 ^ { - 2 }$ of the real number $t$ such that $P ( T \geqslant t ) = 0.066$. Interpret the result.
\end{enumerate}

\textbf{Part C}

The lifetime of an individual set-top box, expressed in years, is modelled by a random variable denoted $S$ which follows an exponential distribution with parameter $\lambda$ strictly positive. The probability density function of $S$ is the function $f$ defined on $[ 0 ; + \infty [$ by
$$f ( x ) = \lambda \mathrm { e } ^ { - \lambda x }$$
The polling institute has observed that one quarter of the set-top boxes have a lifetime between one and two years. The factory that manufactures the set-top boxes claims that their average lifetime is greater than three years. Is the factory's claim correct? The answer must be justified.