In parts A and B of this exercise, we consider a disease; every individual has an equal probability of 0.15 of being affected by this disease.

\section*{Part A}
This part is a multiple choice questionnaire (M.C.Q.). For each question, only one of the four answers is correct. A correct answer earns one point, an incorrect answer or no answer earns or deducts no points.\\
A screening test for this disease has been developed. If the individual is sick, in 94\% of cases the test is positive. For an individual chosen at random from this population, the probability that the test is positive is 0.158.

\begin{enumerate}
  \item An individual chosen at random from the population is tested: the test is positive. A value rounded to the nearest hundredth of the probability that the person is sick is equal to :\\
A: 0.94\\
B: 1\\
C: 0.89\\
D : we cannot know
  \item A random sample is taken from the population, and the test is administered to individuals in this sample. We want the probability that at least one individual tests positive to be greater than or equal to 0.99. The minimum sample size must be equal to :\\
A: 26 people\\
B: 27 people\\
C: 3 people\\
D: 7 people
  \item A vaccine to fight this disease has been developed. It is manufactured by a company in the form of a dose injectable by syringe. The volume $V$ (expressed in millilitres) of a dose follows a normal distribution with mean $\mu = 2$ and standard deviation $\sigma$. The probability that the volume of a dose, expressed in millilitres, is between 1.99 and 2.01 millilitres is equal to 0.997.\\
The value of $\sigma$ must satisfy :\\
A: $\sigma = 0.02$\\
B : $\sigma < 0.003$\\
C: $\sigma > 0.003$\\
D : $\sigma = 0.003$
\end{enumerate}

\section*{Part B}
\begin{enumerate}
  \item A box of a certain medicine can cure a sick person.
\end{enumerate}

The duration of effectiveness (expressed in months) of this medicine is modelled as follows:
\begin{itemize}
  \item during the first 12 months after manufacture, it is certain to remain effective;
  \item beyond that, its remaining duration of effectiveness follows an exponential distribution with parameter $\lambda$.
\end{itemize}

The probability that one of the boxes taken at random from a stock has a total duration of effectiveness greater than 18 months is equal to 0.887.\\
What is the average value of the total duration of effectiveness of this medicine?\\
2. A city of 100,000 inhabitants wants to build up a stock of these boxes in order to treat sick people.\\
What must be the minimum size of this stock so that the probability that it is sufficient to treat all sick people in this city is greater than 95\%?