\section*{Exercise 1 \\ Common to all candidates}
6 POINTS

The three parts are independent. Probability results should be rounded to $10^{-3}$ near.

\section*{Part 1}
It is estimated that in 2013 the world population consists of 4.6 billion people aged 20 to 79 years and that $46.1\%$ of people aged 20 to 79 years live in rural areas and $53.9\%$ in urban areas.\\
In 2013, according to the International Diabetes Federation, $9.9\%$ of the world population aged 20 to 79 years living in urban areas suffers from diabetes and $6.4\%$ of the world population aged 20 to 79 years living in rural areas suffers from diabetes.\\
A person aged 20 to 79 years is randomly selected. We denote:\\
$R$ the event: ``the chosen person lives in a rural area'',\\
$D$ the event: ``the chosen person suffers from diabetes''.

\begin{enumerate}
  \item Translate this situation using a probability tree.
  \item a. Calculate the probability that the interviewed person is diabetic.\\
b. The chosen person is diabetic. What is the probability that they live in a rural area?
\end{enumerate}

\section*{Part 2}
A person is said to be hypoglycemic if their fasting blood glucose is less than $60 \mathrm{mg}.\mathrm{dL}^{-1}$ and they are hyperglycemic if their fasting blood glucose is greater than $110 \mathrm{mg}.\mathrm{dL}^{-1}$. Fasting blood glucose is considered ``normal'' if it is between $70 \mathrm{mg}.\mathrm{dL}^{-1}$ and $110 \mathrm{mg}.\mathrm{dL}^{-1}$. People with a blood glucose level between 60 and $70 \mathrm{mg}.\mathrm{rdL}^{-1}$ are not subject to special monitoring.\\
An adult is randomly chosen from this population. A study established that the probability that they are hyperglycemic is 0.052 to $10^{-3}$ near. In the following, we will assume that this probability is equal to 0.052.\\
We model the fasting blood glucose, expressed in $\mathrm{mg}.\mathrm{dL}^{-1}$, of an adult from a given population, by a random variable $X$ which follows a normal distribution with mean $\mu$ and standard deviation $\sigma$.

\begin{enumerate}
  \item What is the probability that the chosen person has ``normal'' fasting blood glucose?
  \item Determine the value of $\sigma$ rounded to the nearest tenth.
  \item In this question, we take $\sigma = 12$. Calculate the probability that the chosen person is hypoglycemic.
\end{enumerate}

\section*{Part 3}
In order to estimate the proportion, for the year 2013, of people diagnosed with diabetes in the French population aged 20 to 79 years, 10000 people are randomly interviewed.\\
In the sample studied, 716 people were diagnosed with diabetes.

\begin{enumerate}
  \item Using a confidence interval at the $95\%$ confidence level, estimate the proportion of people diagnosed with diabetes in the French population aged 20 to 79 years.
  \item What should be the minimum number of people to interview if we want to obtain a confidence interval with amplitude less than or equal to 0.01?
\end{enumerate}