During a professional examination, each candidate must present a file of type A or a file of type B; $60\%$ of candidates present a file of type A, the others presenting a file of type B.
The jury assigns to each file a mark between 0 and 20. A candidate passes if the mark assigned to their file is greater than or equal to 10.
A file is chosen at random.
It is admitted that the mark assigned to a file of type A can be modeled by a random variable $X$ following the normal distribution with mean 11.3 and standard deviation 3, and the mark assigned to a file of type B by a random variable $Y$ following the normal distribution with mean 12.4 and standard deviation 4.7.
We may denote $A$ the event: ``the file is a file of type A'', $B$ the event: ``the file is a file of type B'', and $R$ the event: ``the file is that of a candidate who passed the examination''.
Probabilities will be rounded to the nearest hundredth.

\begin{enumerate}
  \item The chosen file is of type A. What is the probability that this file is that of a candidate who passed the examination? It is admitted that the probability that the chosen file, given that it is of type B, is that of a candidate who passed is equal to 0.70.
  \item Show that the probability, rounded to the nearest hundredth, that the chosen file is that of a candidate who passed the examination is equal to 0.68.
  \item The jury examines 500 files chosen randomly from files of type B. Among these files, 368 are those of candidates who passed the examination.\\
A jury member claims that this sample is not representative. He justifies his claim by explaining that in this sample, the proportion of candidates who passed is too large.\\
What argument can be put forward to confirm or contest his claims?
  \item The jury awards a ``jury prize'' to files that obtained a mark greater than or equal to $N$, where $N$ is an integer. The probability that a file chosen at random obtains the ``jury prize'' is between 0.10 and 0.15.\\
Determine the integer $N$.
\end{enumerate}