A general knowledge test consists of a multiple choice questionnaire (MCQ) with twenty questions. For each one, the subject proposes four possible answers, of which only one is correct. For each question, the candidate must necessarily choose a single answer. This person earns one point for each correct answer and loses no points if their answer is wrong.

We consider three candidates:
\begin{itemize}
  \item Anselme answers completely at random to each of the twenty questions. In other words, for each of the questions, the probability that he answers correctly is equal to $\frac { 1 } { 4 }$;
  \item Barbara is somewhat better prepared. We consider that for each of the twenty questions, the probability that she answers correctly is $\frac { 1 } { 2 }$;
  \item Camille does even better: for each of the questions, the probability that she answers correctly is $\frac { 2 } { 3 }$.
\end{itemize}

\begin{enumerate}
  \item We denote $X , Y$ and $Z$ the random variables equal to the scores respectively obtained by Anselme, Barbara and Camille.\\
a. What is the probability distribution followed by the random variable $X$? Justify.\\
b. Using a calculator, give the answer rounded to the nearest thousandth of the probability $P ( X \geqslant 10 )$. In the following, we will admit that $P ( Y \geqslant 10 ) \approx 0.588$ and $P ( Z \geqslant 10 ) \approx 0.962$.
  \item We randomly choose the copy of one of these three candidates.
\end{enumerate}

We denote $A , B , C$ and $M$ the events:
\begin{itemize}
  \item $A$: ``the chosen copy is Anselme's'';
  \item $B$: ``the chosen copy is Barbara's'';
  \item $C$: ``the chosen copy is Camille's'';
  \item $M$: ``the chosen copy obtains a score greater than or equal to 10''.
\end{itemize}

We observe, after correcting it, that the chosen copy obtains a score greater than or equal to 10 out of 20.

What is the probability that it is Barbara's copy?\\
Give the answer rounded to the nearest thousandth of this probability.