We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. We define the events:
\begin{itemize}
\item $A _ { n }$: "We obtain $p = q$".
\item $C _ { n }$: "We obtain $p > q$".
\end{itemize}
Calculate $\mathbf { P } \left( A _ { n } \right)$ then $\mathbf { P } \left( C _ { n } \right)$.