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:
$E _ { n }$: "We obtain $( p , q ) \in E _ { 1 } \cup E _ { 2 } \cup E _ { 3 }$".
$A _ { n }$: "We obtain $p = q$".
$B _ { n }$: "We obtain $q > p$ and $q$ is divisible by $p$".
$C _ { n }$: "We obtain $p > q$".
where $E_1 = \{(p,q)\in(\mathbf{N}^*)^2: p=q\}$, $E_2 = \{(p,q)\in(\mathbf{N}^*)^2: pq\}$. Justify that the set $\left\{ A _ { n } , B _ { n } , C _ { n } \right\}$ forms a partition of $E _ { n }$.
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 $E _ { n }$: "We obtain $( p , q ) \in E _ { 1 } \cup E _ { 2 } \cup E _ { 3 }$".
\item $A _ { n }$: "We obtain $p = q$".
\item $B _ { n }$: "We obtain $q > p$ and $q$ is divisible by $p$".
\item $C _ { n }$: "We obtain $p > q$".
\end{itemize}
where $E_1 = \{(p,q)\in(\mathbf{N}^*)^2: p=q\}$, $E_2 = \{(p,q)\in(\mathbf{N}^*)^2: p<q,\, p\mid q\}$, $E_3 = \{(p,q)\in(\mathbf{N}^*)^2: p>q\}$.
Justify that the set $\left\{ A _ { n } , B _ { n } , C _ { n } \right\}$ forms a partition of $E _ { n }$.