We recall that $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ are $n ^ { 2 }$ real random variables that are mutually independent, all following the distribution $\mathcal { R }$, that $M _ { n } = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ is the random matrix taking values in $\mathcal { V } _ { n , n }$ and we denote $$C _ { 1 } = \left( \begin{array} { c } m _ { 11 } \\ \vdots \\ m _ { n 1 } \end{array} \right) , \ldots , C _ { n } = \left( \begin{array} { c } m _ { 1 n } \\ \vdots \\ m _ { n n } \end{array} \right)$$ the random variables taking values in $\mathcal { V } _ { n , 1 }$ constituted by the columns of the matrix $M _ { n }$.
For all $j \in \llbracket 1 , n - 1 \rrbracket$, we denote by $R _ { j }$ the event $$\left( C _ { 1 } , \ldots , C _ { j } \right) \text { is linearly independent and } C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right)$$ and $R _ { n }$ the event $$\left( C _ { 1 } , \ldots , C _ { n } \right) \text { is linearly independent.}$$
Show that $( R _ { 1 } , \ldots , R _ { n } )$ is a complete system of events.

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$. In the case where $n \geqslant 2$, define the events $$A _ { 1 } = \left\{ \left| R _ { 1 } \right| \geqslant 3 x \right\} \quad \text { and } \quad A _ { p } = \left\{ \max _ { 1 \leqslant i \leqslant p - 1 } \left| R _ { i } \right| < 3 x \right\} \cap \left\{ \left| R _ { p } \right| \geqslant 3 x \right\}$$ for $p \in \llbracket 2 , n \rrbracket$.
Express the event $A$ using the events $A _ { 1 } , A _ { 2 } , \ldots , A _ { n }$.

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:

• $E _ { n }$: "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 $E _ { n }$.