On a Saturday morning, four families arrive one after another at the entrance area of an amusement park. Each of the four families pays at one of six cashiers, whereby it is assumed that each cashier is chosen with equal probability. Describe in the context of the problem two events $A$ and $B$ whose probabilities can be calculated using the following terms: $P ( A ) = \frac { 6 \cdot 5 \cdot 4 \cdot 3 } { 6 ^ { 4 } } ; P ( B ) = \frac { 6 } { 6 ^ { 4 } }$

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 $E _ { n }$.