Determine the probability that the fifth family is the first to rent a hand cart.

Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. We denote $q = 1-p$ and $m = n^2$.
Let $i$ and $j$ be in $\{1, \ldots, n\}$. The smallest integer $k \geqslant 1$ such that the coefficient at row $i$, column $j$ of $M_k$ equals 1 is denoted $T_{i,j}$. Give the distribution of $T_{i,j}$.

Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. We denote by $N$ the smallest index $k$ for which the matrix $M_k$ is completely filled.
a) Propose an approach to approximate the expectation of $N$ using a computer simulation with the functions above.
b) Give an expression for the exact value of this expectation involving $q$ and $m$.

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Show that for all $k \in \mathbb{N}^*$, the random variable $\nu_{p_k}(X) + 1$ follows the geometric distribution with parameter $\left(1 - p_k^{-s}\right)$.

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Show that, for $r \in \mathbb{N}^*$, $k_1 < \cdots < k_r$ in $\mathbb{N}^*$ and $\left(n_1, \ldots, n_r\right) \in \mathbb{N}^r$, we have $$\begin{aligned} & P\left(\nu_{p_{k_1}}(X) = n_1, \ldots, \nu_{p_{k_r}}(X) = n_r\right) = \\ & \sum_{\ell=0}^{r}(-1)^{\ell} \sum_{\substack{\left(\varepsilon_1, \ldots, \varepsilon_r\right) \in \{0,1\}^r \\ \varepsilon_1 + \cdots + \varepsilon_r = \ell}} P\left(\nu_{p_{k_1}}(X) \geqslant n_1 + \varepsilon_1, \nu_{p_{k_2}}(X) \geqslant n_2 + \varepsilon_2, \ldots, \nu_{p_{k_r}}(X) \geqslant n_r + \varepsilon_r\right). \end{aligned}$$

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Deduce that the random variables $\nu_{p_1}(X), \ldots, \nu_{p_k}(X), \ldots$ are mutually independent.

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Deduce that there exists a real $K > 0$ independent of $p$ such that
$$\mathbf { E } \left( ( X - \mathbf { E } ( X ) ) ^ { 4 } \right) \leq \frac { K q } { p ^ { 4 } }$$

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.
The probability that $X = 3$ equals
(A) $\frac { 25 } { 216 }$
(B) $\frac { 25 } { 36 }$
(C) $\frac { 5 } { 36 }$
(D) $\frac { 125 } { 216 }$

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.
The probability that $X \geq 3$ equals
(A) $\frac { 125 } { 216 }$
(B) $\frac { 25 } { 36 }$
(C) $\frac { 5 } { 36 }$
(D) $\frac { 25 } { 216 }$

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required.
The conditional probability that $X \geq 6$ given $X > 3$ equals
(A) $\frac { 125 } { 216 }$
(B) $\frac { 25 } { 216 }$
(C) $\frac { 5 } { 36 }$
(D) $\frac { 25 } { 36 }$

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $\mathrm { a } = \mathrm { P } ( \mathrm { X } = 3 ) , \mathrm { b } = \mathrm { P } ( \mathrm { X } \geq 3 )$ and $\mathrm { c } = \mathrm { P } ( \mathrm { X } \geq 6 \mid \mathrm { X } > 3 )$. Then $\frac { \mathrm { b } + \mathrm { c } } { \mathrm { a } }$ is equal to