Let $x \in \mathbb{R}$ such that $x > 1$. Let $X$ and $Y$ be two independent random variables each following a zeta probability distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \mathbb{P}(Y = n) = \frac{1}{\zeta(x) n^{x}}$$ Let $A$ be the event ``No prime number divides $X$ and $Y$ simultaneously''. For all $n \in \mathbb{N}^{*}$, denote by $C_{n}$ the event $$C_{n} = \bigcap_{k=1}^{n} \left((X \notin p_{k}\mathbb{N}^{*}) \cup (Y \notin p_{k}\mathbb{N}^{*})\right)$$ Express the event $A$ using the events $C_{n}$. Deduce that $$\mathbb{P}(A) = \frac{1}{\zeta(2x)}$$

Let $W$ be a random variable on $\mathbb{N}^{*}$ that follows the probability distribution $(\ell_k)_{k \in \mathbb{N}^*}$, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$.
We admit that for all $B \subseteq \mathbb{N}^*$, $\mathbb{P}(W \in B) = \lim_{n \to \infty} \mathbb{P}(W_n \in B)$, and that if $X$ and $Y$ are two random variables taking values in $\mathbb{N}^*$ with $\mathbb{P}(X \in a\mathbb{N}^*) = \mathbb{P}(Y \in a\mathbb{N}^*)$ for all $a \in \mathbb{N}^*$, then $X$ and $Y$ have the same distribution.
Specify the distribution of $W$. By considering $\ell_{1}$, what can we then conclude?

In this part, $d$ equals 1 and we simply denote $0_d = 0$. Moreover, $p$ is an element of $]0,1[$, $q = 1 - p$ and the distribution of $X$ is given by $$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$ We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned} & \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\ & \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n } \end{aligned}$$ For $x \in ]-1,1[$, give a simple expression for $G ( x )$.
Express $P ( R = + \infty )$ as a function of $| p - q |$.
Determine the distribution of $R$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
For $s > 1$ fixed, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}.$$
Let $Z$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ that follows the distribution $\mu_s$. Calculate $P(k \mid Z)$ for $k \in \mathbb{N}^*$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. For fixed $s > 1$, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}$$ Let $Z$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ that follows the distribution $\mu_s$. Calculate $\mathbf{P}(k \mid Z)$ for $k \in \mathbb{N}^*$.

For $N = 1$, among random variables with usual distributions, give without justification one example of a random variable satisfying (8) and two examples of random variables not satisfying (8), where (8) states $\mathbb{P}(|X_n| \leq K) = 1$ for some constant $K \geq 1$, with $\mathbb{E}[X_n] = 0$ and $\operatorname{Var}(X_n) \leq 1$.

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

A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is
(1) $\frac { 5 } { 6 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 5 } { 11 }$
(4) $\frac { 6 } { 11 }$

A department store is preparing many red envelopes for customers to draw during the Lunar New Year period, claiming that the activity will continue until all red envelopes are distributed. The drawing box contains 5 sticks, of which only 1 stick is marked ``Great Fortune'', and each stick has an equal chance of being drawn. Each customer draws one stick from the box, records it, puts it back, and draws again for the next round, drawing at most 3 times. When two consecutive draws result in ``Great Fortune'', the customer stops drawing and receives a red envelope. We can view whether each customer receives a red envelope as a Bernoulli trial. Let $X$ be the position of the first customer to receive a red envelope in the entire activity, and let $E(X)$ denote the expected value of the random variable $X$. Then $E(X) = $ . (Round to the nearest integer)