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 lottery game has a single-play winning probability of 0.1, and each play is an independent event. For each positive integer $n$, let $p_{n}$ be the probability of winning at least once in $n$ plays of this game. Select the correct options.
(1) $p_{n+1} > p_{n}$
(2) $p_{3} = 0.3$
(3) $\langle p_{n} \rangle$ is an arithmetic sequence
(4) Playing this game two or more times, the probability of not winning on the first play and winning on the second play equals $p_{2} - p_{1}$
(5) When playing this game $n$ times with $n \geq 2$, the probability of winning at least 2 times equals $2p_{n}$

A store sells a popular action figure through a lottery. Each lottery draw is independent with a probability of winning of $\frac{2}{5}$. Participants can participate in the lottery using one of the following two methods.
Method 1: Pay 225 yuan to get two lottery chances. Stop drawing as soon as you win and receive one action figure. If you fail to win in both draws, you must pay an additional 75 yuan to receive one action figure.
Method 2: Unlimited number of lottery draws, paying 100 yuan per draw.
If using Method 2 to participate in the lottery until winning one action figure, express the expected value of the number of lottery draws needed using the definition of expected value and the $\sum$ notation, and find its value. (Non-multiple choice question, 4 points)