A random variable $X$ follows a binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 3 } \right)$ and $\mathrm { V} ( 3 X ) = 40$. Find the value of $n$. [3 points]

When rolling a die three times, what is the probability that the number 4 appears exactly once? [3 points]
(1) $\frac { 25 } { 72 }$
(2) $\frac { 13 } { 36 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 7 } { 18 }$
(5) $\frac { 29 } { 72 }$

When rolling a die 3 times, what is the probability that the number 4 appears exactly once? [3 points]
(1) $\frac { 25 } { 72 }$
(2) $\frac { 13 } { 36 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 7 } { 18 }$
(5) $\frac { 29 } { 72 }$

When a coin is tossed 6 times, the probability that the number of heads is greater than the number of tails is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

When the random variable $X$ follows the binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 2 } \right)$ and satisfies $\mathrm { E } \left( X ^ { 2 } \right) = \mathrm { V } ( X ) + 25$, what is the value of $n$? [3 points]
(1) 10
(2) 12
(3) 14
(4) 16
(5) 18

A coin is tossed 7 times. What is the probability of satisfying the following conditions? [4 points] (가) Heads appears at least 3 times. (나) There is a case where heads appears consecutively.
(1) $\frac { 11 } { 16 }$
(2) $\frac { 23 } { 32 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 25 } { 32 }$
(5) $\frac { 13 } { 16 }$

The random variable $X$ follows the binomial distribution $\mathrm { B } ( 80 , p )$ and $\mathrm { E } ( X ) = 20$. Find the value of $\mathrm { V } ( X )$. [3 points]

The random variable $X$ follows a binomial distribution $\mathrm { B } ( 80 , p )$ and $\mathrm { E } ( X ) = 20$. Find the value of $\mathrm { V } ( X )$. [3 points]

A random variable $X$ follows a binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 3 } \right)$ and $\mathrm { V} ( 2 X ) = 40$. What is the value of $n$? [3 points]
(1) 30
(2) 35
(3) 40
(4) 45
(5) 50

Each member of a certain group uses mobile payment with probability $p$. The payment methods of each member are independent. Let $X$ be the number of people among 10 members of the group who use mobile payment. If $D(X) = 2.4$ and $P ( X = 4 ) < P ( X = 6 )$, then $p =$
A. 0.7
B. 0.6
C. 0.4
D. 0.3

During the COVID-19 pandemic prevention and control period, a supermarket opened online sales services and can complete 1200 orders per day. Due to a sharp increase in order volume, orders have accumulated. To solve this problem, many volunteers eagerly signed up to help with order fulfillment. It is known that the supermarket had 500 accumulated unfulfilled orders on a certain day, and the probability that the next day's new orders exceed 1600 is 0.05. Each volunteer can complete 50 orders per day. To ensure that the probability of completing accumulated orders and current day orders within two days is at least 0.95, the minimum number of volunteers needed is
A． 10 people
B． 18 people
C． 24 people
D． 32 people

Two people, A and B, practice table tennis. The winner of each ball scores 1 point, the loser scores 0 points. Let the probability that A wins each ball be $p$ $\left(\frac{1}{2} < p < 1\right)$, the probability that B wins be $q$, with $p + q = 1$. The outcome of each ball is independent. For a positive integer $k \geq 2$, let $p_k$ denote the probability that after $k$ balls, A has scored at least 2 more points than B, and let $q_k$ denote the probability that after $k$ balls, B has scored at least 2 more points than A.
(1) Find $p_3, p_4$ (expressed in terms of $p$).
(2) If $\frac{p_4 - p_3}{q_4 - q_3} = 4$, find $p$.
(3) Prove: For any positive integer $m$, $p_{2m+1} - q_{2m+1} < p_{2m} - q_{2m} < p_{2m+2} - q_{2m+2}$.

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables, defined on a probability space $(\Omega, \mathcal{A}, P)$ and following the same Bernoulli distribution with parameter $p$.
What is the distribution of $S = X_1 + \ldots + X_n$? A proof of the stated result is expected.

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. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$.
Give the distribution of $N_1$, then the conditional distribution of $N_2$ given $(N_1 = i)$ for $i$ in a set to be specified. Are $N_1$ and $N_2$ independent?

Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$.
Let $r \geqslant 1$ be an integer and $S_r = N_1 + \cdots + N_r$. What does $S_r$ represent? Give its distribution (you may use the previous question).

In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$. We have shown that $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\left|\frac{S_{n}}{n}\right| \geqslant \varepsilon\right) \leqslant 2 \exp\left(-n \frac{\varepsilon^{2}}{2c^{2}}\right)$.
Let $n$ be a non-zero natural number, $p$ an element of the interval $]0,1[$ and $Z$ a random variable following a binomial distribution with parameter $(n, p)$. Using the previous question, bound $\mathbb{P}\left(\left|\frac{Z}{n}-p\right| \geqslant \varepsilon\right)$ as a function of $n, p$ and $\varepsilon$.

We introduce a uniformly distributed random variable $Z : \Omega \rightarrow \{-1,1\}^{n}$. For $\omega \in \Omega$, we denote by $Z_{i}(\omega)$ the coordinates of $Z(\omega)$. Show that for all $A = (a_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_{n}(\{-1,1\})$, we have $$\forall i \in \{1, \ldots, n\}, \quad \mathbb{E}\left[\left|\sum_{j=1}^{n} a_{i,j} Z_{j}\right|\right] = \frac{1}{2^{n}} \sum_{k=0}^{n} \binom{n}{k} |n - 2k|,$$ where $\binom{n}{k}$ denotes the binomial coefficient. Deduce $$\mathbb{E}\left[g_{A}(Z)\right] = \frac{n}{2^{n}} \sum_{k=0}^{n} \binom{n}{k} |n - 2k|.$$

Let $E$ be a Euclidean space of dimension $n \geqslant 1$ equipped with an orthonormal basis $(e_{1}, \ldots, e_{n})$. Let $\varepsilon_{1}, \ldots, \varepsilon_{n} : \Omega \rightarrow \{-1, 1\}$ be Rademacher random variables that are independent of each other. We set $X = \sum_{i=1}^{n} \varepsilon_{i} e_{i}$. We assume that $C$ is a closed convex set of $E$ that meets $X(\Omega)$ in a single vector $u$. Show that $\frac{1}{4}d(X, u)^{2}$ follows a binomial distribution with parameters $n$ and $1/2$.

Let $E$ be a Euclidean space of dimension $n \geqslant 1$ equipped with an orthonormal basis $(e_{1}, \ldots, e_{n})$. Let $\varepsilon_{1}, \ldots, \varepsilon_{n} : \Omega \rightarrow \{-1, 1\}$ be Rademacher random variables that are independent of each other. We set $X = \sum_{i=1}^{n} \varepsilon_{i} e_{i}$. We assume that $C$ is a closed convex set of $E$ that meets $X(\Omega)$ in a single vector $u$. Show that $\frac{1}{4} d(X, u)^{2}$ follows a binomial distribution with parameters $n$ and $1/2$.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables (taking values in $\{1,-1\}$ each with probability $1/2$). For every $n \in \mathbb{N}^{*}$, we set $Y_{n} = \frac{1}{2}\left(X_{n}+1\right)$ and $Z_{n} = \sum_{j=1}^{n} Y_{j}$. Determine the distribution of the random variable $Y_{n}$ and that of the random variable $Z_{n}$.

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. Determine the distribution of $X$, its expectation and its variance.

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ With $a = \frac{p\ln(p)}{p-1}$. Let $V$ and $W$ be the Lambert functions defined in Part I. When it exists, express the largest natural integer $r$ satisfying condition (II.2) as a function of $p$, $\alpha$ and $a$ using one of the functions $V$ or $W$.

We consider two urns each containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously, in an equiprobable manner, $n$ balls from the first urn. We denote $Y$ the number of white balls obtained. We also draw, in an equiprobable manner, $n$ balls from the second urn, but successively and with replacement. We denote $Z$ the number of white balls obtained. What is the distribution of the variable $Z$? Give the expectation and variance of $Z$.

Exercise V

A six-sided die is rolled five times. Check TRUE if the proposed random variable follows a binomial distribution and FALSE otherwise. V-A- The random variable corresponding to the number of rolls where an even number appears. V-B- The random variable corresponding to the sum of the results of all rolls.

What is the distribution followed by the random variable $A _ { n }$ representing the number of edges of a graph of $\Omega _ { n }$?