This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, a multiple answer, or the absence of an answer to a question neither awards nor deducts points. The five questions are independent.

1. We consider the function $f$ defined on the interval $]1; +\infty[$ by $$f(x) = 0{,}05 - \frac{\ln x}{x-1}$$ The limit of the function $f$ at $+\infty$ is equal to: a. $+\infty$ b. 0.05 c. $-\infty$ d. 0
   
2. We consider a function $h$ continuous on the interval $[-2;4]$ such that: $$h(-1) = 0, \quad h(1) = 4, \quad h(3) = -1.$$ We can affirm that: a. the function $h$ is increasing on the interval $[-1; 1]$. b. the function $h$ is positive on the interval $[-1; 1]$. c. there exists at least one real number $a$ in the interval $[1; 3]$ such that $h(a) = 1$. d. the equation $h(x) = 1$ has exactly two solutions in the interval $[-2; 4]$.
   
3. We consider two sequences $(u_{n})$ and $(v_{n})$ with strictly positive terms such that $\lim_{n \rightarrow +\infty} u_{n} = +\infty$ and $(v_{n})$ converges to 0. We can affirm that: a. the sequence $\left(\dfrac{1}{v_{n}}\right)$ converges. b. the sequence $\left(\dfrac{v_{n}}{u_{n}}\right)$ converges. c. the sequence $(u_{n})$ is increasing. d. $\lim_{n \rightarrow +\infty} \left(-u_{n}\right)^{n} = -\infty$
   
4. To participate in a game, a player must pay $4\,€$. They then roll a fair six-sided die:
   • if they get 1, they win $12\,€$;
   • if they get an even number, they win $3\,€$;
   • otherwise, they win nothing.
   On average, the player: a. wins $3.50\,€$ b. loses $3\,€$. c. loses $1.50\,€$ d. loses $0.50\,€$.
   
5. We consider the random variable $X$ following the binomial distribution $\mathscr{B}(3; p)$. We know that $P(X = 0) = \dfrac{1}{125}$. We can affirm that: a. $p = \dfrac{1}{5}$ b. $P(X = 1) = \dfrac{124}{125}$ c. $p = \dfrac{4}{5}$ d. $P(X = 1) = \dfrac{4}{5}$

For each of the five questions in this exercise, only one of the four proposed answers is correct. The candidate will indicate on his/her work the number of the question and the chosen answer. No justification is required.
A wrong answer, multiple answers, or the absence of an answer to a question neither gives nor removes points.
An urn contains 15 indistinguishable balls to the touch, numbered from 1 to 15. The ball numbered 1 is red. The balls numbered 2 to 5 are blue. The other balls are green. We choose a ball at random from the urn. We denote $R$ (respectively $B$ and $V$) the event: ``The ball drawn is red'' (respectively blue and green).
Question 1: What is the probability that the ball drawn is blue or numbered with an even number?

Answer A	Answer B	Answer C	Answer D
$\frac { 7 } { 15 }$	$\frac { 9 } { 15 }$	$\frac { 11 } { 10 }$	None of the previous statements is correct.

Question 2: Given that the ball drawn is green, what is the probability that it is numbered 7?

Answer A	Answer B	Answer C	Answer D
$\frac { 1 } { 15 }$	$\frac { 7 } { 15 }$	$\frac { 1 } { 10 }$	None of the previous statements is correct.

A game is set up. To be able to play, the player pays the sum of 10 euros called the stake. This game consists of drawing a ball at random from the urn.

• If the ball drawn is blue, the player wins, in euros, three times the number of the ball.
• If the ball drawn is green, the player wins, in euros, the number of the ball.
• If the ball drawn is red, the player wins nothing.

We denote $G$ the random variable that gives the algebraic gain of the player, that is, the difference between what he wins and his initial stake. For example, if the player draws the blue ball numbered 3, then his algebraic gain is $-1$ euro.
Question 3: What is the value of $P ( G = 5 )$ ?

Answer A	Answer B	Answer C	Answer D
$\frac { 1 } { 15 }$	$\frac { 2 } { 15 }$	$\frac { 1 } { 3 }$	None of the previous statements is correct.

Question 4: What is the value of $P _ { R } ( G = 0 )$ ?

Answer A	Answer B	Answer C	Answer D
0	$\frac { 1 } { 15 }$	1	None of the previous statements is correct.

Question 5: What is the value of $P _ { ( G = - 4 ) } ( V )$ ?

Answer A	Answer B	Answer C	Answer D
$\frac { 1 } { 15 }$	$\frac { 4 } { 15 }$	$\frac { 1 } { 2 }$	None of the previous statements is correct.

Three cubic dice, with faces numbered from 1 to 6, were used in a game. Artur chose two dice, and João got the third. The game consists of both rolling their dice, observing the numbers on the faces facing up, and comparing the largest number obtained by Artur with the number obtained by João. The player who obtains the largest number wins. In case of a tie, the victory goes to João.
The player who has the greatest probability of victory is
(A) Artur, with probability of $\dfrac{2}{3}$
(B) João, with probability of $\dfrac{4}{9}$
(C) Artur, with probability of $\dfrac{91}{216}$
(D) João, with probability of $\dfrac{91}{216}$
(E) Artur, with probability of $\dfrac{125}{216}$

Ten people sitting around a circular table decide to donate some money for charity. You are told that the amount donated by each person was the average of the money donated by the two persons sitting adjacent to him/her. One person donated Rs. 500. Choose the correct option for each of the following two questions. Write your answers as a sequence of two letters (a/b/c/d).
What is the total amount donated by the 10 people?
(a) exactly Rs. 5000
(b) less than Rs. 5000
(c) more than Rs. 5000
(d) not possible to decide among the above three options.
What is the maximum amount donated by an individual?
(a) exactly Rs. 500
(b) less than Rs. 500
(c) more than Rs. 500
(d) not possible to decide among the above three options.

A discrete random variable $X$ can take values $0,1,2,3,4,5,6,7$ and its probability mass function is $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { l l } c , & x = 0,1,2 \\ 2 c , & x = 3,4,5 \\ 5 c ^ { 2 } , & x = 6,7 \end{array} \quad ( \text { where } c \text { is a positive number } ) \right.$$ Let $A$ be the event that the random variable $X$ is at least 6, and let $B$ be the event that the random variable $X$ is at least 3. What is the value of $\mathrm { P } ( A \mid B )$? [3 points]
(1) $\frac { 1 } { 5 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 7 }$
(4) $\frac { 1 } { 8 }$
(5) $\frac { 1 } { 9 }$

Let $0 < x < \frac { 1 } { 6 }$ be a real number. When a certain biased dice is rolled, a particular face $F$ occurs with probability $\frac { 1 } { 6 } - x$ and and its opposite face occurs with probability $\frac { 1 } { 6 } + x$; the other four faces occur with probability $\frac { 1 } { 6 }$. Recall that opposite faces sum to 7 in any dice. Assume that the probability of obtaining the sum 7 when two such dice are rolled is $\frac { 13 } { 96 }$. Then, the value of $x$ is:
(A) $\frac { 1 } { 8 }$
(B) $\frac { 1 } { 12 }$
(C) $\frac { 1 } { 24 }$
(D) $\frac { 1 } { 27 }$.

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 1 to participate in the lottery, what is the probability of paying a total of 300 yuan to obtain one action figure? (Single choice question, 2 points)
(1) $\left(\frac{2}{5}\right)^{2}$
(2) $\left(\frac{2}{5}\right)^{3}$
(3) $\left(\frac{3}{5}\right)^{2}$
(4) $\left(\frac{3}{5}\right)^{3}$
(5) $\left(\frac{2}{5}\right) \times \left(\frac{3}{5}\right)^{2}$