An online gaming company offers a new smartphone application called ``Heart Tickets!''. Each participant generates on their smartphone a ticket containing a $3 \times 3$ grid on which three hearts are placed randomly. The ticket is winning if the three hearts are positioned side by side on the same line, on the same column or on the same diagonal.

1. Justify that there are exactly 84 different ways to position the three hearts on a grid.
2. Show that the probability that a ticket is winning equals $\frac{2}{21}$.
3. When a player generates a ticket, the company deducts \euro{}1 from their bank account. If the ticket is winning, the company then gives the player \euro{}5. Is the game favorable to the player?
4. A player decides to generate 20 tickets on this application. We assume that the generations of tickets are independent of each other. a. Give the probability distribution of the random variable $X$ which counts the number of winning tickets among the 20 tickets generated. b. Calculate the probability, rounded to $10^{-3}$, of the event $(X = 5)$. c. Calculate the probability, rounded to $10^{-3}$, of the event $(X \geqslant 1)$ and interpret the result in the context of the exercise.

Consider the following betting game:
On a ticket with 60 available numbers, a bettor chooses from 6 to 10 numbers. Among the available numbers, only 6 will be drawn. The bettor will be awarded if the 6 drawn numbers are among the numbers chosen by him on the same ticket.
The table presents the price of each ticket, according to the quantity of numbers chosen.

\begin{tabular}{ c } Quantity of numbers

chosen on a ticket

& Ticket price (R\$) \hline 6 & 2.00 \hline 7 & 12.00 \hline 8 & 40.00 \hline 9 & 125.00 \hline 10 & 250.00 \hline \end{tabular}
Five bettors, each with R\$ 500.00 to bet, made the following choices:
Arthur: 250 tickets with 6 numbers chosen; Bruno: 41 tickets with 7 numbers chosen and 4 tickets with 6 numbers chosen; Caio: 12 tickets with 8 numbers chosen and 10 tickets with 6 numbers chosen; Douglas: 4 tickets with 9 numbers chosen; Eduardo: 2 tickets with 10 numbers chosen.
The two bettors with the highest probabilities of being awarded are
(A) Caio and Eduardo. (B) Arthur and Eduardo. (C) Bruno and Caio. (D) Arthur and Bruno. (E) Douglas and Eduardo.

A regular 100-sided polygon is inscribed in a circle. Suppose three of the 100 vertices are chosen at random, all such combinations being equally likely. Find the probability that the three chosen points form vertices of a right angled triangle.

(Probability and Statistics) There are 9 balls in a bag, each labeled with a natural number from 1 to 9. When 4 balls are randomly drawn simultaneously from the bag, what is the probability that the sum of the largest and smallest numbers on the drawn balls is at least 7 and at most 9? [3 points]
(1) $\frac{5}{9}$
(2) $\frac{1}{2}$
(3) $\frac{4}{9}$
(4) $\frac{7}{18}$
(5) $\frac{1}{3}$

There are 2 students each from Korea, China, and Japan. When these 6 students each randomly select and sit in one of 6 seats with assigned seat numbers as shown in the figure, what is the probability that the two students from the same country sit such that the difference in their seat numbers is 1 or 10? [4 points]

11

12

13

21

22

23

(1) $\frac { 1 } { 20 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 3 } { 20 }$
(4) $\frac { 1 } { 5 }$
(5) $\frac { 1 } { 4 }$

In a table tennis competition with 4 male table tennis players and 4 female table tennis players, when 2 people are randomly selected to form 4 teams, what is the probability that exactly 2 teams consist of 1 male and 1 female? [3 points]
(1) $\frac { 3 } { 7 }$
(2) $\frac { 18 } { 35 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 24 } { 35 }$
(5) $\frac { 27 } { 35 }$

In the following seating chart, 4 female students and 4 male students are randomly assigned to 8 seats excluding the seat at row 2, column 2, with one person per seat. Find the value of $70p$, where $p$ is the probability that at least 2 male students are seated adjacent to each other. (Two people are considered adjacent if they are next to each other in the same row or directly in front or behind each other in the same column.) [4 points]

Among all ordered pairs $( x , y , z )$ of non-negative integers satisfying the equation $x + y + z = 10$, one is randomly selected. Find the probability that the selected ordered pair $( x , y , z )$ satisfies $( x - y ) ( y - z ) ( z - x ) \neq 0$. If this probability is $\frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

A bag contains 3 white balls and 4 black balls. When drawing 4 balls simultaneously at random from the bag, what is the probability of drawing 2 white balls and 2 black balls? [3 points]
(1) $\frac { 2 } { 5 }$
(2) $\frac { 16 } { 35 }$
(3) $\frac { 18 } { 35 }$
(4) $\frac { 4 } { 7 }$
(5) $\frac { 22 } { 35 }$

A survey was conducted on the preferences for Subject A and Subject B among 16 students in a class. Each student who participated in the survey chose one of the two subjects. 9 students chose Subject A and 7 students chose Subject B. When 3 students are randomly selected from the 16 students who participated in the survey, what is the probability that at least one of the 3 selected students chose Subject B? [3 points]
(1) $\frac{3}{4}$
(2) $\frac{4}{5}$
(3) $\frac{17}{20}$
(4) $\frac{9}{10}$
(5) $\frac{19}{20}$

In a deck of cards, what is the probability that when drawing 10 cards, none of them are hearts?

17. Two schools A and B organize student teams to participate in a debate competition. School A recommends $3$ male students and $2$ female students, while school B recommends $3$ male students and $4$ female students. The students recommended by both schools participate in training. Since the students' levels are comparable after training, $3$ people are randomly selected from the male students and $3$ people are randomly selected from the female students to form a representative team.
(1) Find the probability that at least $1$ student from school A is selected for the representative team.
(2) Before a certain competition, $4$ people are randomly selected from the $6$ team members to participate. Let $X$ denote the number of male students participating, find the probability distribution and mathematical expectation of $X$.

Five cards numbered $1, 2, 3, 4, 5$ are shuffled and three are drawn in order. The probability that the number on the first card is greater than the number on the third card is
A. $\dfrac{1}{10}$
B. $\dfrac{1}{5}$
C. $\dfrac{3}{10}$
D. $\dfrac{2}{5}$

From 2 male students and 3 female students, 2 people are selected to participate in community service. The probability that both selected are female students is
A. 0.6
B. 0.5
C. 0.4
D. 0.3

6. In ancient Chinese classics, the ``Book of Changes'' uses ``hexagrams'' to describe the changes of all things. Each ``hexagram'' consists of 6 lines arranged from bottom to top, with lines divided into yang lines ``—'' and yin lines ``- -''. The figure shows a hexagram. If a hexagram is randomly selected from all hexagrams, the probability that it has exactly 3 yang lines is
A. $\frac { 5 } { 16 }$
B. $\frac { 11 } { 32 }$
C. $\frac { 21 } { 32 }$
D. $\frac { 11 } { 16 }$

6. In ancient Chinese classics, the ``Book of Changes'' uses ``hexagrams'' to describe the changes of all things. Each ``hexagram'' consists of 6 lines arranged from bottom to top, with lines divided into yang lines ``——'' and yin lines ``——'', as shown in the figure. If a hexagram is randomly selected from all hexagrams, the probability that it has exactly 3 yang lines is
A. $\frac { 5 } { 16 }$
B. $\frac { 11 } { 32 }$
C. $\frac { 21 } { 32 }$
D. $\frac { 11 } { 16 }$

Let $O$ be the center of square $A B C D$. If we randomly select 3 points from $O , A , B , C , D$, the probability that the 3 points are collinear is
A. $\frac { 1 } { 5 }$
B. $\frac { 2 } { 5 }$
C. $\frac { 1 } { 2 }$
D. $\frac { 4 } { 5 }$

If 4 vertices are randomly selected from the 8 vertices of a cube, the probability that these 4 points lie on the same plane is $\_\_\_\_$

From 5 classmates including A and B, 3 are randomly selected to participate in community service. The probability that both A and B are selected is $\_\_\_\_$ .

From 5 classmates including A and B, 3 are randomly selected to participate in community service work. The probability that both A and B are selected is $\_\_\_\_$.

Let $V$ be the set of vertices of a regular polygon with twenty sides. Three distinct vertices are chosen at random from $V$. Then, the probability that the chosen triplet are the vertices of a right angled triangle is
(A) $\frac{7}{19}$
(B) $\frac{3}{19}$
(C) $\frac{3}{38}$
(D) $\frac{1}{38}$.

An office has 8 officers including two who are twins. Two teams, Red and Blue, of 4 officers each are to be formed randomly. What is the probability that the twins would be together in the Red team?
(A) $\frac { 1 } { 6 }$
(B) $\frac { 3 } { 7 }$
(C) $\frac { 1 } { 4 }$
(D) $\frac { 3 } { 14 }$

A box has 13 distinct pairs of socks. Let $p _ { r }$ denote the probability of having at least one matching pair among a bunch of $r$ socks drawn at random from the box. If $r _ { 0 }$ is the maximum possible value of $r$ such that $p _ { r } < 1$, then the value of $p _ { r _ { 0 } }$ is
(A) $1 - \frac { 12 } { { } ^ { 26 } C _ { 12 } }$.
(B) $1 - \frac { 13 } { { } ^ { 26 } C _ { 13 } }$.
(C) $1 - \frac { 2 ^ { 13 } } { { } ^ { 26 } C _ { 13 } }$.
(D) $1 - \frac { 2 ^ { 12 } } { { } ^ { 26 } C _ { 12 } }$.

An urn contains 30 balls out of which one is special. If 6 of these balls are taken out at random, what is the probability that the special ball is chosen?
(A) $\frac { 1 } { 30 }$
(B) $\frac { 1 } { 6 }$
(C) $\frac { 1 } { 5 }$
(D) $\frac { 1 } { 15 }$

There are two bags, A and B. Bag A contains four white balls and one red ball, and bag B contains two white balls and three red balls. Two balls are taken simultaneously out of bag A, then two balls are taken simultaneously out of bag B.
(1) The probability that two white balls are taken out of A, and one white ball and one red ball are taken out of B is $\frac{\mathbf{J}}{\mathbf{J}}$.
(2) The probability that the four balls taken out consist of three white balls and one red ball is $\frac{\mathbf{M}}{\mathbf{M}}$.
(3) The probability that the four balls taken out all have the same color is $\square$ PQ
(4) The probability that of the four balls taken out, two or fewer are white balls is $\frac{\mathbf{RS}}{\mathbf{TU}}$.