There are four people of different heights. When they are arranged in a line, what is the probability that the third person from the front is shorter than the two people adjacent to him? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 3 } { 4 }$

When two dice are rolled simultaneously, what is the probability that the number on one die is a multiple of the number on the other die? [4 points]
(1) $\frac { 7 } { 18 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 11 } { 18 }$
(4) $\frac { 13 } { 18 }$
(5) $\frac { 5 } { 6 }$

For two events $A$ and $B$ in the sample space $S$, if they are mutually exclusive events, $A \cup B = S$, and $\mathrm { P } ( A ) = 2 \mathrm { P } ( B )$, what is the value of $\mathrm { P } ( A )$? [3 points]
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 1 } { 4 }$

There is a regular tetrahedron-shaped box with the numbers $1,1,1,2$ written one on each face. When this box is thrown, if the number on the bottom face is 1, region A in the figure on the right is colored, and if the number is 2, region B is colored. When the box is thrown repeatedly until both regions are colored, find the probability that the process is completed on the 3rd throw. If this probability is $\frac { q } { p }$, find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

There is a regular tetrahedron-shaped box with the numbers $1,1,1,2$ written one on each face. When this box is thrown, if the number on the bottom face is 1, color region A in the figure on the right; if the number is 2, color region B. Continue throwing this box until both regions are colored. Find the probability that the process is completed on the 3rd throw, expressed as $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

Among $3n$ cards labeled with the numbers $1,2,3 , \cdots , 3 n$ ($n$ is a natural number), two cards are randomly drawn, and the two numbers on them are denoted as $a , b$ ($a < b$) respectively. Let $\mathrm { P } _ { n }$ be the probability that $3 a < b$. The following is the process of finding $\lim _ { n \rightarrow \infty } \mathrm { P } _ { n }$. The number of ways to draw 2 cards from $3 n$ cards is ${ } _ { 3 n } \mathrm { C } _ { 2 }$. When $3 a < b$, we have $b \leqq 3 n$, so $1 \leqq a < n$. Therefore, if $a = k$, the number of cases of $b$ satisfying $3 a < b$ is (A), so $$\mathrm { P } _ { n } = \frac { \text { (B) } } { { } _ { 3 n } \mathrm { C } _ { 2 } } \text { . }$$ Therefore, $\lim _ { n \rightarrow \infty } \mathrm { P } _ { n } =$ (C).
What are the correct values for (A), (B), and (C)? [4 points]

	(A)	(B)	(C)
(1)	$3 ( n - k )$	$\frac { 3 } { 2 } n ( n - 1 )$	$\frac { 1 } { 3 }$
(2)	$3 ( n - k )$	$\frac { 3 } { 2 } n ( n - 1 )$	$\frac { 2 } { 3 }$
(3)	$3 ( n - k )$	$3 n ( n - 1 )$	$\frac { 2 } { 3 }$
(4)	$3 ( n - k + 1 )$	$3 n ( n - 1 )$	$\frac { 1 } { 3 }$
(5)	$3 ( n - k + 1 )$	$3 n ( n - 1 )$	$\frac { 2 } { 3 }$

From $3n$ cards labeled with the numbers $1,2,3 , \cdots , 3 n$ ($n$ is a natural number), two cards are randomly drawn. Let the two numbers on the cards be $a , b$ ($a < b$) respectively. Let $\mathrm { P } _ { n }$ be the probability that $3 a < b$. The following is the process of finding $\lim _ { n \rightarrow \infty } \mathrm { P } _ { n }$.
The number of ways to draw 2 cards from $3n$ cards is ${ } _ { 3 n } \mathrm { C } _ { 2 }$. For $3 a < b$, we have $b \leqq 3 n$, so $1 \leqq a < n$. Therefore, if $a = k$, the number of cases of $b$ satisfying $3 a < b$ is (A), so $$\mathrm { P } _ { n } = \frac { \text{(B)} } { { } _ { 3 n } \mathrm { C } _ { 2 } } \text{.}$$ Therefore, $\lim _ { n \rightarrow \infty } \mathrm { P } _ { n } =$ (C).
What are the correct values for (A), (B), and (C) in the above process? [4 points]
(1) (A) $3(n-k)$, (B) $\frac{3}{2}n(n-1)$, (C) $\frac{1}{3}$
(2) (A) $3(n-k)$, (B) $\frac{3}{2}n(n-1)$, (C) $\frac{2}{3}$
(3) (A) $3(n-k)$, (B) $3n(n-1)$, (C) $\frac{2}{3}$
(4) (A) $3(n-k+1)$, (B) $3n(n-1)$, (C) $\frac{1}{3}$
(5) (A) $3(n-k+1)$, (B) $3n(n-1)$, (C) $\frac{2}{3}$

When 3 coins are tossed simultaneously, let $A$ be the event that at most 1 coin shows heads, and let $B$ be the event that all 3 coins show the same face. Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $\mathrm { P } ( A ) = \frac { 1 } { 2 }$ ㄴ. $\mathrm { P } ( A \cap B ) = \frac { 1 } { 8 }$ ㄷ. Events $A$ and $B$ are independent of each other.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Six students $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }$ are to be randomly paired into 3 groups of 2. What is the probability that A and B are in the same group and C and D are in different groups? [4 points]
(1) $\frac { 1 } { 15 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 2 } { 15 }$
(4) $\frac { 1 } { 6 }$
(5) $\frac { 1 } { 5 }$

Bag A and Bag B each contain five marbles with the numbers $1,2,3,4,5$ written on them, one number per marble. Chulsu draws one marble from Bag A, and Younghee draws one marble from Bag B. They check the numbers on the two marbles and do not put them back. This process is repeated. What is the probability that the numbers on the two marbles drawn the first time are different, and the numbers on the two marbles drawn the second time are the same? [4 points]
(1) $\frac { 3 } { 20 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 3 } { 10 }$
(5) $\frac { 7 } { 20 }$

Pouches A and B each contain five marbles with the numbers $1,2,3,4,5$ written on them, one number per marble. Chulsu draws one marble from pouch A, and Younghee draws one marble from pouch B. They check the numbers on the two marbles and do not put them back. This trial is repeated. What is the probability that the numbers on the two marbles drawn the first time are different, and the numbers on the two marbles drawn the second time are the same? [4 points]
(1) $\frac { 3 } { 20 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 3 } { 10 }$
(5) $\frac { 7 } { 20 }$

In information theory, when an event $E$ occurs, the information content $I ( E )$ of event $E$ is defined as follows:
$$I ( E ) = - \log _ { 2 } \mathrm { P } ( E )$$
Which of the following statements in $\langle$Remarks$\rangle$ are correct? (Note: The probability $\mathrm { P } ( E )$ of event $E$ is positive, and the unit of information content is bits.) [4 points]
$\langle$Remarks$\rangle$ ㄱ. If event $E$ is rolling an odd number on a single die, then $I ( E ) = 1$. ㄴ. If two events $A$ and $B$ are independent and $\mathrm { P } ( A \cap B ) > 0$, then $I ( A \cap B ) = I ( A ) + I ( B )$. ㄷ. For two events $A$ and $B$ with $\mathrm { P } ( A ) > 0$ and $\mathrm { P } ( B ) > 0$, we have $2 I ( A \cup B ) \leqq I ( A ) + I ( B )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In information theory, when an event $E$ occurs, the information content $I ( E )$ of the event $E$ is defined as follows. $$I ( E ) = - \log _ { 2 } \mathrm { P } ( E )$$ Which of the following are correct? Select all that apply from . (Note: the probability that event $E$ occurs, $\mathrm { P } ( E )$, is positive, and the unit of information content is bits.) [4 points]
ㄱ. If event $E$ is rolling one die and getting an odd number, then $I ( E ) = 1$. ㄴ. If two events $A , B$ are independent and $\mathrm { P } ( A \cap B ) > 0$, then $I ( A \cap B ) = I ( A ) + I ( B )$. ㄷ. For two events $A , B$ with $\mathrm { P } ( A ) > 0 , \mathrm { P } ( B ) > 0$, we have $2 I ( A \cup B ) \leqq I ( A ) + I ( B )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Chulsu participated in a certain design competition. Participants receive scores in two categories, and the possible scores in each category are one of three types shown in the table. The probability that Chulsu receives score A in each category is $\frac { 1 } { 2 }$, the probability of receiving score B is $\frac { 1 } { 3 }$, and the probability of receiving score C is $\frac { 1 } { 6 }$. When the event of receiving an audience vote score and the event of receiving a judge score are independent, what is the probability that the sum of the two scores Chulsu receives is 70? [3 points]

Category	Score A	Score B	Score C
Audience Vote	40	30	20
Judge	50	40	30

(1) $\frac { 1 } { 3 }$
(2) $\frac { 11 } { 36 }$
(3) $\frac { 5 } { 18 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 2 } { 9 }$

Box A contains 3 red balls and 5 black balls, and box B is empty. When 2 balls are randomly drawn from box A, if a red ball appears, perform [Execution 1], and if no red ball appears, perform [Execution 2]. What is the probability that the number of red balls in box B is 1? [3 points] [Execution 1] Put the drawn balls into box B. [Execution 2] Put the drawn balls into box B, and then randomly draw 2 more balls from box A and put them into box B.
(1) $\frac { 1 } { 2 }$
(2) $\frac { 7 } { 12 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 3 } { 4 }$
(5) $\frac { 5 } { 6 }$

For two events $A , B$, $A ^ { C }$ and $B$ are mutually exclusive events, and $$\mathrm { P } ( A ) = 2 \mathrm { P } ( B ) = \frac { 3 } { 5 }$$ When this condition is satisfied, what is the value of $\mathrm { P } \left( A \cap B ^ { C } \right)$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 3 } { 20 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 3 } { 10 }$
(5) $\frac { 7 } { 20 }$

A survey of 320 students at a school regarding membership in the mathematics club found that 60\% of male students and 50\% of female students joined the mathematics club. Let $p _ { 1 }$ be the probability that a randomly selected student from those who joined the mathematics club is male, and let $p _ { 2 }$ be the probability that a randomly selected student from those who joined the mathematics club is female. When $p _ { 1 } = 2 p _ { 2 }$, what is the number of male students at this school? [4 points]
(1) 170
(2) 180
(3) 190
(4) 200
(5) 210

Two sets $$A = \{ 1,2,3,4,5 \} , B = \{ 2,4,6,8,10 \}$$ What is the value of $n ( A \cup B )$? [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For two events $A$ and $B$, $$\mathrm { P } ( A \cap B ) = \frac { 1 } { 8 } , \mathrm { P } \left( A \cap B ^ { C } \right) = \frac { 3 } { 16 }$$ What is the value of $\mathrm { P } ( A )$? (Here, $B ^ { C }$ is the complement of $B$.) [3 points]
(1) $\frac { 3 } { 16 }$
(2) $\frac { 7 } { 32 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 9 } { 32 }$
(5) $\frac { 5 } { 16 }$

The total number of students at a certain school is 360, and each student chose either experiential learning A or experiential learning B. Among the students at this school, those who chose experiential learning A are 90 male students and 70 female students. When one student is randomly selected from the students at this school who chose experiential learning B, the probability that this student is male is $\frac { 2 } { 5 }$. What is the number of female students at this school? [3 points]
(1) 180
(2) 185
(3) 190
(4) 195
(5) 200

The universal set is $U = \{ x \mid x$ is a natural number not exceeding 9 $\}$, and two subsets of $U$ are $$A = \{ 3,6,7 \} , B = \{ a - 4,8,9 \}$$ If $$A \cap B ^ { C } = \{ 6,7 \}$$ find the value of the natural number $a$. [3 points]

There are two bags A and B, each containing 4 cards with the numbers $1,2,3,4$ written on them. Person 甲 draws two cards from bag A, and person 乙 draws two cards from bag B, each randomly. The probability that the sum of the numbers on the two cards held by 甲 equals the sum of the numbers on the two cards held by 乙 is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

Two sets $A = \{ 2 , a + 1,5 \} , B = \{ 2,3 , b \}$ satisfy $A = B$. Find the value of $a + b$. (Here, $a$ and $b$ are real numbers.) [2 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8

For the universal set $U = \{ 1,2,3,4,5,6,7,8 \}$ and two subsets $$A = \{ 1,2,3 \} , \quad B = \{ 2,4,6,8 \}$$ Find the value of $n \left( A \cup B ^ { C } \right)$. [3 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]