A student named Chulsu participated in a 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 audience voting scores and the event of receiving judge scores are mutually 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 Voting	40	30	20
Judges	50	40	30

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

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) $\frac { 5 } { 6 }$

Bag A contains 2 white balls and 3 black balls, and Bag B contains 1 white ball and 3 black balls. One ball is randomly drawn from Bag A. If it is white, 2 white balls are put into Bag B; if it is black, 2 black balls are put into Bag B. Then one ball is randomly drawn from Bag B. What is the probability that the drawn ball is white? [4 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 7 } { 30 }$
(4) $\frac { 4 } { 15 }$
(5) $\frac { 3 } { 10 }$

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) $\frac { 7 } { 20 }$

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

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

Two sets $$A = \{ 3,5,7,9 \} , B = \{ 3,7 \}$$ For the sets above, when $A - B = \{ a , 9 \}$, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

A bag contains 7 marbles, each labeled with a natural number from 2 to 8. When 2 marbles are drawn simultaneously from the bag, what is the probability that the two natural numbers on the drawn marbles are coprime? [3 points]
(1) $\frac { 8 } { 21 }$
(2) $\frac { 10 } { 21 }$
(3) $\frac { 4 } { 7 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 16 } { 21 }$

A die is rolled three times, and the results are $a$, $b$, and $c$ in order. What is the probability that $a \times b \times c = 4$? [3 points]
(1) $\frac { 1 } { 54 }$
(2) $\frac { 1 } { 36 }$
(3) $\frac { 1 } { 27 }$
(4) $\frac { 5 } { 108 }$
(5) $\frac { 1 } { 18 }$

There are 5 cards with letters $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }$ written on them and 4 cards with numbers $1,2,3,4$ written on them. When all 9 cards are arranged in a line in random order using each card once, what is the probability that the card with letter A has number cards on both sides? [3 points]
(1) $\frac { 5 } { 12 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 6 }$
(5) $\frac { 1 } { 12 }$

A box contains 5 white masks and 9 black masks. When 3 masks are randomly drawn simultaneously from the box, what is the probability that at least one of the 3 masks is white? [3 points]
(1) $\frac { 8 } { 13 }$
(2) $\frac { 17 } { 26 }$
(3) $\frac { 9 } { 13 }$
(4) $\frac { 19 } { 26 }$
(5) $\frac { 10 } { 13 }$

Given sets $A = \{ 1,2,3 \} , B = \{ 1,3 \}$, then $\mathrm { A } \cap \mathrm { B } =$
(A) $\{ 2 \}$
(B) $\{ 1,2 \}$
(C) $\{ 1,3 \}$
(D) $\{ 1,2,3 \}$

1. Given sets $A = \{ 1,2,3 \} , B = \{ 2,4,5 \}$, then the number of elements in set $A \cup B$ is $\_\_\_\_$ .

Given sets $\mathrm { A } = \{ - 2 , - 1,0,2 \} , \mathrm { B } = \{ \mathrm { x } \mid ( \mathrm { x } - 1 ) ( \mathrm { x } + 2 ) < 0 \}$ , then $\mathrm { A } \cap \mathrm { B } =$
(A) $\{ - 1,0 \}$
(B) $\{ 0,1 \}$
(C) $\{ - 1,0,1 \}$
(D) $\{ 0,1,2 \}$

1. Let $M = \left\{ x \mid x ^ { 2 } = x \right\} , N = \{ x \mid \lg x \leq 0 \}$, then $M \bigcup N =$
A. $[ 0,1 ]$
B. $( 0,1 ]$
C. $[ 0,1 )$
D. $( - \infty , 1 ]$

Given the universal set $U = \{1,2,3,4,5,6,7,8\}$, set $A = \{2,3,5,6\}$, set $\mathrm{B} = \{1,3,4,6,7\}$, then $\mathrm{A} \cap \mathrm{C}_{U}\mathrm{B} =$
(A) $\{2,5\}$
(B) $\{3,6\}$
(C) $\{2,5,6\}$
(D) $\{2,3,5,6,8\}$

3. The negation of the proposition ``$\exists x _ { 0 } \in ( 0 , + \infty ) , \ln x _ { 0 } = x _ { 0 } - 1$'' is
A. $\forall x \in ( 0 , + \infty ) , \ln x \neq x - 1$
B. $\forall x \notin ( 0 , + \infty ) , \ln x = x - 1$
C. $\exists x _ { 0 } \in ( 0 , + \infty ) , \ln x _ { 0 } \neq x _ { 0 } - 1$
D. $\exists x _ { 0 } \notin ( 0 , + \infty ) , \ln x _ { 0 } = x _ { 0 } - 1$

5. A bag contains 4 balls of identical shape and size, including 1 white ball, 1 red ball, and 2 yellow balls. If 2 balls are randomly drawn at once, then the probability that the 2 balls have different colors is $\_\_\_\_$ .

11. Given the set $\mathrm { U } = \{ 1,2,3,4 \} , \mathrm { A } = \{ 1,3 \} , \mathrm { B } = \{ 1,3,4 \}$, then $\mathrm { A } \cup ( C \cup B ) =$ $\_\_\_\_$

16. (This question is worth 12 points)
A shopping mall is holding a promotional lottery activity. After customers purchase goods of a certain amount, they can participate in the lottery. The lottery method is as follows: randomly draw 1 ball each from box A containing 2 red balls $\mathrm { A } _ { 1 } , \mathrm { A } _ { 2 }$ and 1 white ball B, and from box B containing 2 red balls $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 }$ and 2 white balls $\mathrm { b } _ { 1 } , \mathrm { b } _ { 2 }$. If both balls drawn are red, the customer wins; otherwise, the customer does not win. (I) List all possible outcomes of drawing balls using the ball labels. (II) Someone claims: Since there are more red balls than white balls in both boxes, the probability of winning is greater than the probability of not winning. Do you agree? Please explain your reasoning.

A supermarket randomly selected 1000 customers and recorded their purchasing of four products: A, B, C, and D. The data is organized in the table below, where ``✓'' indicates purchase and ``×'' indicates no purchase.\n\n\n

\n\n\backslashbox{Number of Customers}{Product}	A	B	C	D
\n\n100	✓	×	✓	✓
\n\n217	×	✓	×	✓
\n\n200	✓	✓	✓	×
\n\n300	✓	×	✓	×
\n\n85	✓	×	×	×
\n\n98	×	✓	×	×
\n\n

\n\n\n(I) Estimate the probability that a customer purchases both B and C\n(II) Estimate the probability that a customer purchases exactly 3 of the four products A, B, C, and D\n(III) If a customer has purchased product A, which of products B, C, and D is the customer most likely to have purchased?