Let $P ( X )$ denote the probability of event $X$ occurring, and $P ( X \mid Y )$ denote the probability of event $X$ occurring given that event $Y$ has occurred. There are 7 balls of the same size: 2 black balls, 2 white balls, and 3 red balls arranged in a row. Let event $A$ be the event that the 2 black balls are adjacent, event $B$ be the event that the 2 black balls are not adjacent, and event $C$ be the event that no two red balls are adjacent. Select the correct options.
(1) $P ( A ) > P ( B )$
(2) $P ( C ) = \frac { 2 } { 7 }$
(3) $2 P ( C \mid A ) + 5 P ( C \mid B ) < 2$
(4) $P ( C \mid A ) > 0.2$
(5) $P ( C \mid B ) > 0.3$

Xiaoming wrote a program to move a robot on a $2 \times 2$ chessboard, as shown in the figure. Each execution, the program selects one direction from ``up, down, left, right'' with equal probability for each direction, and instructs the robot to move one square in that direction. However, if the selected direction would move the robot off the board, the robot stays in place. Each execution starts from the robot's current position with a newly selected direction by the program. Assume the robot's initial position is $A$. Let $a _ { n } , b _ { n } , c _ { n }$, and $d _ { n }$ be the probabilities that the robot is at positions $A , B , C , D$ respectively after executing the program $n$ times. Select the correct options.
(1) $b _ { 1 } = \frac { 1 } { 4 }$
(2) $b _ { 2 } = \frac { 1 } { 8 }$
(3) $a _ { 2 } + d _ { 2 } = \frac { 3 } { 4 }$

$A$	$B$
$C$	$D$

(4) $b _ { 99 } = c _ { 99 }$
(5) $a _ { 100 } + d _ { 100 } > \frac { 1 } { 2 }$

A shopping mall holds a raffle drawing activity with on-site registration. After registration closes, the host places the same number of raffle balls as the number of registrants, of which 10 balls are marked as lucky prizes: 5 balls for a 5000 yuan gift voucher and 5 balls for an 8000 yuan gift voucher. Each ball has an equal probability of being drawn, and balls are not replaced after drawing. Before the drawing, the organizers announce a winning probability of 0.4\% based on the number of prizes and registrants. After the drawing begins, each person draws a ball in order, and each person has only one chance to draw. If among the first 100 participants, exactly 1 person draws a 5000 yuan voucher and no one draws an 8000 yuan voucher, then the expected value of the prize amount that the 101st person can receive is (15－1)(15－2) yuan.

An opaque bag contains blue and green balls, each marked with a number 1 or 2. The quantities are shown in the table below. For example, there are 2 blue balls marked with number 1.

	Blue	Green
Number 1	2	4
Number 2	3	$k$

A ball is randomly drawn from the bag (each ball has an equal probability of being drawn). Given that the event of drawing a blue ball and the event of drawing a ball marked with 1 are independent, what is the value of $k$?
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

There are two fair six-sided dice A and B: The numbers on A are $1, 2, 5, 6, 7, 9$, The numbers on B are $1, 3, 4, 5, 6, 9$. The relationship between the numbers on A and B is recorded in the table below. For example: if the numbers on A and B are 5 and 3 respectively, it is recorded as ``A wins''; if both A and B show 5, it is recorded as ``tie''.

\multirow{2}{*}{}		\multicolumn{6}{|c|}{A}
	Number	1	2	5	6	7	9
\multirow{6}{*}{B}	1	Tie	A wins	A wins	A wins	A wins	A wins
	3	B wins	B wins	A wins	A wins	A wins	A wins
	4	B wins	B wins	A wins	A wins	A wins	A wins
	5	B wins	B wins	Tie	A wins	A wins	A wins
	6	B wins	B wins	B wins	Tie	A wins	A wins
	9	B wins	B wins	B wins	B wins	B wins	Tie

If a person rolls both dice A and B simultaneously, what is the probability that B shows 6 given that A's number is greater than B's number?
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 9 }$
(3) $\frac { 1 } { 16 }$
(4) $\frac { 1 } { 18 }$
(5) $\frac { 1 } { 32 }$

There is a card-drawing prize activity with the following rules: In an opaque box, there are 2 cards marked with ``1000 yuan'' and 3 cards marked with ``0 yuan''. A participant randomly draws one card from the box. Without knowing the amount marked on the drawn card, the host then places a card marked with ``500 yuan'' into the box. At this point, the participant has two choices: (I) Keep the originally drawn card; the amount marked on that card is the prize won. (II) Discard the originally drawn card without returning it, and randomly draw another card from the box; the amount marked on that card is the prize won.
A participant joins this activity. Assume each card has an equal chance of being drawn. Select the correct options.
(1) If the participant chooses (I), the probability of winning 0 yuan is $\frac { 3 } { 5 }$
(2) If the participant chooses (I), the expected value of the prize is 500 yuan
(3) If the participant chooses (II), the probability of winning 1000 yuan is $\frac { 2 } { 5 }$
(4) If the participant chooses (II), the probability of winning 0 yuan is $\frac { 12 } { 25 }$
(5) If the participant chooses (II), the expected value of the prize is 420 yuan

There are two parking lots next to a scenic spot. Assume that on a certain day, the probability that either parking lot has no available spaces is 0.7, and whether the two parking lots have available spaces is independent. If a car arrives at these two parking lots on that day, the probability that at least one parking lot has available spaces is 0.(13--1)(13--2).

A holiday market stall offers ``test your luck—cute dolls regularly priced at 480 yuan can be purchased for as low as 240 yuan''. The rules are: customers flip a fair coin up to 5 times. If 3 consecutive heads are obtained in the first 3 flips, they can purchase a doll for 240 yuan. If 3 heads are accumulated by the 4th flip, they can purchase for 320 yuan. If 3 heads are accumulated by the 5th flip, they can purchase for 400 yuan. If 3 heads are not accumulated after 5 flips, they can purchase for 480 yuan. The expected value of the amount a customer spends to purchase a doll is (15-1) (15-2) (15-3) yuan.

7. A bag contains $n$ red balls, $n$ yellow balls, and $n$ blue balls.
One ball is selected at random and not replaced.
A second ball is then selected at random and not replaced. Each ball is equally likely to be chosen. The probability that the two balls are not the same colour is
A $\frac { n - 1 } { 3 n - 1 }$
B $\frac { 2 n - 2 } { 3 n - 1 }$
C $\frac { 2 n } { 3 n - 1 }$
D $\quad \frac { ( n - 1 ) ^ { 3 } } { 27 ( 3 n - 1 ) ^ { 3 } }$
E $\quad \frac { 3 ( n - 1 ) } { 3 n - 1 }$ F $\quad \frac { n ^ { 3 } } { 27 ( 3 n - 1 ) ^ { 3 } }$

P, Q, and R are each mixtures of red and white paint. The percentage by volume of red paint in P is $30 \%$. The percentage by volume of red paint in Q is 20\%. The mixtures P, Q, and R are combined in the proportion $12 : 5 : 3$ respectively. If the resulting mixture contains $25 \%$ by volume of red paint, what percentage by volume of mixture $R$ is red paint?
A $25 \%$ B 23\% C $13 \frac { 1 } { 3 } \%$ D $19 \frac { 1 } { 2 } \%$ E $9 \frac { 3 } { 4 } \%$ F It is impossible to achieve this result.

60\% of a sports club's members are women and the remainder are men. This sports club offers the opportunity to play tennis or cricket. Every member plays exactly one of the two sports. $\frac { 2 } { 5 }$ of the male members of the club play cricket; $\frac { 2 } { 3 }$ of the cricketing members of the club are women. What is the probability that a member of the club, chosen at random, is a woman who plays tennis?
A $\frac { 1 } { 5 }$ B $\frac { 7 } { 25 }$ C $\frac { 1 } { 3 }$ D $\frac { 11 } { 25 }$ E $\frac { 3 } { 5 }$

Most students in a large college study Mathematics. A teacher chooses three different students at random, one after the other.
Consider these three probabilities:
$R = \mathrm { P }$ (At least one of the students chosen studies Mathematics)
$S = \mathrm { P }$ (The second student chosen studies Mathematics)
$T = \mathrm { P }$ (All three of the students chosen study Mathematics)
Which of the following is true?

A student is chosen at random from a class. Each student is equally likely to be chosen.
Which of the following conditions is/are necessary for the probability that the student wears glasses to equal $\frac { 4 } { 15 }$ ?
I Exactly 11 students in the class do not wear glasses. II The number of students in the class is divisible by 3 . III The class contains 30 students, and 8 of them wear glasses.
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III

2

Balls are drawn one at a time from a bag containing 3 black balls, 4 red balls, and 5 white balls, and all 12 drawn balls are arranged in a horizontal row in the order they were drawn. Each ball in the bag is equally likely to be drawn.

1. [(1)] Find the probability $p$ that no two red balls are adjacent to each other.
2. [(2)] Given that no two red balls are adjacent to each other, find the conditional probability $q$ that no two black balls are adjacent to each other.

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Let $A = \{1,2,3,4\}$ and $B = \{-2,-1,0\}$. For any element $(a,b)$ taken from the Cartesian product set $A \times B$, what is the probability that the sum $a + b$ equals zero?
A) $\frac{1}{4}$
B) $\frac{1}{5}$
C) $\frac{1}{6}$
D) $\frac{1}{7}$
E) $\frac{2}{7}$

A bag contains 2 red, 2 white, and 1 yellow marble.
When 4 marbles are randomly drawn from the bag, what is the probability that the remaining marble in the bag is red?
A) $\frac { 1 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 2 } { 5 }$
E) $\frac { 3 } { 5 }$

From a group of 6 girls and 7 boys, 2 representatives are selected.
What is the probability that one of the two selected representatives is a girl and the other is a boy?
A) $\frac { 3 } { 4 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 2 } { 13 }$
D) $\frac { 7 } { 13 }$
E) $\frac { 9 } { 13 }$

Four students of different heights line up randomly in a row.
According to this, what is the probability that the shortest and tallest students are at the ends?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 6 }$
E) $\frac { 1 } { 12 }$

A bag contains nine balls numbered from 1 to 9. Ayşe will choose a number from 1 to 9 and then draw a ball randomly from the bag. Ayşe wins the game if the sum of the number on the ball and the number she chose is at most 9 and their product is at least 9.
Which number should Ayşe choose so that her probability of winning the game is highest?
A) 2
B) 3
C) 4
D) 5
E) 6

A fair cubic die is rolled and it is known that one of its faces is in contact with the ground.
Given this, what is the probability that only one of the corners A and B is in contact with the ground?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 1 } { 6 }$
E) $\frac { 5 } { 6 }$

For positive integers $n$, the subsets of the set $R$ of real numbers are defined as
$$A _ { n } = \left\{ x \in R : \frac { ( - 1 ) ^ { n } } { n } < x < \frac { 2 } { n } \right\}$$
Accordingly, $$A _ { 1 } \cap A _ { 2 } \cap A _ { 3 }$$
the intersection set is equal to which of the following?
A) $\left( \frac { 1 } { 2 } , \frac { 2 } { 3 } \right)$
B) $\left( \frac { 1 } { 2 } , 2 \right)$
C) $\left( \frac { - 1 } { 3 } , \frac { 2 } { 3 } \right)$
D) $\left( \frac { - 1 } { 3 } , 1 \right)$
E) $\left( - 1 , \frac { 2 } { 3 } \right)$

Four identical matches are taken, each with only one flammable end. These matches are randomly arranged along all sides of a square whose side length is the same as the length of one match, with the ends touching each other.
What is the probability that there are no flammable ends in contact with each other in this arrangement?
A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 8 }$
C) $\frac { 3 } { 8 }$
D) $\frac { 1 } { 16 }$
E) $\frac { 3 } { 16 }$

Deniz randomly colored two of the following four points that are the vertices of a square red and the other two blue, and drew line segments connecting the points she colored the same color.
What is the probability that these line segments intersect?
A) $\frac { 1 } { 6 }$ B) $\frac { 1 } { 4 }$ C) $\frac { 1 } { 3 }$ D) $\frac { 2 } { 3 }$ E) $\frac { 3 } { 4 }$

In the figure, 3 of the 6 edges of a regular tetrahedron are randomly painted.
Accordingly, what is the probability that all three painted edges are on the same face?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 5 }$
E) $\frac { 1 } { 6 }$

In a cube, 6 of its 8 vertices are randomly painted white and the other 2 are painted black.
What is the probability that there is an edge with both endpoints painted black in this cube?
A) $\frac { 1 } { 7 }$
B) $\frac { 2 } { 7 }$
C) $\frac { 3 } { 7 }$
D) $\frac { 4 } { 7 }$
E) $\frac { 5 } { 7 }$