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 $\frac{\mathbf{O}}{\mathbf{PQ}}$.
(2) The probability that the four balls taken out consist of three white balls and one red ball is $\frac{\mathbf{R}}{\mathbf{S}}$.
(3) The probability that the four balls taken out all have the same color is $\square$ T UV
(4) The probability that of the four balls taken out, two or fewer are white balls is $\frac{\mathbf{WX}}{\mathbf{W}}$.

Let us simultaneously throw three dice which are different in size and denote the number on the large, medium and small dice by $x , y$ and $z$, respectively.
Let $A$ be the event where $x = y = z$; let $B$ be the event where $x + y + z = 6$; let $C$ be the event where $x + y = z$.
(1) The numbers of outcomes in event $A$ is $\mathbf { J }$, in event $B$ is $\mathbf { K } \mathbf { L }$, and in event $C$ is $\mathbf { M N }$.
(2) The numbers of outcomes in event $A \cap B$ is $\mathbf { O }$, in event $B \cap C$ is $\mathbf { P }$, and in event $C \cap A$ is $\mathbf { Q }$.
(3) The probability of event $B \cup C$ is
$$P ( B \cup C ) = \frac { \mathbf { R S } } { \mathbf { T U V } } .$$

Let us simultaneously throw three dice which are different in size and denote the number on the large, medium and small dice by $x , y$ and $z$, respectively.
Let $A$ be the event where $x = y = z$; let $B$ be the event where $x + y + z = 6$; let $C$ be the event where $x + y = z$.
(1) The numbers of outcomes in event $A$ is $\mathbf { J }$, in event $B$ is $\mathbf { K } \mathbf { L }$, and in event $C$ is $\mathbf { M N }$.
(2) The numbers of outcomes in event $A \cap B$ is $\mathbf { O }$, in event $B \cap C$ is $\mathbf { P }$, and in event $C \cap A$ is $\mathbf { Q }$.
(3) The probability of event $B \cup C$ is
$$P ( B \cup C ) = \frac { \mathbf { R S } } { \mathbf { T U V } } .$$

In a box there are ten cards on which the numbers from 0 to 9 have been written successively. We take three cards out of the box using two methods and consider the probabilities.
(1) We take out three cards simultaneously.
(i) The probability that each number on the three cards is 2 or more and 6 or less is $\dfrac{\mathbf{KL}}{\mathbf{MN}}$.
(ii) The probability that the smallest number is 2 or less and the greatest number is 8 or more is $\dfrac { \mathbf { N O } } { \mathbf { P Q } }$.
(2) Three times we take out one card from the box, check its number, and then return it to the box. The probability that the smallest number is 2 or more and the greatest number is 6 or less is $\dfrac { \mathbf { R } } { \mathbf { S } }$.

Q2 A triangle ABC is drawn on a plane, and a ball is placed on vertex A. A dice is rolled, and the ball is moved according to the following rules:
(i) when the ball is on A, if the number on the dice is 1 the ball is moved to B, otherwise it stays on A;
(ii) when the ball is on B, if the number on the dice is less than or equal to 4 the ball is moved to C, otherwise it stays on B.
If the ball is moved to C, the trials are stopped.
We are to find the probability that the ball is moved to C within 4 rolls of the dice.
(1) The probability that the ball is moved to C on the second roll of the dice is $\frac { 1 } { \mathbf{N} }$
(2) The probability that the ball is moved to C on the third roll of the dice is $\frac{\mathbf{O}}{\mathbf{PQ}}$
(3) The probability that the ball is moved to C on the fourth roll of the dice is $\frac{\mathbf{RS}}{\mathbf{TUV}}$
Therefore, find the probability that the ball is moved to C within 4 rolls of the dice.

Consider four natural numbers $a , b , c$ and $d$ satisfying $1 < a < b < c < d$. Suppose that two sets using these numbers, $A = \{ a , b , c , d \}$ and $B = \left\{ a ^ { 2 } , b ^ { 2 } , c ^ { 2 } , d ^ { 2 } \right\}$, satisfy the following two conditions:
(i) Just two elements belong to the intersection $A \cap B$, and the sum of these two elements is greater than or equal to 15, and less than or equal to 25.
(ii) The sum of all the elements belonging to the union $A \cup B$, is less than or equal to 300.
We are to find the values of $a , b , c$ and $d$.
First, set $A \cap B = \{ x , y \}$, where $x < y$. Since $x \in B$ and $y \in B$, it follows from (i) that $y = \mathbf { A B }$ and that $x$ is either $\mathbf{C}$ or $\mathbf { D }$. (Write the answers in the order $\mathbf { C } < \mathbf { D }$.) Here, when we consider (ii), we see that $x = \mathbf { E }$. Hence $A$ includes the elements $\mathbf { F }$, $\mathbf{F}$ and $\mathbf{F}$.
Furthermore, when we denote the remaining element of $A$ by $z$, from (ii) we see that $z$ satisfies
$$z ^ { 2 } + z \leqq \mathbf { G H } .$$
Hence we have $z = \mathbf { I }$. From the above we obtain
$$a = \mathbf { J } , \quad b = \mathbf { K } , \quad c = \mathbf { L } \text { and } d = \mathbf { M N } .$$

Q2 A triangle ABC is drawn on a plane, and a ball is placed on vertex A. A dice is rolled, and the ball is moved according to the following rules:
(i) when the ball is on A, if the number on the dice is 1 the ball is moved to B, otherwise it stays on A;
(ii) when the ball is on B, if the number on the dice is less than or equal to 4 the ball is moved to C, otherwise it stays on B.
If the ball is moved to C, the trials are stopped.
We are to find the probability that the ball is moved to C within 4 rolls of the dice.
(1) The probability that the ball is moved to C on the second roll of the dice is $\frac{1}{\mathbf{N}}$
(2) The probability that the ball is moved to C on the third roll of the dice is $\frac{\mathbf{O}}{\mathbf{PQ}}$
(3) The probability that the ball is moved to C on the fourth roll of the dice is $\frac{\mathbf{RS}}{\mathbf{TUV}}$
Therefore, find the probability that the ball is moved to C within 4 rolls of the dice.

There is a lottery consisting of 4 tickets in total: 2 winning tickets and 2 losing tickets. Three people A, B, and C draw one ticket each in that order. Once a ticket is drawn, it is not returned.

Q2 Consider dice-X with the following property: when it is rolled, for numbers 1 through 5 the probabilities of that number's coming up are all the same, but the probability that 6 comes up is twice that of any other number.
(1) Denote by $p$ the probability that a particular number 1 through 5 comes up. The probability that number 6 comes up is $\mathbf { L }$ p. Since the probability of the whole event is $\mathbf { M }$, we have $p = \frac { \mathbf { N } } { \mathbf { O } }$.
(2) Dice-X is rolled twice in succession. Let us denote by $A$ the event that for both rolls the number that comes up is 1 through 5, and by $B$ the event that number 6 comes up at least once. Then the probability of event $A$, $P(A)$, and that of event $B$, $P(B)$, are
$$P ( A ) = \frac { \mathbf { PQ } } { \mathbf { RS } } , \quad P ( B ) = \frac { \mathbf { TU } } { \mathbf { VW } } .$$
Hence, we see that $\mathbf { X }$. (For $\mathbf { X }$, choose the correct answer from among choices (0) $\sim$ (4) below.) (0) $P ( A )$ is less than $P ( B )$ and the difference between them is not less than $\frac { 1 } { 36 }$.
(1) $P ( A )$ is less than $P ( B )$ and the difference between them is less than $\frac { 1 } { 36 }$.
(2) $P ( A )$ and $P ( B )$ are the same.
(3) $P ( A )$ is greater than $P ( B )$ and the difference between them is not less than $\frac { 1 } { 36 }$.
(4) $P ( A )$ is greater than $P ( B )$ and the difference between them is less than $\frac { 1 } { 36 }$.
(3) Next, dice-X is rolled three times in succession. Let us denote by $C$ the event that for all three rolls the number which comes up is 1 through 5, and by $D$ the event that the number 6 comes up at least once. When the probability $P(C)$ is compared with the probability $P(D)$, we see that $\mathbf { Y }$. (For $\mathbf { Y }$, choose the correct answer from among choices (0) $\sim$ (4) below.) (0) $P ( C )$ is less than $P ( D )$ and $P ( D )$ is not less than twice $P ( C )$.
(1) $P ( C )$ is less than $P ( D )$ and $P ( D )$ is less than twice $P ( C )$.
(2) $P ( C )$ and $P ( D )$ are the same.
(3) $P ( C )$ is greater than $P ( D )$ and $P ( C )$ is not less than twice $P ( D )$.
(4) $P ( C )$ is greater than $P ( D )$ and $P ( C )$ is less than twice $P ( D )$.

(Course 2) Q2 Consider dice-X with the following property: when it is rolled, for numbers 1 through 5 the probabilities of that number's coming up are all the same, but the probability that 6 comes up is twice that of any other number.
(1) Denote by $p$ the probability that a particular number 1 through 5 comes up. The probability that number 6 comes up is $\mathbf { L }$ p. Since the probability of the whole event is $\mathbf { M }$, we have $p = \frac { \mathbf { N } } { \mathbf { O } }$.
(2) Dice-X is rolled twice in succession. Let us denote by $A$ the event that for both rolls the number that comes up is 1 through 5, and by $B$ the event that number 6 comes up at least once. Then the probability of event $A$, $P(A)$, and that of event $B$, $P(B)$, are
$$P ( A ) = \frac { \mathbf { PQ } } { \mathbf { RS } } , \quad P ( B ) = \frac { \mathbf { TU } } { \mathbf { VW } }$$
Hence, we see that $\mathbf { X }$. (For $\mathbf { X }$, choose the correct answer from among choices (0) $\sim$ (4) below.) (0) $P ( A )$ is less than $P ( B )$ and the difference between them is not less than $\frac { 1 } { 36 }$.
(1) $P ( A )$ is less than $P ( B )$ and the difference between them is less than $\frac { 1 } { 36 }$.
(2) $P ( A )$ and $P ( B )$ are the same.
(3) $P ( A )$ is greater than $P ( B )$ and the difference between them is not less than $\frac { 1 } { 36 }$.
(4) $P ( A )$ is greater than $P ( B )$ and the difference between them is less than $\frac { 1 } { 36 }$.
(3) Next, dice-X is rolled three times in succession. Let us denote by $C$ the event that for all three rolls the number which comes up is 1 through 5, and by $D$ the event that the number 6 comes up at least once. When the probability $P(C)$ is compared with the probability $P(D)$, we see that $\mathbf { Y }$. (For $\mathbf { Y }$, choose the correct answer from among choices (0) $\sim$ (4) below.) (0) $P ( C )$ is less than $P ( D )$ and $P ( D )$ is not less than twice $P ( C )$.
(1) $P ( C )$ is less than $P ( D )$ and $P ( D )$ is less than twice $P ( C )$.
(2) $P ( C )$ and $P ( D )$ are the same.
(3) $P ( C )$ is greater than $P ( D )$ and $P ( C )$ is not less than twice $P ( D )$.
(4) $P ( C )$ is greater than $P ( D )$ and $P ( C )$ is less than twice $P ( D )$.

For a game, each of two people, A and B , has a bag containing three cards on which the numbers 1, 2, and 3 are written, each number on a different card. In the game, A and B each take out one card from their own bag and compare the numbers. If the numbers are the same, the game is a draw. If the numbers are different, the person with the greater number wins.
(1) For a single game the probability of a draw is $\frac { \mathbf { L } } { \mathbf { M } }$.
(2) If this game is successively played four times, replacing the cards after each game, let us find the probabilities for the following.
(i) The probability that A wins three times or more is $\frac { \mathbf { N } } { \mathbf{O} }$.
(ii) The probability that A wins once and loses once and two games are draws is $\frac { \mathrm { P } } { \mathrm { QR } }$.
(iii) The probability that the number of times that A wins and the number of times that B wins are the same is $\frac { \mathbf { S T } } { \mathbf { U V } }$. Hence, the probability that the number of times that A wins is greater than the number of times that B wins is $\frac { \mathbf { W } \mathbf { X } } { \mathbf{UV} }$.

For a game, each of two people, A and B , has a bag containing three cards on which the numbers 1, 2, and 3 are written, each number on a different card. In the game, A and B each take out one card from their own bag and compare the numbers. If the numbers are the same, the game is a draw. If the numbers are different, the person with the greater number wins.
(1) For a single game the probability of a draw is $\frac { \mathbf { L } } { \mathbf { M } }$.
(2) If this game is successively played four times, replacing the cards after each game, let us find the probabilities for the following.
(i) The probability that A wins three times or more is $\frac { \mathbf { N } } { \mathbf{O} }$.
(ii) The probability that A wins once and loses once and two games are draws is $\frac { \mathrm { P } } { \mathrm { QR } }$.
(iii) The probability that the number of times that A wins and the number of times that B wins are the same is $\frac { \mathbf { S T } } { \mathbf { U V } }$. Hence, the probability that the number of times that A wins is greater than the number of times that B wins is $\frac { \mathbf { W } \mathbf { X } } { \mathbf{UV} }$.

60\% of sales in a large department store correspond to items with reduced prices. Customers return 15\% of the discounted items they purchase, a percentage that decreases to 8\% if the items were purchased at full price.
a) (1.25 points) Determine the overall percentage of returned items.
b) ( 1.25 points) What percentage of returned items were purchased at reduced prices?

There are three urns $A$, $B$ and $C$. Urn $A$ contains 4 red balls and 2 black balls, urn $B$ contains 3 balls of each color and urn $C$ contains 6 black balls. An urn is chosen at random and two balls are drawn from it consecutively and without replacement. Find:\ a) (1 point) Calculate the probability that the first ball drawn is red.\ b) (1 point) Calculate the probability that the first ball drawn is red and the second is black.\ c) (0.5 points) Given that the first ball drawn is red, calculate the probability that the second is black.

An amateur archer has 4 arrows and shoots at a balloon placed in the center of a target. The probability of hitting the target on the first shot is 30\%. In successive launches the aim improves, so on the second it is 40\%, on the third 50\% and on the fourth 60\%. It is requested:\ a) (1 point) Calculate the probability that the balloon has burst without needing to make the fourth shot.\ b) (0.5 points) Calculate the probability that the balloon remains intact after the fourth shot.\ c) (1 point) In an exhibition ten professional archers participate, who hit 85\% of their shots. Calculate the probability that among the 10 exactly 6 balloons have burst on the first shot.

Consider two events $A$ and $B$ such that $P ( A ) = 0.5 , P ( B ) = 0.25$ and $P ( A \cap B ) = 0.125$. Answer in a reasoned manner or calculate what is requested in the following cases:\ a) (0.5 points) Let $C$ be another event, incompatible with $A$ and with $B$. Are events $C$ and $A \cup B$ compatible?\ b) (0.5 points) Are $A$ and $B$ independent?\ c) (0.75 points) Calculate the probability $P ( \bar { A } \cap \bar { B } )$ (where $\bar { A }$ denotes the event complementary to event A).\ d) (0.75 points) Calculate $P ( \bar { B } / A )$.

From a basket with 6 white hats and 3 black hats, one is chosen at random. If the hat is white, a handkerchief is taken at random from a drawer that contains 2 white, 2 black and 5 with white and black checks. If the hat is black, a handkerchief is chosen at random from another drawer that contains 2 white handkerchiefs, 4 black and 4 with white and black checks. It is requested: a) (1 point) Calculate the probability that the handkerchief shows some color that is not the color of the hat. b) (0.5 points) Calculate the probability that in at least one of the accessories (hat or handkerchief) the color black appears. c) (1 point) Calculate the probability that the hat was black, knowing that the handkerchief was checked.

2. In the checkerboard grid shown on the right, two squares are randomly selected. The probability that the two selected squares are not in the same row (whether or not they are in the same column is irrelevant) is
(1) $\frac { 1 } { 20 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 3 } { 5 }$
(5) $\frac { 4 } { 5 }$

7. A high school has 20 classes, each with 40 students: 25 boys and 15 girls. If 80 students are selected from the school's 800 students using simple random sampling, which of the following statements are correct?
(1) At least one student from each class will be selected
(2) The number of boys selected will definitely be greater than the number of girls selected
(3) Given that Xiaowén is a boy and Xiaomei is a girl, the probability that Xiaowén is selected is greater than the probability that Xiaomei is selected
(4) If students A and B are in the same class, and student C is in another class, then the probability that both A and B are selected equals the probability that both A and C are selected
(5) If students A and B are brothers, the probability that both are selected is less than $\frac { 1 } { 100 }$

4. Three high schools A, B, and C have 3, 4, and 5 classes respectively in their first year. One class is randomly selected from these 12 classes to participate in a Chinese language test, and then one class is randomly selected from the remaining 11 classes to participate in an English test. What is the probability that the two classes participating in the tests are from the same school closest to which of the following options?
(1) $21\%$
(2) $23\%$
(3) $25\%$
(4) $27\%$
(5) $29\%$

There are two boxes $A$ and $B$. Box $A$ contains 6 white balls and 4 red balls, and box $B$ contains 8 white balls and 2 blue balls. There are three lottery methods (each ball in each box has equal probability of being drawn): (I) First draw one ball from box $A$; if a red ball is drawn, stop; if a white ball is drawn, then draw one ball from box $B$; (II) First draw one ball from box $B$; if a blue ball is drawn, stop; if a white ball is drawn, then draw one ball from box $A$; (III) Simultaneously draw one ball from each of boxes $A$ and $B$. The prize rules are: Among red and blue balls, if only a red ball is drawn, win 50 yuan; if only a blue ball is drawn, win 100 yuan; if both colors are drawn, still win only 100 yuan; if neither color is drawn, win nothing. Let $E_{1}$, $E_{2}$, $E_{3}$ denote the expected values of winnings for methods (I), (II), (III) respectively. Select the correct option.
(1) $E_{1} > E_{2} > E_{3}$
(2) $E_{1} = E_{2} > E_{3}$
(3) $E_{2} = E_{3} > E_{1}$
(4) $E_{1} = E_{3} > E_{2}$
(5) $E_{3} > E_{2} > E_{1}$

A company holds a year-end lottery. Each person randomly draws two cards from six cards numbered 1 to 6. Assume each card has an equal chance of being drawn, and the rules are as follows: (I) If the sum of the numbers on the two cards is odd, the person wins 100 yuan and the lottery ends; (II) If the sum is even, the two cards are discarded, and two cards are randomly drawn from the remaining four cards. If the sum of their numbers is odd, the person wins 50 yuan; otherwise, there is no prize and the lottery ends. According to the above rules, what is the expected value of the prize money for each person participating in this lottery?
(1) 50
(2) 70
(3) 72
(4) 80
(5) 100

A scratch-off lottery game with 12 boxes labeled 1 to 12. Each game involves tossing a fair coin four times to determine which boxes to scratch. The rules are as follows: (I) On the first coin toss, if heads, scratch box 1; if tails, scratch box 3. (II) On the second, third, and fourth coin tosses, if heads, the number of the box to scratch is the number of the previous box plus 1; if tails, the number of the box to scratch is the number of the previous box plus 3, and so on. Example: If the results of four coin tosses are ``heads, tails, tails, heads'' in order, then boxes numbered 1, 4, 7, and 8 will be scratched. Let $p _ { m }$ denote the probability that box $m$ is scratched in each game. Select the correct options.
(1) $p _ { 2 } = \frac { 1 } { 4 }$
(2) $p _ { 3 } = \frac { 1 } { 2 }$
(3) $p _ { 4 } = \frac { 1 } { 2 } p _ { 1 } + \frac { 1 } { 2 } p _ { 3 }$
(4) $p _ { 8 } > p _ { 10 }$
(5) Given that box 4 is scratched, the probability that box 3 is scratched is $\frac { 1 } { 2 }$

A psychologist conducted an experiment with 1000 subjects in a dark room, where each subject had to observe and identify three digit cards: 6, 8, and 9. The probability of mistaking the actual digit for another digit is shown in the following table:

\backslashbox{Actual Digit}{Seen as}	6	8	9	Other
6	0.4	0.3	0.2	0.1
8	0.3	0.4	0.1	0.2
9	0.2	0.2	0.5	0.1

For example: The actual digit 6 is seen as 6, 8, 9 with probabilities 0.4, 0.3, 0.2 respectively, and is seen as another digit with probability 0.1. Based on the above experimental results, select the correct options.
(1) If the actual digit is 8, then there is at least a 50\% chance it will be seen as 8
(2) If the actual digit is 6, then there is a 60\% chance it will be seen as not 6
(3) Among the three digits 6, 8, 9, the digit 9 has the lowest probability of being misidentified
(4) If the digit seen is 6, then the probability that it is actually 6 is less than 50\%
(5) If the digit seen is 9, then the probability that it is actually 9 is greater than $\frac { 2 } { 3 }$

A village mayor election has two polling stations. The proportion of valid votes received by the two candidates at each polling station is shown in the following table (invalid votes are not counted):

	Candidate A	Candidate B
First Polling Station	$40 \%$	$60 \%$
Second Polling Station	$55 \%$	$45 \%$

Assume the number of valid votes at the first and second polling stations are $x$ and $y$ respectively (where $x > 0 , y > 0$), and the candidate with the higher total votes wins. Based on the above table, select the correct options.
(1) When the total number of valid votes $x + y$ is known, the winner can be determined
(2) When the ratio $x : y$ is less than $\frac { 1 } { 2 }$, the winner can be determined
(3) When $x > y$, the winner can be determined
(4) When Candidate A's valid votes at the first polling station exceed those at the second polling station, the winner can be determined
(5) When Candidate B's valid votes at the second polling station exceed those at the first polling station, the winner can be determined