1. In order to promote the spirit of "May Fourth" school held a speech contest, after big data analysis, found that the results of this speech contest obey ${ } ^ { N ( 70,64 ) }$, according to which it is estimated that the results of the contest is not less than 86 students accounted for a percentage of ( ) Reference data: $P ( \mu - \sigma < X < \mu + \sigma ) \approx 0.6827 , ~ P ( \mu - 2 \sigma < X < \mu + 2 \sigma ) \approx 0.9545$, $P ( \mu - 3 \sigma < X < \mu + 3 \sigma ) \approx 0.9973$

• A. A. $0.135 \%$
• B. B. $0.27 \%$
• C. C. $2.275 \%$
• D. D. $3.173 \%$

Answer: C

2. Zhang Ming and Li Hua are playing a game, then the following rules of the game are unfair ( )

• A. A. Throw a uniformly textured die. If the upward point is odd, Zhang Ming wins, and if the upward point is even, Li Hua wins.
• B. B. Toss two even-textured coins at the same time, if exactly one of them is heads up, Zhang Ming wins, if both of them are heads up, Li Hua wins.
• C. C. Draw a card from a deck of playing cards that does not contain a king or a queen. If the card is red, Zhang Ming wins, and if the card is black, Li Hua wins.
• D. D. Zhang Ming and Li Hua each write a number 0 or 1, if they write the same number then Zhang Ming wins, otherwise Li Hua wins.

Answer: B

3. A, B two people play a card game, each round, two people at the same time randomly selected from their own cards to each other. At the beginning of the game, a hand of the two card numbers were 1,3, b hand of the two card numbers were 2, 4. Then a round, a hand of card numbers and greater than the hands of the card numbers and the probability of b hand of card numbers for ( )

• A. A. $\frac { 1 } { 2 }$
• B. B. $\frac { 1 } { 4 }$
• C. C. $\frac { 3 } { 4 }$
• D. D. $\frac { 3 } { 8 }$

Answer: B

4. The use of stochastic simulation to solve problems is known as the Monte Carlo method, with which a large number of repetitive trials can be carried out quickly, and the probability can be estimated by frequency. A, B two players to play a game, using the system of two out of three to determine the winner, if the probability of winning each game A is 0.4, the probability of winning B is 0.6. The use of computers to generate $1 \sim 5$ between the random integers, agreed to appear random number 1 or 2 when the game that a game A win, due to the game 3 innings, so 3 random numbers for a group, is now producing Since there are 3 games, 3 random numbers are a group. 20 groups of random numbers are generated as follows: $\begin{array} { l l l l l l l l l l l l l l l l l } 354 & 151 & 314 & 432 & 125 & 334 & 541 & 112 & 443 & 534 & 312 & 324 & 252 & 525 & 453 & 114 & 344 \end{array}$ 423 123243, then according to this can be estimated that the probability that player A finally win the game ()

• A. A. 0.40
• B. B. 0.35
• C. C. 0.30
• D. D. 0.25

Answer: B

5. If the distribution of the random variable $X$ is shown in the table on the right, then the mathematical expectation of $X$ $E ( X )$ is ( ) | $X$ | -1 | 0 | 1 | | :--- | :--- | :--- | :--- | | $P$ | $\frac { 1 } { 4 }$ | $\frac { 1 } { 2 }$ | $\frac { 1 } { 4 }$ | $\frac { 1 } { 4 }$ |

• A. A. 0
• B. B. $\frac { 1 } { 4 }$
• C. C. $\frac { 1 } { 2 }$
• D. D. 1

Answer: A

6. a basketball player each shot missed the probability of 0.3, the probability of hitting a 2-point shot is 0.4, the probability of hitting a 3-point shot is 0.3, then the athlete shoots a score of mathematical expectation is

• A. A. 1.5
• B. B. 1.6
• C. C. 1.7
• D. D. 1.8

Answer: C

7. The $A , B$ two places of state-owned enterprises employees late for work statistics, it can be seen that the two places of state-owned enterprises employees late for work time are consistent with the normal distribution, in which the $A$ place employees late for work time for $X$ (unit: $\min ) , \quad X \sim N ( 3,4 )$, corresponding to the curvature of the $\min ) , \quad X \sim N ( 3,4 )$ and the $\min ) , \quad X \sim N ( 3,4 )$, the corresponding curve. FORMULA_3]], the corresponding qu INLINE_FORMULA_4]] and the corresponding curve is $Y$ (in $\min$), $Y \sim N \left( 2 , \frac { 1 } { 9 } \right)$ and the corresponding curve is $C _ { 2 }$. The correct image is ( )

• A. A. 
• B. B. 
• C. C. 
• D. D. 

Answer: D

8. The random variables $X$ and $Y$ satisfy $Y = 2 X + 1$, and if $D ( X ) = 2$ , then $D ( Y ) =$

• A. A. 8
• B. B. 5
• C. C. 4
• D. D. 2

Answer: A

9. The probability that the event "the number of points on die (1) is greater than the number of points on die (2)" occurs when two uniformly textured dice (labeled as (1) and (2)) are thrown is

• A. A. $\frac { 5 } { 12 }$
• B. B. $\frac { 1 } { 2 }$
• C. C. $\frac { 5 } { 6 }$
• D. D. $\frac { 5 } { 9 }$

Answer: A

10. bag has the same size and shape of the red ball, black ball each one, the existing put back to randomly touch 3 times, each time to feel a ball. If you touch the red ball to get 2 points, touch the black ball to get 1 point, then 3 times to touch the ball to get a total of 5 points of probability is .

• A. A. $\frac { 1 } { 3 }$
• B. B. $\frac { 3 } { 8 }$
• C. C. $\frac { 1 } { 2 }$
• D. D. $\frac { 5 } { 8 }$

Answer: B

11. The probability of choosing two of the five boys and four girls to participate in a singing competition is

• A. A. $\frac { 4 } { 9 }$
• B. B. $\frac { 5 } { 9 }$
• C. C. $\frac { 6 } { 9 }$
• D. D. $\frac { 7 } { 9 }$

Answer: B

12. There are 3 white rabbits and 2 gray rabbits in a cage, now let them run out of the cage one by one, assuming that each of them has the same probability of running out of the cage, then the probability that one of the two rabbits that run out of the cage first is a white rabbit and the other is a gray rabbit is

• A. A. $\frac { 3 } { 5 }$
• B. B. $\frac { 4 } { 5 }$
• C. C. $\frac { 2 } { 3 }$
• D. D. $\frac { 3 } { 4 }$

Answer: A

13. Toss a uniformly textured coin $n$ three times, noting that the event $A =$ "$n$ has both heads and tails", and the event $B =$ "$n$ has heads at most once", then the event $B =$ has both heads and tails at most once. _FORMULA_3]]"$n$ has both front-facing and back-facing at most one time", the event $B =$

• A. A. When $n = 2$, $P ( A \bar { B } ) = \frac { 1 } { 2 }$
• B. B. When $n = 2 ^ { \text {时,事件 } } { } _ { A }$ is independent from event ${ } _ { B }$
• C. C. When $n = 3$, $P ( A + B ) = \frac { 7 } { 8 }$
• D. D. When $n = 3 ^ { \text {时,事件 } } { } _ { A }$ and event ${ } _ { B }$ are mutually exclusive

Answer: C

14. A deck of 52 playing cards without a king or a queen is drawn three times, one at a time, without return. If $K$ is drawn in the first two draws, the probability that $A$ is drawn in the third draw is .

• A. A. $\frac { 1 } { 25 }$
• B. B. $\frac { 2 } { 25 }$
• C. C. $\frac { 3 } { 25 }$
• D. D. $\frac { 3 } { 50 }$

Answer: B

15. The probability that one of the four numbers is twice as large as the other is given by taking two of the four numbers 1, 2, 3, and 4 in turn at random.

• A. A. $\frac { 1 } { 6 }$
• B. B. $\frac { 1 } { 3 }$
• C. C. $\frac { 1 } { 9 }$
• D. D. $\frac { 1 } { 2 }$

Answer: B

16. The following statements are correct

• A. A. In order to find out the intention of students to participate in a social practice activity, a high school uses stratified sampling to draw a sample of 60 students from the three grades in the school. The ratio of the number of students in the first, second and third grades is $5 : 4 : 3$, and 14 students should be drawn from the third grade.
• B. B. There are 8 genuine products and 2 defective products out of 10, if we take any 2 out of these 10 products, the probability of getting exactly 1 defective product is $\frac { 1 } { 3 }$
• C. C. If the random variable $X$ obeys the normal distribution ${ } ^ { N \left( 2 , \sigma ^ { 2 } \right) } , P ( X < 5 ) = 0.86$, then $P ( X \leq - 1 ) = 0.14$
• D. D. Let the weight ${ } ^ { y }$ (unit: kg) and height $x$ (unit: cm) of a male student in a school have a linear correlation, and according to a set of sample data $\left( x _ { i } , y _ { i } \right) ( i = 1,2 , \cdots , n )$, the regression equation established by the least squares method is $\hat { y } = 0.85 x - 82$, if the height of a male student in this school is 170 cm, it can be concluded that his weight is 62.5 kg. FORMULA_3]], if the height of a male student in the school is 170 cm, then his weight can be concluded to be 62.5 kg.

Answer: C

17. The random variable $X \sim B ( n , p )$, if $E ( X ) = 1 , D ( X ) = \frac { 3 } { 4 }$, then $P ( X = 3 ) =$

• A. A. $\frac { 1 } { 16 }$
• B. B. $\frac { 3 } { 64 }$
• C. C. $\frac { 1 } { 64 }$
• D. D. $\frac { 3 } { 256 }$

Answer: B

18. In an opaque bag containing 2 red balls and 3 white balls, which are all the same color except for the color, from which 2 balls are randomly touched, the probability that the touched balls are all white balls is

• A. A. $\frac { 3 } { 10 }$
• B. B. $\frac { 1 } { 10 }$
• C. C. $\frac { 1 } { 5 }$
• D. D. $\frac { 2 } { 5 }$

Answer: A

19. After class, there are still 2 female students and 1 male student left in the classroom, if they go out of the classroom in turn, the probability that the second out of the male students is

• A. A. $\frac { 1 } { 6 }$
• B. B. $\frac { 1 } { 5 }$
• C. C. $\frac { 1 } { 2 }$
• D. D. $\frac { 1 } { 3 }$

Answer: D

20. The following statement about the random variable $X$ is correct ( ). Using $X$ to represent the number of times the event $A$ occurs, the distribution of $X$ is $P ( X = k ) = p ^ { k } ( 1 - p ) ^ { n - k } , ~ k = 0,1,2 , \mathrm {~L} , n$. (back), using $X$ to denote the number of defective products in the sampled $n$, the distribution of $X$ is $P ( X = k ) = \frac { \mathrm { C } _ { M } ^ { k } \mathrm { C } _ { N - M } ^ { n - k } } { \mathrm { C } _ { N } ^ { n } }$, and the distribution of $k = m , m + 1 , m + 2 , \cdots , r$ is $n , N , M \in \mathrm {~N} ^ { * } , M \leq N , n \leq N \quad , m = \max \{ 0 , n + M - N \}$. $k = m , m + 1 , m + 2 , \cdots , r$, where $n , N , M \in \mathrm {~N} ^ { * } , M \leq N , n \leq N \quad , m = \max \{ 0 , n + M - N \}$ $r = \min \{ n , M \}$ $E ( X ) = n p$ $A$ the number of experiments required for the first success, then $P ( X = k ) = \mathrm { C } _ { n } ^ { k } p ^ { k } ( 1 - p ) ^ { n - k } , k = 1,2 , \cdots , n$

• A. A. In general, in ${ } ^ { n }$ re-Bernoulli trials, let the probability of the event $A$ occurring in each trial be $p ( 0 < p < 1 )$, and
• B. B. Suppose there are $N$ pieces in a batch, among which there are $M$ pieces of defective products, and $n$ pieces are randomly selected from $N$ pieces of products (without putting
• C. C. If the distribution of the random variable $X$ is $P ( X = 2 k ) = \mathrm { C } _ { n } ^ { k } p ^ { k } ( 1 - p ) ^ { n - k } , k = 0,1,2 , \mathrm {~L} , n$, then
• D. D. If ${ } ^ { n }$ reproduces a Bernoulli test, let the probability of each experimental event $A$ occurring be $p ( 0 < p < 1 )$, and denote the event with $X$

Answer: B

21. If the random variable $\xi _ { \text {满足 } } E ( 1 - \xi ) = 4 , D ( 1 - \xi ) = 4$ . then the following statement is correct ( )

• A. A. $E ( \xi ) = - 4 , D ( \xi ) = 4$
• B. B. $E ( \xi ) = - 3 , D ( \xi ) = 3$
• C. C. $E ( \xi ) = - 4 , D ( \xi ) = - 4$
• D. D. $E ( \xi ) = - 3 , D ( \xi ) = 4$

Answer: D

22. The Chinese Character Museum collects the essence of Chinese character samples of all times, and shows the world with detailed information the Chinese character and brilliant civilization of the Chinese nation. The museum collection of important collections are mainly divided into bronze, stone tablets, coins, pottery, jade and stoneware, oracle bones, bamboo and wood, paper, porcelain, a total of nine categories. Xiaoming went to visit the Chinese character museum, and arbitrarily selected three important collections to focus on visiting, then Xiaoming in the tablets, oracle bones, porcelain in at least one of the three types of probability of visiting ( )

• A. A. $\frac { 6 } { 7 }$
• B. B. $\frac { 16 } { 21 }$
• C. C. $\frac { 5 } { 7 }$
• D. D. $\frac { 2 } { 3 }$

Answer: B

23. It is known that the winning rate of a raffle is $\frac { 1 } { 2 }$ and that each drawing does not affect each other. Construct the series $\left\{ c _ { n } \right\}$ such that $c _ { n } = \left\{ \begin{array} { l } 1 , \text { 第 } n \text { 次中奖,} \\ - 1 , \text { 第 } n \text { 次未中奖 } \end{array} \right.$, noting $S _ { n } = c _ { 1 } + c _ { 2 } + \cdots + c _ { n } \left( n \in \mathbf { N } ^ { * } \right)$, then the probability of $\left| S _ { 5 } \right| = 1$ is ( ).

• A. A. $\frac { 5 } { 8 }$
• B. B. $\frac { 1 } { 2 }$
• C. C. $\frac { 5 } { 16 }$
• D. D. $\frac { 3 } { 4 }$

Answer: A

24. The random variable $\xi$ is known to satisfy $P ( \xi = 0 ) = 1 - p , P ( \xi = 1 ) = p$ and $0 < p < \frac { 1 } { 2 }$. Let the random variable $\eta = \xi - E ( \xi ) \mid$, then

• A. A. $E ( \eta ) < E ( \xi )$
• B. B. $E ( \eta ) > E ( \xi )$
• C. C. $E ( \eta ) = E ( \xi )$
• D. D. ${ } ^ { E ( \eta ) }$ and $^ { E ( \xi ) }$ are of indeterminate size.

Answer: B

25. If the random variable $X$ obeys the normal distribution $N \left( 2 , \sigma ^ { 2 } \right)$ and $2 P ( X \geq 3 ) = P ( 1 \leq x \leq 2 ) , P ( X < 3 ) =$

• A. A. $\frac { 1 } { 3 }$
• B. B. $\frac { 5 } { 6 }$
• C. C. $\frac { 1 } { 6 }$
• D. D. $\frac { 2 } { 3 }$

Answer: B

26. The probability that two of the three couples selected at random will participate in the interview is

• A. A. $\frac { 1 } { 6 }$
• B. B. $\frac { 1 } { 5 }$
• C. C. $\frac { 1 } { 4 }$
• D. D. $\frac { 1 } { 3 }$

Answer: B

27. A child plays a coin-flipping checkerboard game with the following rules: Flip a coin, and if heads is up, jump two squares forward, if tails is up, jump one square forward. Remember to jump to the $n$ frame may have $a _ { n }$ situation, $\left\{ a _ { n } \right\}$ of the previous $n$ sum of $S _ { n }$, then $S _ { 8 } =$, then $A$, and $B$, then $B$. FORMULA_5]]

• A. A. 56
• B. B. 68
• C. C. 87
• D. D. 95

Answer: C

28. Books are the ladder of human progress, mathematical masterpieces are even more so, "nine chapters of arithmetic", "Sun Tzu arithmetic", "Zhou Thigh arithmetic The Nine Chapters of Mathematics, Sun Tzu's Book of Arithmetic, Zhou Thighs' Book of Arithmetic, and Sea Island's Book of Arithmetic are the four most influential works in the field of mathematics in ancient China, and the Original Geometry, the Complete Works of Archimedes, and the Theory of the Conic Curve are known as the "Three Great Books of Mathematics in Ancient Greece", which represent the highest achievements in European mathematics before the Renaissance, and they have a far-reaching and wide-ranging impact on the mathematical development of the later generations. Now choose any three of these seven books. The probability that at least two of them are famous Chinese mathematical works is

• A. A. $\frac { 1 } { 7 }$
• B. B. $\frac { 18 } { 35 }$
• C. C. $\frac { 22 } { 35 }$
• D. D. $\frac { 4 } { 15 }$

Answer: C

29. Existing A, B, C, D, E 5 kinds of extracurricular reading books, a school to be randomly selected from 2 kinds of winter vacation reading as students, then A, B at least 1 kind of probability of being selected for the

• A. A. $\frac { 3 } { 10 }$
• B. B. $\frac { 1 } { 2 }$
• C. C. $\frac { 7 } { 10 }$
• D. D. $\frac { 9 } { 10 }$

Answer: C

30. Zu Chongzhi was an outstanding mathematician and astronomer during the Northern and Southern Dynasties in China. He spent his life studying natural sciences, and his main contributions were in mathematics, astronomical calendars and mechanical engineering, especially in exploring the accuracy of pi $\pi$, which was the first time that it was accurate to the seventh decimal place, i.e., $\pi$, and on the basis of which we have learned from the third to the seventh place of pi, i.e., [[[[]]]. INLINE_FORMULA_2]], based on which we randomly take two digits from the third to eighth valid digits of "Pi" $a , ~ b$, then the event "[[INLINE_FORMULA_4 ]]" has a probability of

• A. A. $\frac { 1 } { 3 }$
• B. B. $\frac { 8 } { 15 }$
• C. C. $\frac { 2 } { 3 }$
• D. D. $\frac { 7 } { 15 }$

Answer: B

31. If there are three students in class A $A , B , C$ and two students in class B $D , E$, and two students are selected to participate in a certain activity, the probability that the two selected students are from different classes is

• A. A. $\frac { 1 } { 5 }$
• B. B. $\frac { 3 } { 10 }$
• C. C. $\frac { 2 } { 5 }$
• D. D. $\frac { 3 } { 5 }$

Answer: D

33. The abacus is a traditional Chinese calculation tool, its shape rectangular, around the wooden frame, inside the straight column, commonly known as the "file", the file across the beam, two beads on the beam, each bead for the number of five, five beads under the beam, each bead for the number of one, the abacus bead beam on the upper part of the bead, beam under the part of the bead, for example, in the hundreds of files to dial a lower bead, the ten files to dial a bead on the bead and the two lower beads, then the number 170, if the number 170, the number 170, the number 170, and the number 170, and the number 170, the number 170. For example, if you dial one lower bead in the hundreds place, one upper bead and two lower beads in the tens place, the number 170 will be represented. In the individual, tens, hundreds, thousands of gears, first randomly select a gear to dial an upper bead, and then randomly select two gears to dial a lower bead, then the probability that the dialed number is greater than 500 ( ) Thousands, hundreds, and thousands of digits 

• A. A. $\frac { 1 } { 2 }$
• B. B. $\frac { 2 } { 3 }$
• C. C. $\frac { 1 } { 4 }$
• D. D. $\frac { 3 } { 4 }$

Answer: D

34. There are 410 male students in the senior class of a school, with school number $001,002 , ~ \mathrm {~L} , ~ 410$, and 290 female students, with school numbers 411, 412, L, and 700. Questionnaire survey of senior students, according to the school number using a systematic sampling method, from the 700 students selected 10 people to carry out the questionnaire survey (the first group of simple random sampling, the number drawn to 030); and then from the 10 students randomly selected 3 people to analyze the data, then the probability of these 3 people are both boys and girls are

• A. A. $\frac { 1 } { 5 }$
• B. B. $\frac { 3 } { 10 }$
• C. C. $\frac { 7 } { 10 }$
• D. D. $\frac { 4 } { 5 }$

Answer: D

35. In the People's Teaching Society of Mathematics, A version of the compulsory one of the editor's message, "learning mathematics while young" this sentence touched the heart of the Jiangxia Experimental High School students, if the six words are arranged arbitrarily, exactly the composition of the "while young to learn mathematics" probability of

• A. A. $\frac { 1 } { 6 \times 5 \times 4 \times 3 }$
• B. B. $\frac { 1 } { 6 \times 5 \times 4 \times 3 \times 2 \times 1 }$
• C. C. $\frac { 1 } { 6 ^ { 4 } }$
• D. D. $\frac { 2 } { 6 ^ { 6 } }$

Answer: A

There is a box with 1 red ball, now put ${ } ^ { n } \left( n \in N ^ { * } \right)$ black balls into the box, and then take a ball from the box at random, the number of red balls is ${ } ^ { \xi }$, then as ${ } ^ { n } \left( n \in N ^ { * } \right)$ increases, the following statement is correct

• A. A. $E ( \xi ) _ { \text {减小 } } , D ( \xi ) _ { \text {增加 } }$
• B. B. $E ( \xi ) _ { \text {增加 } } , D ( \xi ) _ { \text {减小 } }$
• C. C. $E ( \xi ) _ { \text {增加,} } D ( \xi ) _ { \text {增加 } }$
• D. D. $E ( \xi ) _ { \text {减小 } } , D ( \xi ) _ { \text {减小 } }$

Answer: D

37. If the random variables $\xi \sim B ( 12 , p )$ and $E ( 2 \xi - 3 ) = 5$ are known, then $D ( 3 \xi ) _ { \text {等于( )} }$

• A. A. 24
• B. B. 36
• C. C. 48
• D. D. 72

Answer: A

38. The quality of the product is the root of the enterprise, product testing is an indispensable part of the production of important work, a factory in order to ensure product quality, the use of two different methods of testing, two employees randomly from the production line to extract the same number of a batch of products, known as a batch of products in two people extracted are 5 pieces of defective products, employee A from the batch of products in the return to randomly extract 3 pieces of products, employee B from the batch of products without return to randomly select 3 pieces of products. A batch of products from this batch of products in the non-return to randomly selected 3 products, let employee A extracted to the number of defective 3 products for $X$, employee B extracted to the number of defective 3 products for $Y , k = 0,1,2,3$, then the following The incorrect judgment is ( ) (Ref: hypergeometric distribution with its mean $E ( X ) = \frac { n M } { N }$)

• A. A. The random variable $X$ obeys the binomial distribution
• B. B. Random variable $Y$ obeys a hypergeometric distribution
• C. C. $E ( X ) = E ( Y )$
• D. D. $P ( X = k ) < P ( Y = k )$

Answer: D

39. The probability that a person who spells the English word "every" with one letter $\mathrm { v } , \mathrm { r } , \mathrm { y }$ and two letters e each is ( )

• A. A. $\frac { 119 } { 120 }$
• B. B. $\frac { 9 } { 10 }$
• C. C. $\frac { 19 } { 20 }$
• D. D. $\frac { 59 } { 60 }$

Answer: D

40. The following four propositions: (1) from the uniform transmission of products on the production line, every 30 minutes from which a product is taken for testing, such sampling is stratified sampling; (2) a city conducted a city high school boys height statistics survey, the data show that a city 30,000 high school boys height $\xi$ (unit: $c m )$ obeys the normal distribution $N \left( 172 , \sigma ^ { 2 } \right)$ and $P ( 172 < \xi \leq 180 ) = 0.4$, then the number of high school boys in the city whose heights are higher than $180 c m$ is about 3000; (3) The random quantity $X$ obeys the binomial distribution $B ( 100,0.4 )$, and if the random variable $Y = 2 X + 1$, then the mathematical expectation of $Y$ is $E ( Y ) = 81$ and the variance is $D ( Y ) = 48$; (4) The categorical variables $X$ and $Y$, and their random variables $K ^ { 2 }$ have the observation $k$, and when $k$ is smaller the smaller the $k$, the greater the certainty that $X$ is related to $Y$, the greater the number of correct statements ( ). High School Mathematics Assignment, October 29, 2025

• A. A. 1
• B. B. 2
• C. C. 3
• D. D. 4

Answer: A