The probability mass function of a discrete random variable $X$ is $$\mathrm { P } ( X = x ) = \frac { a x + 2 } { 10 } ( x = - 1,0,1,2 )$$ What is the value of the variance $\mathrm { V } ( 3 X + 2 )$ of the random variable $3 X + 2$? (where $a$ is a constant.) [3 points]
(1) 9
(2) 18
(3) 27
(4) 36
(5) 45

The probability distribution of a random variable $X$ is shown in the table below.

$X$	0	1	2	Total
$\mathrm { P } ( X = x )$	$\frac { 1 } { 4 }$	$a$	$2a$	1

What is the value of $\mathrm { E } ( 4X + 10 )$? [3 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15

There are 5 drawers, each labeled with a natural number from 1 to 5. Two drawers are randomly assigned to Younghee. Let $X$ be the random variable representing the smaller of the two natural numbers on the assigned drawers. Find the value of $\mathrm { E } ( 10 X )$. [4 points]

A bag contains 1 ball with the number 1, 2 balls with the number 2, and 5 balls with the number 3. One ball is randomly drawn from the bag, the number on the ball is confirmed, and then it is returned. This trial is repeated 2 times. Let $\bar { X }$ be the average of the numbers on the drawn balls. What is the value of $\mathrm { P } ( \bar { X } = 2 )$? [4 points]
(1) $\frac { 5 } { 32 }$
(2) $\frac { 11 } { 64 }$
(3) $\frac { 3 } { 16 }$
(4) $\frac { 13 } { 64 }$
(5) $\frac { 7 } { 32 }$

The probability distribution of a discrete random variable $X$ is shown in the table below.

$X$	- 5	0	5	Total
$\mathrm { P } ( X = x )$	$\frac { 1 } { 5 }$	$\frac { 1 } { 5 }$	$\frac { 3 } { 5 }$	1

Find the value of $\mathrm { E } ( 4 X + 3 )$. [3 points]

The probability distribution of the random variable $X$ is shown in the table below.

$X$	0.121	0.221	0.321	Total
$\mathrm { P } ( X = x )$	$a$	$b$	$\frac { 2 } { 3 }$	1

The following is the process of finding $\mathrm { V } ( X )$ when $\mathrm { E } ( X ) = 0.271$. Let $Y = 10 X - 2.21$. The probability distribution of the random variable $Y$ is shown in the table below.

$Y$	$-1$	0	1	Total
$\mathrm { P } ( Y = y )$	$a$	$b$	$\frac { 2 } { 3 }$	1

Since $\mathrm { E } ( Y ) = 10 \mathrm { E } ( X ) - 2.21 = 0.5$, $a =$ (가), $b =$ (나) and $\mathrm { V } ( Y ) = \frac { 7 } { 12 }$. On the other hand, since $Y = 10 X - 2.21$, we have $\mathrm { V } ( Y ) =$ (다) $\times \mathrm { V } ( X )$. Therefore, $\mathrm { V } ( X ) = \frac { 1 } { \text{(다)} } \times \frac { 7 } { 12 }$. When the values in (가), (나), and (다) are $p$, $q$, and $r$ respectively, find the value of $pqr$. (Here, $a$ and $b$ are constants.) [4 points]
(1) $\frac { 13 } { 9 }$
(2) $\frac { 16 } { 9 }$
(3) $\frac { 19 } { 9 }$
(4) $\frac { 22 } { 9 }$
(5) $\frac { 25 } { 9 }$

There are 6 weights of 1 unit, 3 weights of 2 units, and 1 empty bag. Using one die, the following trial is performed. (Here, the unit of weight is g.)
Roll the die once. If the number shown is 2 or less, put one weight of 1 unit into the bag. If the number shown is 3 or more, put one weight of 2 units into the bag.
Repeat this trial until the total weight of the weights in the bag is first greater than or equal to 6. Let $X$ be the random variable representing the number of weights in the bag. The following is the process of finding the probability mass function $\mathrm { P } ( X = x ) ( x = 3,4,5,6 )$ of $X$.
(i) The event $X = 3$ is the case where 3 weights of 2 units are in the bag, so $$\mathrm { P } ( X = 3 ) = \text{ (a) }$$ (ii) The event $X = 4$ can be divided into the case where the total weight of weights put in by the third trial is 4 and a weight of 2 units is put in on the fourth trial, and the case where the total weight of weights put in by the third trial is 5. Therefore, $$\mathrm { P } ( X = 4 ) = \left( \text{ (b) } + { } _ { 3 } \mathrm { C } _ { 1 } \left( \frac { 1 } { 3 } \right) ^ { 1 } \left( \frac { 2 } { 3 } \right) ^ { 2 } \right) \times \frac { 2 } { 3 }$$ (iii) The event $X = 5$ can be divided into the case where the total weight of weights put in by the fourth trial is 4 and a weight of 2 units is put in on the fifth trial, and the case where the total weight of weights put in by the fourth trial is 5. Therefore, $$\mathrm { P } ( X = 5 ) = { } _ { 4 } \mathrm { C } _ { 4 } \left( \frac { 1 } { 3 } \right) ^ { 4 } \left( \frac { 2 } { 3 } \right) ^ { 0 } \times \frac { 2 } { 3 } + \text{ (c) }$$ (iv) The event $X = 6$ is the case where the total weight of weights put in by the fifth trial is 5, so $$\mathrm { P } ( X = 6 ) = \left( \frac { 1 } { 3 } \right) ^ { 5 }$$ If the values corresponding to (a), (b), (c) are $a , b , c$ respectively, what is the value of $\frac { a b } { c }$? [4 points]
(1) $\frac { 4 } { 9 }$
(2) $\frac { 7 } { 9 }$
(3) $\frac { 10 } { 9 }$
(4) $\frac { 13 } { 9 }$
(5) $\frac { 16 } { 9 }$

A die is rolled 5 times, and let $a$ be the number of times an odd number appears. A coin is tossed 4 times, and let $b$ be the number of times heads appears. If the probability that $a - b = 3$ is $\frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [3 points]

Point P is at the origin of the coordinate plane. The following trial is performed using one die.
When the die is rolled and the number shown is
2 or less, point P is moved 3 units in the positive direction of the $x$-axis,
3 or more, point P is moved 1 unit in the positive direction of the $y$-axis.
This trial is repeated 15 times, and the distance between the moved point P and the line $3 x + 4 y = 0$ is the random variable $X$. What is the value of $\mathrm { E } ( X )$? [4 points]
(1) 13
(2) 15
(3) 17
(4) 19
(5) 21

There is a bag containing 5 balls with the numbers $3,3,4,4,4$ written on them, one each. Using this bag and one die, a trial is performed to obtain a score according to the following rule.
A ball is randomly drawn from the bag. If the number on the drawn ball is 3, the die is rolled 3 times and the sum of the three numbers shown is the score. If the number on the drawn ball is 4, the die is rolled 4 times and the sum of the four numbers shown is the score.
What is the probability that the score obtained from one trial is 10 points? [4 points]
(1) $\frac { 13 } { 180 }$
(2) $\frac { 41 } { 540 }$
(3) $\frac { 43 } { 540 }$
(4) $\frac { 1 } { 12 }$
(5) $\frac { 47 } { 540 }$

A bag contains 5 balls labeled with the numbers $3, 3, 4, 4, 4$, one each. Using this bag and one die, a trial is performed to obtain a score according to the following rule:
If the ball drawn from the bag is labeled 3, roll the die 3 times and the sum of the three results is the score. If the ball drawn from the bag is labeled 4, roll the die 4 times and the sum of the four results is the score.
What is the probability that the score obtained from one trial is 10 points? Express this as $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [1 point]

When 4 coins are tossed simultaneously, let $X$ be the random variable representing the number of coins showing heads. Define the discrete random variable $Y$ as $$Y = \begin{cases} X & (\text{if } X \text{ takes the value } 0 \text{ or } 1) \\ 2 & (\text{if } X \text{ takes a value of } 2 \text{ or more}) \end{cases}$$ Find the value of $\mathrm{E}(Y)$. [3 points]
(1) $\frac{25}{16}$
(2) $\frac{13}{8}$
(3) $\frac{27}{16}$
(4) $\frac{7}{4}$
(5) $\frac{29}{16}$

There is a bag containing 5 cards with the numbers $1, 3, 5, 7, 9$ written on them, one number per card. A trial is performed by randomly drawing one card from the bag, confirming the number on the card, and putting it back. This trial is repeated 3 times, and let $\bar{X}$ be the average of the three numbers confirmed. When $\mathrm{V}(a\bar{X} + 6) = 24$, what is the value of the positive number $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

A discrete random variable $X$ takes values that are integers from 0 to 4, and $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { | 2 x - 1 | } { 12 } & ( x = 0,1,2,3 ) \\ a & ( x = 4 ) \end{array} \right.$$ What is the value of $\mathrm { V } \left( \frac { 1 } { a } X \right)$? (Here, $a$ is a nonzero constant.) [3 points]
(1) 36
(2) 39
(3) 42
(4) 45
(5) 48

16. (This question is worth 13 points) Groups $A$ and $B$ each have 7 patients. Their recovery time (in days) after taking a certain drug is recorded as follows: Group A: $10,11,12,13,14,15,16$ Group B: $12,13,15,16,17,14 , a$ Assume that the recovery times of all patients are mutually independent. Randomly select 1 person from each of groups A and B. The person selected from group A is denoted as patient 甲, and the person selected from group B is denoted as patient 乙. (I) Find the probability that the recovery time of patient 甲 is at least 14 days; (II) If $a = 25$, find the probability that the recovery time of patient 甲 is longer than that of patient 乙; (III) For what value of $a$ are the variances of recovery times for groups A and B equal? (Proof of the conclusion is not required)

16. A bank stipulates that if a bank card has 3 incorrect password attempts in one day, the card will be locked. Xiaowang went to the bank to withdraw money and found that he forgot his bank card password, but he is certain that the correct password is one of his 6 commonly used passwords. Xiaowang decides to randomly select one without replacement to try. If the password is correct, he stops trying; otherwise, he continues trying until the card is locked.
(1) Find the probability that Xiaowang's bank card is locked that day;
(2) Let $X$ denote the number of password attempts Xiaowang makes that day. Find the probability distribution of $X$ and its mathematical expectation.

To promote the development of table tennis, a certain table tennis competition allows athletes from different associations to form teams. There are 3 athletes from Association A, of which 2 are seeded players, and 5 athletes from Association B, of which 3 are seeded players. Randomly select 4 people from these 8 athletes to participate in the competition.
(I) Let A be the event ``exactly 2 seeded players are selected, and these 2 seeded players are from the same association''. Find the probability of this event.
(II) Let X be the number of seeded players among the 4 selected people. Find the probability distribution and mathematical expectation of the random variable X.

18. (This question is worth 12 points)
A shopping mall is holding a promotional lottery. After purchasing goods of a certain amount, customers can participate in a lottery. Each lottery involves randomly drawing one ball from box A (containing 4 red balls and 6 white balls) and one ball from box B (containing 5 red balls and 5 white balls). If both balls drawn are red, the customer wins the first prize; if exactly one ball is red, the customer wins the second prize; if neither ball is red, the customer wins no prize.
(1) Find the probability that a customer wins a prize in one lottery;
(2) If a customer has 3 lottery chances, let X denote the number of times the customer wins the first prize in the 3 lotteries. Find the probability distribution and mathematical expectation of X.

19. (12 points) The one-way driving time T between a school's new and old campuses depends only on road conditions. A sample of 100 observations was collected with the following results:

T (minutes)

25

30

35

40

Frequency

20

30

40

10

(I) Find the probability distribution of T and the mathematical expectation ET; (II) Professor Liu drives from the old campus to the new campus for a 50-minute lecture, then immediately returns to the old campus. Find the probability that the total time from leaving the old campus to returning is no more than 120 minutes.

13. The defect rate of a batch of products is 0.02. Drawing one item at a time with replacement from this batch, 100 times total, let $X$ denote the number of defective items drawn. Then $D X = $ ______

15. Teams A and B are playing a best-of-seven basketball series (the series ends when one team wins four games). Based on previous results, Team A's home and away arrangement is ``home, home, away, away, home, away, home'' in order. The probability that Team A wins at home is 0.6, and the probability that Team A wins away is 0.5. Each game is independent. The probability that Team A wins 4-1 is $\_\_\_\_$.

15. Teams A and B are in a basketball championship series using a best-of-seven format (the series ends when one team wins four games). Based on previous results, Team A's home/away schedule is ``home, home, away, away, home, away, home'' in order. Team A's probability of winning at home is 0.6, and away is 0.5. Each game is independent. The probability that Team A wins 4-1 is $\_\_\_\_$ .

18. (12 points) In an 11-point table tennis match, each point won scores 1 point. When the score reaches 10:10, players alternate serves, and the first player to score 2 more points wins the match. Two students, A and B, play a singles match. Assume that when A serves, A scores with probability 0.5; when B serves, A scores with probability 0.4. The results of each point are independent. After a certain match reaches 10:10 with A serving first, the two players play $X$ more points before the match ends.
(1) Find $P ( X = 2 )$;
(2) Find the probability of the event ``$X = 4$ and A wins''.

0-1 periodic sequences have important applications in communication technology. If a sequence $a _ { 1 } a _ { 2 } \cdots a _ { n } \cdots$ satisfies $a _ { i } \in \{ 0,1 \} ( i = 1,2 , \cdots )$ and there exists a positive integer $m$ such that $a _ { i + m } = a _ { i } ( i = 1,2 , \cdots )$ , then it is called a 0-1 periodic sequence, and the smallest positive integer $m$ satisfying $a _ { i + m } = a _ { i } ( i = 1,2 , \cdots )$ is called the period of this sequence. For a 0-1 sequence $a _ { 1 } a _ { 2 } \cdots a _ { n } \cdots$ with period $m$ , $C ( k ) = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } a _ { i } a _ { i + k } ( k = 1,2 , \cdots , m - 1 )$ is an important index describing its properties. Among the following 0-1 sequences with period 5, the one satisfying $C ( k ) \leqslant \frac { 1 } { 5 } ( k = 1,2,3,4 )$ is
A． $11010 \ldots$
B． $11011 \cdots$
C． $10001 \cdots$
D． $11001 \cdots$

A factory accepted a processing contract. The processed products (unit: pieces) are classified into four grades: A, B, C, and D according to standards. According to the contract: for grade A, B, and C products, the customer pays processing fees of 90 yuan, 50 yuan, and 20 yuan per piece respectively; for grade D products, the factory must compensate 50 yuan per piece for raw material loss. The factory has two branch factories, Factory A and Factory B, that can undertake the processing contract. Factory A has a processing cost of 25 yuan per piece, and Factory B has a processing cost of 20 yuan per piece. To decide which branch factory should undertake the contract, the factory conducted trial processing of 100 pieces of this product at each branch factory and recorded the grades of these products, as shown below:
Frequency Distribution Table of Product Grades for Factory A:

Grade	A	B	C	D
Frequency	40	20	20	20

Frequency Distribution Table of Product Grades for Factory B:

Grade	A	B	C	D
Frequency	28	17	34	21

(1) Estimate the probability that a product from Factory A and Factory B respectively is grade A;
(2) Find the average profit for 100 products from Factory A and Factory B respectively. Based on average profit, which branch factory should the factory choose to undertake the contract?