20. The statement " $\lim _ { x \rightarrow a } f ( x ) = L$ " means that for each $\varepsilon > 0$, there exists a $\delta > 0$ such that
(A) if $0 < | x - a | < \varepsilon$, then $| f ( x ) - L | < \delta$
(B) if $0 < | f ( x ) - L | < \varepsilon$, then $| x - a | < \delta$
(C) if $| f ( x ) - L | < \delta$, then $0 < | x - a | < \varepsilon$
(D) $\quad 0 < | x - a | < \delta$ and $| f ( x ) - L | < \varepsilon$
(E) if $0 < | x - a | < \delta$, then $| f ( x ) - L | < \varepsilon$

Suppose $A$, $B$ and $C$ are three events and $P ( A ) = a , P ( B ) = b , P ( C ) = c$ are known. Let $P ( A \cup B \cup C ) = p$. The statements below are about whether we can find the value of $p$ if we know some additional information. (Note: $\cup$ is the same as OR. Similarly $\cap$ is the same as AND.)
Statements
(33) We can find the value of $p$ if we know that at least one of $a , b , c$ is 1. (34) We can find the value of $p$ if we know that at least one of $a , b , c$ is 0. (35) We can find the value of $p$ if we know that any two of $A , B$ and $C$ are mutually exclusive. (36) We can find the value of $p$ if we know that any two of $A , B$ and $C$ are independent and we know the value of $P ( A \cap B \cap C )$.

15. A game being offered in a casino consists of guessing the outcomes of two tosses of a fair coin. The gambler wins if she/he has correctly guessed at least one of the two tosses. To play a game, the gambler has to pay a fee of Rs. 80, and the winner gets a reward of Rs. 100 on winning the game (and nothing otherwise). Which of the following statements are correct?
(a) In the first 10 minutes on a given day exactly three gamblers play the game, one after the other. The probability that the casino owner makes a profit in the first 10 minutes equals $1 / 4$.
(b) One gambler plays the game three times. The probability that she wins exactly two of the three games is $27 / 64$.
(c) Three friends go together and play the game with each playing once. The probability that all three win equals $27 / 64$.
(d) If 1200 players play the game on a given day, the expected profit of the casino owner for the day equals Rs. 6000.

When 3 coins are tossed simultaneously, let $A$ be the event that at most 1 coin shows heads, and let $B$ be the event that all 3 coins show the same face. Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $\mathrm { P } ( A ) = \frac { 1 } { 2 }$ ㄴ. $\mathrm { P } ( A \cap B ) = \frac { 1 } { 8 }$ ㄷ. Events $A$ and $B$ are independent of each other.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In information theory, when an event $E$ occurs, the information content $I ( E )$ of event $E$ is defined as follows:
$$I ( E ) = - \log _ { 2 } \mathrm { P } ( E )$$
Which of the following statements in $\langle$Remarks$\rangle$ are correct? (Note: The probability $\mathrm { P } ( E )$ of event $E$ is positive, and the unit of information content is bits.) [4 points]
$\langle$Remarks$\rangle$ ㄱ. If event $E$ is rolling an odd number on a single die, then $I ( E ) = 1$. ㄴ. If two events $A$ and $B$ are independent and $\mathrm { P } ( A \cap B ) > 0$, then $I ( A \cap B ) = I ( A ) + I ( B )$. ㄷ. For two events $A$ and $B$ with $\mathrm { P } ( A ) > 0$ and $\mathrm { P } ( B ) > 0$, we have $2 I ( A \cup B ) \leqq I ( A ) + I ( B )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In information theory, when an event $E$ occurs, the information content $I ( E )$ of the event $E$ is defined as follows. $$I ( E ) = - \log _ { 2 } \mathrm { P } ( E )$$ Which of the following are correct? Select all that apply from . (Note: the probability that event $E$ occurs, $\mathrm { P } ( E )$, is positive, and the unit of information content is bits.) [4 points]
ㄱ. If event $E$ is rolling one die and getting an odd number, then $I ( E ) = 1$. ㄴ. If two events $A , B$ are independent and $\mathrm { P } ( A \cap B ) > 0$, then $I ( A \cap B ) = I ( A ) + I ( B )$. ㄷ. For two events $A , B$ with $\mathrm { P } ( A ) > 0 , \mathrm { P } ( B ) > 0$, we have $2 I ( A \cup B ) \leqq I ( A ) + I ( B )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

2. Let $A$ and $B$ be two sets. Then ``$A \cap B = A$'' is ``$A \subseteq B$'' a
A. sufficient but not necessary condition
B. necessary but not sufficient condition
C. necessary and sufficient condition
D. neither sufficient nor necessary condition

8. B

Solution: Let the probabilities of events A, B, C, D be $P ( a ) , P ( b ) , P ( c ) , P ( d )$ respectively. Then $P ( a ) = \frac { 1 } { 6 } , P ( b ) = \frac { 5 } { 36 } , P ( d ) = \frac { 1 } { 36 }$. The probability that A and C occur simultaneously is $P ( a c ) = 0$; the probability that A and D occur simultaneously is $P ( a d ) = 0$; the probability that B and C occur simultaneously is $P ( b c ) = \frac { 1 } { 36 }$; the probability that C and D occur simultaneously is $P ( c d ) = 0$. The condition $P ( x y ) = P ( x ) P ( y )$ is satisfied by option B.

II. Multiple Selection Questions

Consider the system of equations $a x + b y = 0 , c x + d y = 0$, where $a , b , c , d \in \{ 0,1 \}$. STATEMENT-1 : The probability that the system of equations has a unique solution is $\frac { 3 } { 8 }$. and STATEMENT-2 : The probability that the system of equations has a solution is 1.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Q68. Let $A = \{ 2,3,6,8,9,11 \}$ and $B = \{ 1,4,5,10,15 \}$. Let $R$ be a relation on $A \times B$ defined by ( $a , b$ ) $R ( c , d )$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is
(1) an equivalence relation.
(2) reflexive and symmetric but not transitive.
(3) transitive but not symmetric.
(4) reflexive but not symmetric.

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 $\frac { 1 } { 100 }$

7. Which of the following options contain rational numbers?
(1) 3.1416
(2) $\sqrt{3}$
(3) $\log_{10}\sqrt{5} + \log_{10}\sqrt{2}$
(4) $\frac{\sin 15^{\circ}}{\cos 15^{\circ}} + \frac{\cos 15^{\circ}}{\sin 15^{\circ}}$
(5) The unique real root of the equation $x^{3} - 2x^{2} + x - 1 = 0$

A person's probability of hitting a dart each time is $\frac { 1 } { 2 }$, and the results of each dart throw are independent. From the following options, select the events with probability $\frac { 1 } { 2 }$.
(1) Throwing darts 2 times consecutively, hitting exactly 1 time
(2) Throwing darts 4 times consecutively, hitting exactly 2 times
(3) Throwing darts 4 times consecutively, the total number of hits is odd
(4) Throwing darts 6 times consecutively, given that the first throw misses, the second throw hits
(5) Throwing darts 6 times consecutively, given that exactly 1 hit in the first 2 throws, exactly 2 hits in the last 4 throws

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