Let $A = \{ 1,2,3,4,5 \}$. Let R be a relation on A defined by $x \mathrm { R } y$ if and only if $4 x \leq 5 \mathrm { y }$. Let m be the number of elements in R and n be the minimum number of elements from $\mathrm { A } \times \mathrm { A }$ that are required to be added to R to make it a symmetric relation. Then $\mathrm { m } + \mathrm { n }$ is equal to:
(1) 25
(2) 24
(3) 26
(4) 23

An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :
(1) $\frac { 5 } { 256 }$
(2) $\frac { 5 } { 715 }$
(3) $\frac { 3 } { 715 }$
(4) $\frac { 3 } { 256 }$

Let the sum of two positive integers be 24. If the probability, that their product is not less than $\frac { 3 } { 4 }$ times their greatest possible product, is $\frac { m } { n }$, where $\operatorname { gcd } ( m , n ) = 1$, then $n - m$ equals
(1) 10
(2) 9
(3) 11
(4) 8

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:
(1) $\frac { 18 } { 25 }$
(2) $\frac { 12 } { 25 }$
(3) $\frac { 6 } { 25 }$
(4) $\frac { 4 } { 25 }$

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i ^ { \text {th} }$ roll than the number obtained in the $( i - 1 ) ^ { \text {th} }$ roll, $i = 2,3$, is equal to
(1) $3 / 54$
(2) $2 / 54$
(3) $1 / 54$
(4) $5 / 54$

The number of symmetric relations defined on the set $\{1, 2, 3, 4\}$ which are not reflexive is $\underline{\hspace{1cm}}$.

Define a relation R on the interval $\left[0, \frac{\pi}{2}\right)$ by $x\mathrm{R}y$ if and only if $\sec^2 x - \tan^2 y = 1$. Then R is:
(1) both reflexive and transitive but not symmetric
(2) an equivalence relation
(3) reflexive but neither symmetric nor transitive
(4) both reflexive and symmetric but not transitive

Two numbers $\mathrm { k } _ { 1 }$ and $\mathrm { k } _ { 2 }$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $\mathrm { i } ^ { \mathrm { k } _ { 1 } } + \mathrm { i } ^ { \mathrm { k } _ { 2 } } , ( \mathrm { i } = \sqrt { - 1 } )$ is non-zero, equals
(1) $\frac { 1 } { 2 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 2 } { 3 }$

Let $A = \left[ a_{ij} \right]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $\mathrm{P}(\mathrm{E})$ is:
(1) $\frac{3}{16}$
(2) $\frac{5}{8}$
(3) $\frac{3}{8}$
(4) $\frac{1}{8}$

Let $\mathbf { R } = \{ ( 1,2 ) , ( 2,3 ) , ( 3,3 ) \}$ be a relation defined on the set $\{ 1,2,3,4 \}$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:
(1) 10
(2) 7
(3) 8
(4) 9

A board has 16 squares as shown in the figure (a $4 \times 4$ grid of squares). Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :
(1) $7/10$
(2) $4/5$
(3) $23/30$
(4) $3/5$

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is $29 / 45$, then $n$ is equal to :
(1) 6
(2) 3
(3) 5
(4) 4

There are nine cards on which the integers from 1 to 9 are written in a box. Two cards are taken simultaneously from this box. Let $S$ denote the sum of the numbers written on the two cards.
(1) The probability that $S$ is 5 or less is $\frac { \mathbf { A } } { \mathbf { B } }$. Let us assign a score to the result $S$.
When $S$ is 5 or less the score is $10 - S$, and when it is greater than 5 the score is 2. Then the expected value of the score is $\frac { \mathbf { C D } } { \mathbf { E F } }$.
(2) Let us perform the above trial twice, returning the two cards to the box before the second trial.
(i) The probability that $S$ is 5 or less in both trials is $\frac { \mathbf { G } } { \mathbf { H } }$.
(ii) The probability that $S$ is 5 or less in at least one trial is $\frac { \mathbf { J K } } { \mathbf { L M } }$.

Consider the sets $A = \{ 4 m \mid m$ is a natural number $\}$ and $B = \{ 6 m \mid m$ is a natural number $\}$.
(1) For each of the following $\mathbf { L } \sim \mathbf{O}$, choose the correct answer from among (0) $\sim$ (3) below.
Let $n$ be a natural number.
(i) $n \in A$ is $\mathbf { L }$ for $n$ to be divisible by 2 .
(ii) $n \in B$ is $\mathbf { M }$ for $n$ to be divisible by 24 .
(iii) $n \in A \cup B$ is $\mathbf { N }$ for $n$ to be divisible by 3 .
(iv) $n \in A \cap B$ is $\square\mathbf{O}$ for $n$ to be divisible by 12 . (0) a necessary and sufficient condition
(1) a necessary condition but not a sufficient condition
(2) a sufficient condition but not a necessary condition
(3) neither a necessary condition nor a sufficient condition
(2) Let $C = \{ m \mid m$ is a natural number satisfying $1 \leqq m \leqq 100 \}$.
The number of elements which belong to $( \bar { A } \cup \bar { B } ) \cap C$ is $\mathbf { P Q }$, and the number of elements which belong to $\bar { A } \cap \bar { B } \cap C$ is $\mathbf { R S }$. Note that $\bar { A }$ and $\bar { B }$ denote the complements of $A$ and $B$, where the universal set is the set of all natural numbers.

Consider the sets $A = \{ 4 m \mid m$ is a natural number $\}$ and $B = \{ 6 m \mid m$ is a natural number $\}$.
(1) For each of the following $\mathbf { L } \sim \square \mathbf { O }$, choose the correct answer from among (0) $\sim$ (3) below.
Let $n$ be a natural number.
(i) $n \in A$ is $\mathbf { L }$ for $n$ to be divisible by 2 .
(ii) $n \in B$ is $\mathbf { M }$ for $n$ to be divisible by 24 .
(iii) $n \in A \cup B$ is $\mathbf { N }$ for $n$ to be divisible by 3 .
(iv) $n \in A \cap B$ is $\mathbf{O}$ for $n$ to be divisible by 12 . (0) a necessary and sufficient condition
(1) a necessary condition but not a sufficient condition
(2) a sufficient condition but not a necessary condition
(3) neither a necessary condition nor a sufficient condition
(2) Let $C = \{ m \mid m$ is a natural number satisfying $1 \leqq m \leqq 100 \}$.
The number of elements which belong to $( \bar { A } \cup \bar { B } ) \cap C$ is $\mathbf { P Q }$, and the number of elements which belong to $\bar { A } \cap \bar { B } \cap C$ is $\mathbf { R S }$. Note that $\bar { A }$ and $\bar { B }$ denote the complements of $A$ and $B$, where the universal set is the set of all natural numbers.

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
$$P ( B \cup C ) = \frac { \mathbf { R S } } { \mathbf { T U V } } .$$

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 $P ( D )$.

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 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 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

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).

$$\begin{aligned} & A = \{ a , b , e \} \\ & B = \{ a , b , c , d \} \end{aligned}$$
Given this, how many sets $K$ satisfy the condition $( A \cap B ) \subseteq K \subseteq ( A \cup B )$?
A) 3
B) 4
C) 5
D) 8
E) 9

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) $\frac{2}{7}$