Let $G = ( S , A ) \in \Omega _ { n }$. Determine the probability $\mathbf { P } ( \{ G \} )$ of the elementary event $\{ G \}$ in terms of $p _ { n } , q _ { n } , N$ and $a = \operatorname { card } ( A )$. Then recover the fact that $\mathbf { P } \left( \Omega _ { n } \right) = 1$.

We fix $n \in \mathbf { N } ^ { * }$ and draw successively and with replacement two integers $p$ and $q$ according to a uniform distribution on $\llbracket 1 , n \rrbracket$. We define the events:

• $A _ { n }$: "We obtain $p = q$".
• $C _ { n }$: "We obtain $p > q$".

Calculate $\mathbf { P } \left( A _ { n } \right)$ then $\mathbf { P } \left( C _ { n } \right)$.

A box $B _ { 1 }$ contains 1 white ball, 3 red balls and 2 black balls. Another box $B _ { 2 }$ contains 2 white balls, 3 red balls and 4 black balls. A third box $B _ { 3 }$ contains 3 white balls, 4 red balls and 5 black balls.
If 1 ball is drawn from each of the boxes $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$, the probability that all 3 drawn balls are of the same colour is
(A) $\frac { 82 } { 648 }$
(B) $\frac { 90 } { 648 }$
(C) $\frac { 558 } { 648 }$
(D) $\frac { 566 } { 648 }$

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is
(A) $\frac{1}{2}$
(B) $\frac{1}{3}$
(C) $\frac{2}{3}$
(D) $\frac{3}{4}$

Let $S = \left\{x \in (-\pi, \pi) : x \neq 0, \pm\frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3}\sec x + \operatorname{cosec} x + 2(\tan x - \cot x) = 0$ in the set $S$ is equal to
(A) $-\frac{7\pi}{9}$
(B) $-\frac{2\pi}{9}$
(C) $0$
(D) $\frac{5\pi}{9}$

Two players, $P _ { 1 }$ and $P _ { 2 }$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P _ { 1 }$ and $P _ { 2 }$, respectively. If $x > y$, then $P _ { 1 }$ scores 5 points and $P _ { 2 }$ scores 0 point. If $x = y$, then each player scores 2 points. If $x < y$, then $P _ { 1 }$ scores 0 point and $P _ { 2 }$ scores 5 points. Let $X _ { i }$ and $Y _ { i }$ be the total scores of $P _ { 1 }$ and $P _ { 2 }$, respectively, after playing the $i ^ { \text {th } }$ round.
List-I (I) Probability of $\left( X _ { 2 } \geq Y _ { 2 } \right)$ is (II) Probability of $\left( X _ { 2 } > Y _ { 2 } \right)$ is (III) Probability of $\left( X _ { 3 } = Y _ { 3 } \right)$ is (IV) Probability of $\left( X _ { 3 } > Y _ { 3 } \right)$ is
List-II (P) $\frac { 3 } { 8 }$ (Q) $\frac { 11 } { 16 }$ (R) $\frac { 5 } { 16 }$ (S) $\frac { 355 } { 864 }$ (T) $\frac { 77 } { 432 }$
The correct option is:
(A) (I) → (Q); (II) → (R); (III) → (T); (IV) → (S)
(B) (I) → (Q); (II) → (R); (III) → (T); (IV) → (T)
(C) (I) → (P); (II) → (R); (III) → (Q); (IV) → (S)
(D) (I) → (P); (II) → (R); (III) → (Q); (IV) → (T)

Consider the $6 \times 6$ square in the figure. Let $A _ { 1 } , A _ { 2 } , \ldots , A _ { 49 }$ be the points of intersections (dots in the picture) in some order. We say that $A _ { i }$ and $A _ { j }$ are friends if they are adjacent along a row or along a column. Assume that each point $A _ { i }$ has an equal chance of being chosen.
Two distinct points are chosen randomly out of the points $A _ { 1 } , A _ { 2 } , \ldots , A _ { 49 }$. Let $p$ be the probability that they are friends. Then the value of $7p$ is

The probability that a randomly chosen $2 \times 2$ matrix with all the entries from the set of first 10 primes, is singular, is equal to
(1) $\frac { 133 } { 10 ^ { 4 } }$
(2) $\frac { 19 } { 10 ^ { 3 } }$
(3) $\frac { 18 } { 10 ^ { 3 } }$
(4) $\frac { 271 } { 10 ^ { 4 } }$

Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N - 2 , \sqrt { 3 N } , N + 2$ are in geometric progression be $\frac { k } { 48 }$. Then the value of $k$ is
(1) 2
(2) 4
(3) 16
(4) 8

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is
(1) $\frac { 5 } { 24 }$
(2) $\frac { 2 } { 15 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 5 } { 36 }$

Let $R$ be a relation on $Z \times Z$ defined by $( a , b ) R ( c , d )$ if and only if $a d - b c$ is divisible by 5 . Then R is
(1) Reflexive and symmetric but not transitive
(2) Reflexive but neither symmetric not transitive
(3) Reflexive, symmetric and transitive
(4) Reflexive and transitive but not symmetric

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) $\frac { 4 } { 25 }$

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.

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) $5 / 54$

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) $3/5$

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

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 $\frac { 1 } { 2 }$

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) $\frac{2}{7}$

Four students of different heights line up randomly in a row.
According to this, what is the probability that the shortest and tallest students are at the ends?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 6 }$
E) $\frac { 1 } { 12 }$

Deniz randomly colored two of the following four points that are the vertices of a square red and the other two blue, and drew line segments connecting the points she colored the same color.
What is the probability that these line segments intersect?
A) $\frac { 1 } { 6 }$ B) $\frac { 1 } { 4 }$ C) $\frac { 1 } { 3 }$ D) $\frac { 2 } { 3 }$ E) $\frac { 3 } { 4 }$

In the figure, 3 of the 6 edges of a regular tetrahedron are randomly painted.
Accordingly, what is the probability that all three painted edges are on the same face?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 5 }$
E) $\frac { 1 } { 6 }$

In a cube, 6 of its 8 vertices are randomly painted white and the other 2 are painted black.
What is the probability that there is an edge with both endpoints painted black in this cube?
A) $\frac { 1 } { 7 }$
B) $\frac { 2 } { 7 }$
C) $\frac { 3 } { 7 }$
D) $\frac { 4 } { 7 }$
E) $\frac { 5 } { 7 }$

There is an ant at each of the vertices $K$ and $L$ of a regular tetrahedron.
Each of these ants starts walking along one of the edges emanating from their respective corners, chosen at random, and stops when reaching the other end of that edge.
Accordingly, what is the probability that the ants meet?
A) $\frac { 1 } { 3 }$ B) $\frac { 2 } { 3 }$ C) $\frac { 1 } { 4 }$ D) $\frac { 3 } { 4 }$ E) $\frac { 1 } { 9 }$