Two newspapers $A$ and $B$ are published in a city. It is known that $25 \%$ of the city population reads $A$ and $20 \%$ reads $B$ while $8 \%$ reads both $A$ and $B$. Further, 30\% of those who read $A$ but not $B$ look into advertisements and $40 \%$ of those who read $B$ but not $A$ also look into advertisements, while $50 \%$ of those who read both $A$ and $B$ look into advertisements. Then the percentage of the population who look into advertisements is:
(1) 13.5
(2) 12.8
(3) 13.9
(4) 13

If $R = \{(x, y) : x, y \in Z, x^{2} + 3y^{2} \leq 8\}$ is a relation on the set of integers $Z$, then the domain of $R^{-1}$ is
(1) $\{-2, -1, 1, 2\}$
(2) $\{0, 1\}$
(3) $\{-2, -1, 0, 1, 2\}$
(4) $\{-1, 0, 1\}$

In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of 6 before $B$ throws a total of 7 and $B$ wins the game if he throws a total of 7 before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is:
(1) $\frac { 5 } { 31 }$
(2) $\frac { 31 } { 61 }$
(3) $\frac { 5 } { 6 }$
(4) $\frac { 30 } { 61 }$

Let $A$ be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of $A$ leaves remainder 2 when divided by 5 is:
(1) $\frac { 1 } { 5 }$
(2) $\frac { 122 } { 297 }$
(3) $\frac { 97 } { 297 }$
(4) $\frac { 2 } { 9 }$

Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be $\frac { 1 } { 2 }$ and probability of occurrence of 0 at the odd place be $\frac { 1 } { 3 }$. Then the probability that 10 is followed by 01 is equal to:
(1) $\frac { 1 } { 18 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 1 } { 9 }$

Let $A , B$ and $C$ be three events such that the probability that exactly one of $A$ and $B$ occurs is $( 1 - k )$, the probability that exactly one of $B$ and $C$ occurs is $( 1 - 2k )$, the probability that exactly one of $C$ and $A$ occurs is $( 1 - k )$ and the probability of all $A , B$ and $C$ occur simultaneously is $k ^ { 2 }$, where $0 < k < 1$. Then the probability that at least one of $A , B$ and $C$ occur is:
(1) greater than $\frac { 1 } { 8 }$ but less than $\frac { 1 } { 4 }$
(2) greater than $\frac { 1 } { 2 }$
(3) greater than $\frac { 1 } { 4 }$ but less than $\frac { 1 } { 2 }$
(4) exactly equal to $\frac { 1 } { 2 }$

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8 . The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is $p$, then $98p$ is equal to

Let $E _ { 1 } , E _ { 2 } , E _ { 3 }$ be three mutually exclusive events such that $P \left( E _ { 1 } \right) = \frac { 2 + 3 p } { 6 } , P \left( E _ { 2 } \right) = \frac { 2 - p } { 8 }$ and $P \left( E _ { 3 } \right) = \frac { 1 - p } { 2 }$. If the maximum and minimum values of $p$ are $p _ { 1 }$ and $p _ { 2 }$ then $\left( p _ { 1 } + p _ { 2 } \right)$ is equal to:
(1) $\frac { 2 } { 3 }$
(2) $\frac { 5 } { 3 }$
(3) $\frac { 5 } { 4 }$
(4) 1

Let $S$ be the sample space of all five digit numbers. If $p$ is the probability that a randomly selected number from $S$, is a multiple of 7 but not divisible by 5 , then $9 p$ is equal to
(1) 1.0146
(2) 1.2085
(3) 1.0285
(4) 1.1521

Let $A$ and $B$ be two events such that $P ( B \mid A ) = \frac { 2 } { 5 } , P ( A \mid B ) = \frac { 1 } { 7 }$ and $P ( A \cap B ) = \frac { 1 } { 9 }$. Consider $( S1 )\; P \left( A ^ { \prime } \cup B \right) = \frac { 5 } { 6 }$, $( S2 )\; P \left( A ^ { \prime } \cap B ^ { \prime } \right) = \frac { 1 } { 18 }$. Then
(1) Both $(S1)$ and $(S2)$ are true
(2) Both $(S1)$ and $(S2)$ are false
(3) Only $(S1)$ is true
(4) Only $(S2)$ is true

Let $S = \left\{ E _ { 1 } , E _ { 2 } \ldots E _ { 8 } \right\}$ be a sample space of a random experiment such that $P \left( E _ { n } \right) = \frac { n } { 36 }$ for every $n = 1,2 \ldots 8$. Then the number of elements in the set $\left\{ A \subset S : P ( A ) \geq \frac { 4 } { 5 } \right\}$ is $\_\_\_\_$.

Let $A = \{ 0 , 3 , 4 , 6 , 7 , 8 , 9 , 10 \}$ and $R$ be the relation defined on $A$ such that $R = \{ ( x , y ) \in A \times A : x - y$ is odd positive integer or $x - y = 2 \}$. The minimum number of elements that must be added to the relation $R$, so that it is a symmetric relation, is equal to $\_\_\_\_$

Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^N < N!$ is $\frac{m}{n}$ where $m$ and $n$ are coprime, then $4m - 3n$ is equal to
(1) 6
(2) 12
(3) 10
(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 M be the maximum value of the product of two positive integers when their sum is 66 . Let the sample space $S = \left\{ x \in Z : x ( 66 - x ) \geq \frac { 5 } { 9 } M \right\}$ and the event $\mathrm { A } = \{ \mathrm { x } \in \mathrm { S } : \mathrm { x }$ is a multiple of $3 \}$. Then $\mathrm { P } ( \mathrm { A } )$ is equal to
(1) $\frac { 15 } { 44 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 7 } { 22 }$

Let $S = \left\{ w _ { 1 } , w _ { 2 } , \ldots \right\}$ be the sample space associated to a random experiment. Let $P \left( w _ { n } \right) = \frac { P \left( w _ { n - 1 } \right) } { 2 } , n \geq 2$ . Let $A = \{ 2 k + 3 l ; k , l \in \mathbb { N } \}$ and $B = \left\{ w _ { n } ; n \in A \right\}$. Then $P ( B )$ is equal to (1) $\frac { 3 } { 32 }$ (2) $\frac { 3 } { 64 }$ (3) $\frac { 1 } { 16 }$ (4) $\frac { 1 } { 32 }$

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $p$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is $q$. If $p : q = m : n$, where $m$ and $n$ are co-prime, then $m + n$ is equal to

If $R$ is the smallest equivalence relation on the set $\{ 1,2,3,4 \}$ such that $\{ ( 1,2 ) , ( 1,3 ) \} \subset R$, then the number of elements in $R$ is
(1) 10
(2) 12
(3) 8
(4) 15

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

Let the relations $R _ { 1 }$ and $R _ { 2 }$ on the set $X = \{ 1,2,3 , \ldots , 20 \}$ be given by $R _ { 1 } = \{ ( x , y ) : 2 x - 3 y = 2 \}$ and $R _ { 2 } = \{ ( x , y ) : - 5 x + 4 y = 0 \}$. If $M$ and $N$ be the minimum number of elements required to be added in $R _ { 1 }$ and $R _ { 2 }$, respectively, in order to make the relations symmetric, then $M + N$ equals
(1) 12
(2) 16
(3) 8
(4) 10

Let Ajay will not appear in JEE exam with probability $p = \frac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $q = \frac{1}{5}$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is:
(1) $\frac{9}{35}$
(2) $\frac{18}{35}$
(3) $\frac{24}{35}$
(4) $\frac{3}{35}$

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) $5 / 54$