## MCQ (Single Correct Answer)

Two dice are thrown independently. Let $$\mathrm{A}$$ be the event that the number appeared on the $$1^{\text {st }}$$ die is less than the number app...

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black b...

If an unbiased die, marked with $$-2,-1,0,1,2,3$$ on its faces, is thrown five times, then the probability that the product of the outcomes is positiv...

Let $$\mathrm{S} = \{ {w_1},{w_2},......\} $$ be the sample space associated to a random experiment. Let $$P({w_n}) = {{P({w_{n - 1}})} \over 2},n \ge...

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, ...

Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that $$N-2,\sqrt{3N},N+2$$ are in geometric progression...

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 \mathbb{Z}:x(66 - x...

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations
$$x + y + z = 1$$
$$2x + \mathrm{N}y ...

Let $$\Omega$$ be the sample space and $$\mathrm{A \subseteq \Omega}$$ be an event.
Given below are two statements :
(S1) : If P(A) = 0, then A = $$\p...

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and...

Let $$S=\{1,2,3, \ldots, 2022\}$$. Then the probability, that a randomly chosen number n from the set S such that $$\mathrm{HCF}\,(\mathrm{n}, 2022)=1...

Let $$\mathrm{A}$$ and $$\mathrm{B}$$ 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} \cdot...

Out of $$60 \%$$ female and $$40 \%$$ male candidates appearing in an exam, $$60 \%$$ candidates qualify it. The number of females qualifying the exam...

Let X have a binomial distribution B(n, p) such that the sum and the product of the mean and variance of X are 24 and 128 respectively. If $$P(X>n-3)=...

A six faced die is biased such that
$$3 \times \mathrm{P}($$a prime number$$)\,=6 \times \mathrm{P}($$a composite number$$)\,=2 \times \mathrm{P}(1)$$...

Let $$S$$ be the sample space of all five digit numbers. It $$p$$ is the probability that a randomly selected number from $$S$$, is a multiple of 7 bu...

Let $$X$$ be a binomially distributed random variable with mean 4 and variance $$\frac{4}{3}$$. Then, $$54 \,P(X \leq 2)$$ is equal to

The mean and variance of a binomial distribution are $$\alpha$$ and $$\frac{\alpha}{3}$$ respectively. If $$\mathrm{P}(X=1)=\frac{4}{243}$$, then $$\m...

Let $$\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3}$$ be three mutually exclusive events such that $$\mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{2+3 ...

If $$A$$ and $$B$$ are two events such that $$P(A)=\frac{1}{3}, P(B)=\frac{1}{5}$$ and $$P(A \cup B)=\frac{1}{2}$$, then $$P\left(A \mid B^{\prime}\ri...

If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes i...

If the numbers appeared on the two throws of a fair six faced die are $$\alpha$$ and $$\beta$$, then the probability that $$x^{2}+\alpha x+\beta>0$$, ...

If a random variable X follows the Binomial distribution B(5, p) such that P(X = 0) = P(X = 1), then $${{P(X = 2)} \over {P(X = 3)}}$$ is equal to :...

The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to

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 :

The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfies f(a) + 2f(b) $$-$$ f(c) = f(d) ...

The probability, that in a randomly selected 3-digit number at least two digits are odd, is

If a point A(x, y) lies in the region bounded by the y-axis, straight lines 2y + x = 6 and 5x $$-$$ 6y = 30, then the probability that y

Five numbers $${x_1},{x_2},{x_3},{x_4},{x_5}$$ are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order $$...

Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of get...

A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of getting a face with mark n is $${1 \over n}$$. If the die ...

Let E1 and E2 be two events such that the conditional probabilities $$P({E_1}|{E_2}) = {1 \over 2}$$, $$P({E_2}|{E_1}) = {3 \over 4}$$ and $$P({E_1} \...

A random variable X has the following probability distribution:
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Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn fro...

If a random variable X follows the Binomial distribution B(33, p) such that $$3P(X = 0) = P(X = 1)$$, then the value of $${{P(X = 15)} \over {P(X = 18...

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is :...

Let S = {1, 2, 3, 4, 5, 6}. Then the probability that a randomly chosen onto function g from S to S satisfies g(3) = 2g(1) is :

Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is :

When a certain biased die is rolled, a particular face occurs with probability $${1 \over 6} - x$$ and its opposite face occurs with probability $${1 ...

A fair die is tossed until six is obtained on it. Let x be the number of required tosses, then the conditional probability P(x $$\ge$$ 5 | x > 2) i...

Two fair dice are thrown. The numbers on them are taken as $$\lambda$$ and $$\mu$$, and a system of linear equationsx + y + z = 5x + 2y + 3z = $$\mu$$...

Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P (exactly one of A, B occurs) = $${5 \over 9}$$, i...

A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability.
the smallest ...

The probability that a randomly selected 2-digit number belongs to the set {n $$\in$$ N : (2n $$-$$ 2) is a multiple of 3} is equal to

Let X be a random variable such that the probability function of a distribution is given by $$P(X = 0) = {1 \over 2},P(X = j) = {1 \over {{3^j}}}(j = ...

Let 9 distinct balls be distributed among 4 boxes, B1, B2, B3 and B4. If the probability than B3 contains exactly 3 balls is $$k{\left( {{3 \over 4}} ...

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 $$\times$$ 2 matrices. The probability that such formed matr...

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 ...

Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth ...

The probability of selecting integers a$$\in$$[$$-$$ 5, 30] such that x2 + 2(a + 4)x $$-$$ 5a + 64 > 0, for all x$$\in$$R, is :

Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then...

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 $${1 ...

Two dies are rolled. If both dices have six faces numbered 1, 2, 3, 5, 7 and 11, then the probability that the sum of the numbers on the top faces is ...

Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then probability of event A is equ...

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, i...

A seven digit number is formed using digits 3, 3, 4, 4, 4, 5, 5. The probability, that number so formed is divisible by 2, is :

A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to probability of getting 9 heads, then the probability ...

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 ...

In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. The...

When a missile is fired from a ship, the probability that it is intercepted is $${1 \over 3}$$ and the probability that the missile hits the target, g...

The coefficients a, b and c of the quadratic equation, ax2 + bx + c = 0 are obtained by throwing a dice three times. The probability that this equatio...

The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is :

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd
number 2 times is equal to the probability of getting a...

The probabilities of three events A, B and C are
given by P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5.
If P(A$$ \cup $$B) = 0.8, P(A$$ \cap $$C) = 0.3, P(A$...

Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with po...

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 ...

The probability that a randomly chosen 5-digit
number is made from exactly two digits is :

A dice is thrown two times and the sum of the
scores appearing on the die is observed to be
a multiple of 4. Then the conditional probability
that the...

Let EC denote the complement of an event E.
Let E1
, E2
and E3
be any pairwise independent
events with P(E1) > 0
and P(E1 $$ \cap $$ E2 $$ \cap ...

Box I contains 30 cards numbered 1 to 30 and
Box II contains 20 cards numbered 31 to 50. A
box is selected at random and a card is drawn
from it. The ...

A random variable X has the following
probability distribution :
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If 10 different balls are to be placed in 4 distinct
boxes at random, then the probability that two
of these boxes contain exactly 2 and 3 balls is :

In a box, there are 20 cards, out of which 10
are lebelled as A and the remaining 10 are
labelled as B. Cards are drawn at random, one
after the other...

Let A and B be two events such that the
probability that exactly one of them occurs is $${2 \over 5}$$ and the probability that A or B occurs is $${1 ...

Let A and B be two independent events such
that P(A) = $${1 \over 3}$$ and P(B) = $${1 \over 6}$$. Then, which of
the following is TRUE ?

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $${{1 \over 4}}$$
. If the probability ...

An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value of k when k
consecutive heads are obtained for k = 3, 4, 5, otherw...

A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice), wins
Rs. 12 when the throw results in the sum of ...

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability
that the candidate solve any problem i...

If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle
formed with these chosen vertices is eq...

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is
more than 99% is :

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each,
then the conditional probability that all...

Four persons can hit a target correctly with
probabilities
$${1 \over 2}$$, $${1 \over 3}$$, $${1 \over 4}$$ and
$${1 \over 8}$$ respectively. if all ...

The minimum number of times one has to toss a
fair coin so that the probability of observing at least
one head is at least 90% is:

Let A and B be two non-null events such that
A $$ \subset $$ B . Then, which of the following statements
is always correct ?

In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to th...

In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then ...

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth thr...

Let S = {1, 2, . . . . . ., 20}. A subset B of S is said to be "nice", if the sum of the elements of B is 203. Then the probability that a...

A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If X be the number of white ball...

Two integers are selected at random from the set {1, 2, ...., 11}. Given that the sum of selected numbers is even, the conditional probability that bo...

If the probability of hitting a target by a shooter, in any shot, is $${1 \over 3}$$, then the minimum number of independent
shots at the target requi...

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If t...

An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and i...

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained i...

Let A, B and C be three events, which are pair-wise independent and $$\overrightarrow E $$ denotes the completement of an event E. If $$P\left( {A \ca...

Two different families A and B are blessed with equal numbe of children. There are 3 tickets to be distributed amongst the children of these families ...

A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and
this ball along with two additional balls o...

A player X has a biased coin whose probability of showing heads is p and a player Y has a fair coin. They start playing a game with their own coins an...

A box 'A' contains $$2$$ white, $$3$$ red and $$2$$ black balls. Another box 'B' contains $$4$$ white, $$2$$ red and $$3$$ black balls. If two balls a...

From a group of 10 men and 5 women, four member committees are to be formed each of which must contain at least one woman. Then the probability for th...

Let E and F be two independent events. The probability that both E and F happen is $${1 \over {12}}$$ and the probability that neither E nor F happens...

Three persons P, Q and R independently try to hit a target. I the probabilities of their hitting the target are $${3 \over 4},{1 \over 2}$$ and $${5 \...

An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :

For three events A, B and C, P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P
(Exactly one of C or A occurs) = $${1 \over 4}$$
a...

If two different numbers are taken from the set {0, 1, 2, 3, ........, 10}; then the probability that their sum as
well as absolute difference are bot...

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with
replacement, then the variance of the number of green ba...

An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is :

If A and B are any two events such that P(A) = $${2 \over 5}$$ and P (A $$ \cap $$ B) = $${3 \over {20}}$$, hen the conditional probability, P(A $$\l...

Let two fair six-faced dice $$A$$ and $$B$$ be thrown simultaneously. If $${E_1}$$ is the event that die $$A$$ shows up four, $${E_2}$$ is the event t...

If $$12$$ different balls are to be placed in $$3$$ identical boxes, then the probability that one of the boxes contains exactly $$3$$ balls is:

Let $$A$$ and $$B$$ be two events such that $$P\left( {\overline {A \cup B} } \right) = {1 \over 6},\,P\left( { {A \cap B} } \right) = {1 \over 4}$$ a...

A multiple choice examination has $$5$$ questions. Each question has three alternative answers of which exactly one is correct. The probability that a...

Three numbers are chosen at random without replacement from $$\left\{ {1,2,3,..8} \right\}.$$ The probability that their minimum is $$3,$$ given that ...

Consider $$5$$ independent Bernoulli's trials each with probability of success $$p.$$ If the probability of at least one failure is greater than or eq...

If $$C$$ and $$D$$ are two events such that $$C \subset D$$ and $$P\left( D \right) \ne 0,$$ then the correct statement among the following is

Four numbers are chosen at random (without replacement) from the set $$\left\{ {1,2,3,....20} \right\}.$$
Statement - 1: The probability that the cho...

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. ...

In a binomial distribution $$B\left( {n,p = {1 \over 4}} \right),$$ if the probability of at least one success is greater than or equal to $${9 \over...

One ticket is selected at random from $$50$$ tickets numbered $$00, 01, 02, ...., 49.$$ Then the probability that the sum of the digits on the selecte...

It is given that the events $$A$$ and $$B$$ are such that
$$P\left( A \right) = {1 \over 4},P\left( {A|B} \right) = {1 \over 2}$$ and $$P\left( {B|A}...

A die is thrown. Let $$A$$ be the event that the number obtained is greater than $$3.$$ Let $$B$$ be the event that the number obtained is less than $...

Two aeroplanes $${\rm I}$$ and $${\rm I}$$$${\rm I}$$ bomb a target in succession. The probabilities of $${\rm I}$$ and $${\rm I}$$$${\rm I}$$ scoring...

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly $$9$$ twice is

At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of $$5$$ phone calls du...

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability th...

A random variable $$X$$ has Poisson distribution with mean $$2$$.
Then $$P\left( {X > 1.5} \right)$$ equals

Let $$A$$ and $$B$$ two events such that $$P\left( {\overline {A \cup B} } \right) = {1 \over 6},$$ $$P\left( {A \cap B} \right) = {1 \over 4}$$ and $...

The mean and the variance of a binomial distribution are $$4$$ and $$2$$ respectively. Then the probability of $$2$$ successes is

The probability that $$A$$ speaks truth is $${4 \over 5},$$ while the probability for $$B$$ is $${3 \over 4}.$$ The probability that they contradict ...

Events $$A, B, C$$ are mutually exclusive events such that $$P\left( A \right) = {{3x + 1} \over 3},$$ $$P\left( B \right) = {{1 - x} \over 4}$$ and $...

The mean and variance of a random variable $$X$$ having binomial distribution are $$4$$ and $$2$$ respectively, then $$P(X=1)$$ is

Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is

A problem in mathematics is given to three students $$A,B,C$$ and their respective probability of solving the problem is $${1 \over 2},{1 \over 3}$$ a...

$$A$$ and $$B$$ are events such that $$P\left( {A \cup B} \right) = 3/4$$,$$P\left( {A \cap B} \right) = 1/4,$$
$$P\left( {\overline A } \right) = 2/...

A dice is tossed $$5$$ times. Getting an odd number is considered a success. Then the variance of distribution of success is

## Numerical

Let A be the event that the absolute difference between two randomly choosen real numbers in the sample space $[0,60]$ is less than or equal to a . If...

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...

25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer...

Three urns A, B and C contain 4 red, 6 black; 5 red, 5 black; and $$\lambda$$ red, 4 black balls respectively. One of the urns is selected at random a...

The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. Then the number of trials in the binomial dist...

A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let $$\mathrm{X}$$ be the number of white balls, among the dra...

The probability distribution of X is :
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Let S = {E1, E2, ........., E8} be a sample space of a random experiment such that $$P({E_n}) = {n \over {36}}$$ for every n = 1, 2, ........, 8. Then...

If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is p, then 96 p is equal to _________...

In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability $${3 \o...

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit...

A fair coin is tossed n-times such that the probability of getting at least one head is at least 0.9. Then the minimum value of n is ______________.

Let there be three independent events E1, E2 and E3. The probability that only E1 occurs is $$\alpha$$, only E2 occurs is $$\beta$$ and only E3 occurs...

Let Bi (i = 1, 2, 3) be three independent events in a sample space. The probability that only B1 occur is $$\alpha $$, only B2 occurs is $$\beta $$ an...

In a bombing attack, there is 50% chance that
a bomb will hit the target. Atleast two
independent hits are required to destroy the
target completely. ...

The probability of a man hitting a target is $${1 \over {10}}$$. The least number of shots required, so that the
probability of his hitting the target...