How many questions from Probability appear in JEE?

Probability has appeared in 176 of the last 186 JEE Main and JEE Advanced papers — about 95% of them — contributing 226 questions in total across those papers.

Is Probability an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 95% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Probability questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Probability JEE Previous Year Questions with Answers

Q1·MathematicsMultiple correctJEE Advanced 2026

Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let

E_1

be the event that the ball chosen belonged to Box I and let

E_2

be the event that the ball chosen belonged to Box II. Let

F_1

be the event that the ball chosen is red and let

F_2

be the event that the ball chosen is green. Then which of the following statements is (are) TRUE ?

(A)The events $E_1$ and $F_1$ are independent
(B)The events $E_2$ and $F_2$ are dependent
(C)The conditional probability $P(F_1 \mid E_1)$ is equal to the conditional probability $P(F_1 \mid E_2)$
(D)The conditional probability $P(F_1 \mid E_1)$ is greater than the conditional probability $P(F_2 \mid E_2)$

Q2·MathematicsNumericalJEE Advanced 2026

A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let

X

be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If

\alpha

is the mean of the random variable

X

, then the value of

77\alpha

is _____.

Q3·MathematicsSingle correctJEE Main 2026

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are

\frac{2}{5}

,

\frac{1}{5}

and

\frac{2}{5}

. The probabilities that the candidate reaches late at the examination centre are

\frac{1}{5}

,

\frac{1}{3}

and

\frac{1}{4}

if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:

(A) $\frac{11}{37}$
(B) $\frac{12}{37}$
(C) $\frac{13}{37}$
(D) $\frac{14}{37}$

Q4·MathematicsSingle correctJEE Main 2026

A bag contains (N + 1) coins − N fair coins, and one coin with 'Head' on both sides. A coin is selected at random and tossed. If the probability of getting 'Head' is

\frac{9}{16}

, then N is equal to:

(A)5
(B)7
(C)8
(D)9

Q5·MathematicsSingle correctJEE Main 2026

A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is :

(A) $\frac{63}{925}$
(B) $\frac{17}{231}$
(C) $\frac{16}{231}$
(D) $\frac{64}{925}$

Q6·MathematicsSingle correctJEE Main 2026

A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:

(A) $\frac{7}{10}$
(B) $\frac{10}{17}$
(C) $\frac{12}{19}$
(D) $\frac{7}{19}$

Q7·MathematicsSingle correctJEE Main 2026

The probabilities that players A and B of a team are selected for the captaincy for a tournament are 0.6 and 0.4, respectively. If A is selected the captain, the probability that the team wins the tournament is 0.8 and if B is selected the captain, the probability that the team wins the tournament is 0.7. Then the probability, that the team wins the tournament, is :

(A)0.74
(B)0.76
(C)0.72
(D)0.78

Q8·MathematicsNumericalJEE Main 2026

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to

\frac{a}{b}

, where

a, b \in \mathbb{N}

and

\gcd(a, b) = 1

, then

a + b

is equal to ______

Q9·MathematicsNumericalJEE Main 2026

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is

p

, then

96p

is equal to _______.

Q10·MathematicsNumericalJEE Main 2026

Let

a, b, c \in \{1, 2, 3, 4\}

. If the probability, that

ax^{2} + 2\sqrt{2}\,bx + c > 0

for all

x \in \mathbb{R}

, is

\frac{m}{n}

,

\gcd(m, n) = 1

, then

m + n

is equal to ________.

Q11·MathematicsSingle correctJEE Main 2026

A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is

\frac{m}{n}

,

\gcd(m, n) = 1

, then

m + n

is equal to:

(A) $53$
(B) $55$
(C) $107$
(D) $105$

Q12·MathematicsSingle correctJEE Main 2026

The probability distribution of a random variable X is given below :

\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline X & 4k & \frac{30}{7}k & \frac{32}{7}k & \frac{34}{7}k & \frac{36}{7}k & \frac{38}{7}k & \frac{40}{7}k & 6k \\ \hline P(X) & \frac{2}{15} & \frac{1}{15} & \frac{2}{15} & \frac{1}{5} & \frac{1}{15} & \frac{2}{15} & \frac{1}{5} & \frac{1}{15} \\ \hline \end{array}

If

E(X) = \dfrac{263}{15}

, then

P(X < 20)

is equal to :

(A) $\dfrac{3}{5}$
(B) $\dfrac{8}{15}$
(C) $\dfrac{11}{15}$
(D) $\dfrac{14}{15}$

Q13·MathematicsNumericalJEE Main 2026

Let S be a set of 5 elements and P(S) denote the power set of S. Let E be an event of choosing an ordered pair (A, B) from the set

P(S) \times P(S)

such that

A \cap B = \varnothing

. If the probability of the event E is

\dfrac{3^{p}}{2^{q}}

, where p, q

\in \mathbb{N}

, then

p + q

is equal to

Q14·MathematicsSingle correctJEE Main 2026

From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is :-

(A) $\frac{7}{10^{7}}$
(B) $\frac{81}{10^{8}}$
(C) $\frac{67}{10^{8}}$
(D) $\frac{73}{10^{8}}$

Q15·MathematicsSingle correctJEE Main 2026

Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, is

\dfrac{p}{q}

, gcd(p, q) = 1, then

p + q

is equal to

(A)22
(B)23
(C)24
(D)21

Q16·MathematicsNumericalJEE Main 2026

From the first 100 natural numbers, two numbers first a and then b are selected randomly without replacement. If the probability that

a - b \ge 10

is

\frac{m}{n}

, gcd (m, n) = 1, then

m + n

is equal to _______.

Q17·MathematicsSingle correctJEE Main 2026

Two distinct numbers a and b are selected at random from 1, 2, 3,......, 50. The probability, that their product ab is divisible by 3, is

(A) $\frac{561}{1225}$
(B) $\frac{664}{1225}$
(C) $\frac{272}{1225}$
(D) $\frac{8}{25}$

Q18·MathematicsSingle correctJEE Main 2026

If a random variable x has the probability distribution x : 0, 1, 2, 3, 4, 5, 6, 7 p(x) : 0, 2k, k, 3k,

2k^2

, 2k,

k^2 + k

,

7k^2

then

P(3 < x \le 6)

is equal to

(A)0.34
(B)0.22
(C)0.64
(D)0.33

Q19·MathematicsSingle correctJEE Main 2026

Let the mean and variance of 7 observations 2, 4, 10, x, 12, 14, y,

x > y

, be 8 and 16 respectively. Two numbers are chosen from

\{1, 2, 3, x–4, y, 5\}

one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4, is:

(A) $\frac{3}{5}$
(B) $\frac{4}{5}$
(C) $\frac{2}{5}$
(D) $\frac{1}{3}$

Q20·MathematicsSingle correctJEE Main 2026

A random variable X takes values 0, 1, 2, 3 with probabilities

\frac{2a+1}{30}, \frac{8a-1}{30}, \frac{4a+1}{30}

, b respectively, where a, b ∈

\mathbf{R}

. Let μ and σ respectively be the mean and standard deviation of X such that

\sigma^2 + \mu^2 = 2

. Then

\frac{a}{b}

is equal to :

(A)30
(B)3
(C)60
(D)12

Q21·MathematicsNumericalJEE Advanced 2025

A factory has a total of three manufacturing units,

M_1

,

M_2

, and

M_3

, which produce bulbs independent of each other. The units

M_1

,

M_2

, and

M_3

produce bulbs in the proportions of 2 : 2 : 1. respectively. It is known that 20% of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by

M_1

, 15% are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by

M_2

is

\frac{2}{5}

. If a bulb is chosen randomly from the bulbs produced by

M_3

, then the probability that it is defective is ______

Q22·MathematicsSingle correctJEE Advanced 2025

Three students

S_1

,

S_2

and

S_3

are given a problem to solve. Consider the following events: U: At least one of

S_1

,

S_2

, and

S_3

can solve the problem, V:

S_1

can solve the problem, given that neither

S_2

nor

S_3

can solve the problem, W:

S_2

can solve the problem and

S_3

cannot solve the problem, T:

S_3

can solve the problem. for any event E, let P(E) denote the probability of E. If

P(U) = \frac{1}{2}

,

P(V) = \frac{1}{10}

, and

P(W) = \frac{1}{12}

, then P(T) is equal to

(A) $\frac{13}{36}$
(B) $\frac{1}{3}$
(C) $\frac{19}{60}$
(D) $\frac{1}{4}$

Q23·MathematicsSingle correctJEE Main 2025

If

A

and

B

are two events such that

P(A)=0.7

,

P(B)=0.4

and

P(A\cap\bar B)=0.5

, where

\bar B

denotes the complement of

B

, then

P\big(B\,|\,(A\cup\bar B)\big)

is equal to:

(A) $\dfrac{1}{4}$
(B) $\dfrac{1}{2}$
(C) $\dfrac{1}{6}$
(D) $\dfrac{1}{3}$

Q24·MathematicsSingle correctJEE Main 2025

Let a random variable

X

take values

0,1,2,3

with

P(X=0)=P(X=1)=p

,

P(X=2)=P(X=3)

and

E(X^2)=2E(X)

. Then the value of

8p-1

is:

(A)0
(B)2
(C)1
(D)3

Q25·MathematicsSingle correctJEE Main 2025

A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is

\dfrac{m}{n}

,

\gcd(m,n)=1

, then

n^2-m^2

is equal to:

(A)80
(B)60
(C)72
(D)64

Q26·MathematicsSingle correctJEE Main 2025

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is:

(A) $\dfrac{129}{182}$
(B) $\dfrac{103}{182}$
(C) $\dfrac{17}{26}$
(D) $\dfrac{19}{26}$

Q27·MathematicsSingle correctJEE Main 2025

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let

X

denote the number of defective pens. Then the variance of

X

is

(A) $\dfrac{11}{15}$
(B) $\dfrac{28}{75}$
(C) $\dfrac{2}{15}$
(D) $\dfrac{3}{5}$

Q28·MathematicsIntegerJEE Main 2025

A card from a pack of 52 cards is lost. From the remaining 51 cards,

n

cards are drawn and are found to be spades. If the probability of the lost card to be a spade is

\dfrac{11}{50}

, then

n

is equal to ______.

Q29·MathematicsSingle correctJEE Main 2025

If the probability that the random variable

X

takes the value

x

is given by

P(X=x)=k(x+1)3^{-x}

,

x=0,1,2,3,\ldots

, where

k

is a constant, then

P(X\ge 3)

is equal to:

(A) $\dfrac{7}{27}$
(B) $\dfrac{4}{9}$
(C) $\dfrac{8}{27}$
(D) $\dfrac{1}{9}$

Q30·MathematicsIntegerJEE Main 2025

Three distinct numbers are selected randomly from the set

\{1,2,3,\ldots,40\}

. If the probability, that the selected numbers are in an increasing G.P. is

\dfrac{m}{n}

,

\gcd(m,n)=1

, then

m+n

is equal to ______.

Q31·MathematicsSingle correctJEE Main 2025

Three identical bags, each containing 10 balls, have colours as follows — Bag I: 3 Red, 2 Blue, 5 Green; Bag II: 4 Red, 3 Blue, 3 Green; Bag III: 5 Red, 1 Blue, 4 Green. A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from Bag I is

p

, and if the ball is Green, the probability that it is from Bag III is

q

, then the value of

\left(\dfrac{1}{p}+\dfrac{1}{q}\right)

is:

(A)6
(B)9
(C)7
(D)8

Q32·MathematicsSingle correctJEE Main 2025

Let

A=[a_{ij}]

be a

2\times 2

matrix such that

a_{ij}\in\{0, 1\}

for all i and j. Let the random variable X denote the possible values of the determinant of the matrix A. Then the variance of X is:

(A) $\dfrac{1}{4}$
(B) $\dfrac{3}{8}$
(C) $\dfrac{5}{8}$
(D) $\dfrac{3}{4}$

Q33·MathematicsSingle correctJEE Main 2025

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 from Bag 2, is white, is

\dfrac{29}{45}

, then n is equal to:

(A)3
(B)4
(C)5
(D)6

Q34·MathematicsSingle correctJEE Main 2025

Bag

B_1

contains 6 white and 4 blue balls, Bag

B_2

contains 4 white and 6 blue balls, and Bag

B_3

contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag

B_2

, is:

(A) $\frac{1}{3}$
(B) $\frac{4}{15}$
(C) $\frac{2}{3}$
(D) $\frac{2}{5}$

Q35·MathematicsSingle correctJEE Main 2025

Two number

k_1

and

k_2

are randomly chosen from the set of natural numbers. Then, the probability that the value of

i^{k_1}+i^{k_2}

,

(i=\sqrt{-1})

is non-zero, equals

(A) $\frac{1}{2}$
(B) $\frac{1}{4}$
(C) $\frac{3}{4}$
(D) $\frac{2}{3}$

Q36·MathematicsSingle correctJEE Main 2025

Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If x denote the number of defective oranges, then the variance of x is:

(A) $\frac{28}{75}$
(B) $\frac{14}{25}$
(C) $\frac{26}{75}$
(D) $\frac{18}{25}$

Q37·MathematicsSingle correctJEE Main 2025

A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability, that A wins if A makes the first throw, is

(A) $\dfrac{9}{17}$
(B) $\dfrac{9}{19}$
(C) $\dfrac{8}{17}$
(D) $\dfrac{8}{19}$

Q38·MathematicsSingle correctJEE Main 2025

Let

A=[a_{ij}]

be a square matrix of order 2 with entries 0 or 1. Let E be the event that A is an invertible matrix. Then the probability

P(E)

is :

(A) $\dfrac{5}{8}$
(B) $\dfrac{3}{16}$
(C) $\dfrac{1}{8}$
(D) $\dfrac{3}{8}$

Q39·MathematicsSingle correctJEE Main 2025

A board has 16 squares as shown in the figure : Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :

(A) $\dfrac{4}{5}$
(B) $\dfrac{7}{10}$
(C) $\dfrac{3}{5}$
(D) $\dfrac{23}{30}$

Q40·MathematicsSingle correctJEE Main 2025

One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is

(A) $\dfrac{1}{2}$
(B) $\dfrac{3}{5}$
(C) $\dfrac{2}{3}$
(D) $\dfrac{4}{9}$

Q41·MathematicsSingle correctJEE Main 2025

A coin is tossed three times. Let

X

be the number of times a tail follows a head. If

\mu,\sigma^2

are the mean and variance of

X

, then

64(\mu+\sigma^2)

equals:

(A)51
(B)48
(C)32
(D)64

Q42·MathematicsSingle correctJEE Main 2025

Two balls are drawn one by one without replacement from a bag of 4 white and 6 black balls. If the probability that the first ball is black given that the second ball is black is

\tfrac{m}{n}

(

\gcd(m,n)=1

), then

m+n

equals:

(A)14
(B)4
(C)11
(D)13

Q43·MathematicsSingle correctJEE Main 2025

If

A

and

B

are two events such that

P(A\cap B)=0.1

, and

P(A|B)

and

P(B|A)

are the roots of the equation

12x^2-7x+1=0

, then the value of

\dfrac{P\left(\bar{A}\cup\bar{B}\right)}{P\left(\bar{A}\cap\bar{B}\right)}

is:

(A) $\dfrac{5}{3}$
(B) $\dfrac{4}{3}$
(C) $\dfrac{9}{4}$
(D) $\dfrac{7}{4}$

Q44·MathematicsIntegerJEE Advanced 2024

A bag contains N balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For

i = 1, 2, 3

, let

W_i

,

G_i

, and

B_i

denote the events that the ball drawn in the

i^{th}

draw is a white ball, green ball, and blue ball, respectively. If the probability

P\left(W_1 \cap G_2 \cap B_3\right) = \frac{2}{5N}

and the conditional probability

P\left(B_3 \mid W_1 \cap G_2\right) = \frac{2}{9}

, then N equals ______

Q45·MathematicsIntegerJEE Advanced 2024

Let

X

be a random variable, and let

P(X = x)

denote the probability that

X

takes the values

x

. Suppose that the points

(x, P(X = x))

,

x = 0, 1, 2, 3, 4

, lie on a fixed straight line in the xy-plane, and

P(X = x) = 0

for all

x \in R - \{0, 1, 2, 3, 4\}

. If the mean of

X

is

\frac{5}{2}

, and the variance of

X

is

\alpha

, then the value of

24\alpha

is ______ .

Q46·MathematicsSingle correctJEE Advanced 2024

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guess it, is

\frac{1}{2}

. Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is

\frac{1}{6}

. Then the probability that the student knows the answer of a randomly chosen question is

(A) $\frac{1}{12}$
(B) $\frac{1}{7}$
(C) $\frac{5}{7}$
(D) $\frac{5}{12}$

Q47·MathematicsSingle correctJEE Main 2024

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the

i^{th}

roll than the number obtained in the

(i-1)^{th}

roll,

i=2, 3

, is equal to:

(A) $\dfrac{3}{54}$
(B) $\dfrac{2}{54}$
(C) $\dfrac{5}{54}$
(D) $\dfrac{1}{54}$

Q48·MathematicsNumericalJEE Main 2024

Let

a

,

b

and

c

denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked

1, 2, 3, 4

. If the probability that

ax^2 + bx + c = 0

has all real roots is

\frac{m}{n}

,

\gcd(m, n) = 1

, then

m + n

is equal to ________.

Q49·MathematicsNumericalJEE Main 2024

Three balls are drawn at random from a bag containing

5

blue and

4

yellow balls. Let the random variables

X

and

Y

respectively denote the number of blue and yellow balls. If

\bar X

and

\bar Y

are the means of

X

and

Y

respectively, then

7\bar X+4\bar Y

is equal to ___

Q50·MathematicsSingle correctJEE Main 2024

Let the sum of two positive integers be

24

. If the probability, that their product is not less than

\dfrac34

times their greatest possible product, is

\dfrac{m}{n}

, where

\gcd(m,n)=1

, then

n-m

equals:

(A) $9$
(B) $11$
(C) $8$
(D) $10$

Q51·MathematicsSingle correctJEE Main 2024

There are three bags X, Y and Z. Bag X contains 5 one-rupee coins and 4 five-rupee coins; Bag Y contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y, is

(A) $\tfrac{1}{3}$
(B) $\tfrac{1}{2}$
(C) $\tfrac{1}{4}$
(D) $\tfrac{5}{12}$

Q52·MathematicsSingle correctJEE Main 2024

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:

(A) $\tfrac{12}{25}$
(B) $\tfrac{18}{25}$
(C) $\tfrac{4}{25}$
(D) $\tfrac{6}{25}$

Q53·MathematicsSingle correctJEE Main 2024

A company has two plants

A

and

B

to manufacture motorcycles. 60% motorcycles are manufactured at plant

A

and the remaining at plant

B

. 80% of the motorcycles manufactured at plant

A

are of standard quality, while 90% of those manufactured at plant

B

are of standard quality. If

p

is the probability that a motorcycle, found to be of standard quality, is manufactured at plant

B

, then

126p

is

(A) $54$
(B) $64$
(C) $66$
(D) $56$

Q54·MathematicsNumericalJEE Main 2024

From a lot of

12

items containing

3

defectives, a sample of

5

items is drawn at random. Let the random variable

X

denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of

X

is

\frac{m}{n}

, where

\gcd(m,n)=1

, then

n-m

is equal to ______.

Q55·MathematicsNumericalJEE Main 2024

Let the mean and the standard deviation of the probability distribution with

X=\{\alpha,1,0,-3\}

and corresponding

P(X)=\left\{\dfrac{1}{3},K,\dfrac{1}{6},\dfrac{1}{4}\right\}

(in the same order) be

\mu

and

\sigma

respectively. If

\sigma-\mu=2

, then

\sigma+\mu

is equal to __________.

Q56·MathematicsSingle correctJEE Main 2024

The coefficients

a, b, c

in the quadratic equation

ax^2 + bx + c = 0

are chosen from the set

\{1, 2, 3, 4, 5, 6, 7, 8\}

. The probability of this equation having repeated roots is:

(A) $\dfrac{3}{256}$
(B) $\dfrac{1}{128}$
(C) $\dfrac{1}{64}$
(D) $\dfrac{3}{128}$

Q57·MathematicsNumericalJEE Main 2024

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. If the variance of X is

\sigma^2

, then

96\sigma^2

is equal to ___.

Q58·MathematicsSingle correctJEE Main 2024

The coefficients

a,b,c

in the quadratic equation

ax^2+bx+c=0

are from the set

\{1,2,3,4,5,6\}

. If the probability of this equation having one real root bigger than the other is

p

, then

216p

equals:

(A)57
(B)38
(C)19
(D)76

Q59·MathematicsNumericalJEE Main 2024

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as

\dfrac{1}{3}

and

\dfrac{2}{3}

respectively. Let

x

be the number of matches that the team wins, and

y

be the number of matches that team loses. If the probability

P(|x-y|\le 2)

is

p

, then

3^9 p

equals

Q60·MathematicsSingle correctJEE Main 2024

Three urns

A

,

B

and

C

contain

7

red,

5

black;

5

red,

7

black and

6

red,

6

black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn

A

is:

(A) $\dfrac{4}{17}$
(B) $\dfrac{5}{18}$
(C) $\dfrac{7}{18}$
(D) $\dfrac{5}{16}$

Q61·MathematicsSingle correctJEE Main 2024

If the mean of the following probability distribution of a random variable X: (

X

: 0, 2, 4, 6, 8 with

P(X)

:

2a

,

3a

,

a+b

,

2b

,

3b

respectively) is

\dfrac{46}{9}

, then the variance of the distribution is

(A) $\tfrac{581}{81}$
(B) $\tfrac{566}{81}$
(C) $\tfrac{173}{27}$
(D) $\tfrac{151}{27}$

Q62·MathematicsSingle correctJEE Main 2024

Let Ajay will not appear in JEE exam with probability

p=\dfrac{2}{7}

, while both Ajay and Vijay will appear in the exam with probability

q=\dfrac{1}{5}

. Then the probability, that Ajay will appear in the exam and Vijay will not appear is:

(A) $\dfrac{9}{35}$
(B) $\dfrac{18}{35}$
(C) $\dfrac{24}{35}$
(D) $\dfrac{3}{35}$

Q63·MathematicsSingle correctJEE Main 2024

A bag contains

8

balls, whose colours are either white or black.

4

balls are drawn at random without replacement and it was found that

2

balls are white and

2

balls are black. The probability that the bag contains equal number of white and black balls is:

(A) $\dfrac{2}{5}$
(B) $\dfrac{2}{7}$
(C) $\dfrac{1}{7}$
(D) $\dfrac{1}{5}$

Q64·MathematicsSingle correctJEE Main 2024

Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is

(A) $\dfrac{2}{25}$
(B) $\dfrac{4}{25}$
(C) $\dfrac{2}{3}$
(D) $\dfrac{4}{75}$

Q65·MathematicsSingle correctJEE Main 2024

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable x to be the number of rotten apples in a draw of two apples, the variance of x is

(A) $\dfrac{37}{153}$
(B) $\dfrac{57}{153}$
(C) $\dfrac{47}{153}$
(D) $\dfrac{40}{153}$

Q66·MathematicsSingle correctJEE Main 2024

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is

(A) $\dfrac{2}{9}$
(B) $\dfrac{1}{9}$
(C) $\dfrac{2}{27}$
(D) $\dfrac{1}{27}$

Q67·MathematicsSingle correctJEE Main 2024

Bag A contains 3 white, 7 red balls and bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from bag A, if the ball drawn is white, is:

(A) $\dfrac{1}{4}$
(B) $\dfrac{1}{9}$
(C) $\dfrac{1}{3}$
(D) $\dfrac{3}{10}$

Q68·MathematicsSingle correctJEE Main 2024

Two integers

x

and

y

are chosen with replacement from the set

\{0,1,2,3,\ldots,10\}

. Then the probability that

|x-y|>5

is:

(A) $\dfrac{30}{121}$
(B) $\dfrac{62}{121}$
(C) $\dfrac{60}{121}$
(D) $\dfrac{31}{121}$

Q69·MathematicsSingle correctJEE Main 2024

A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is

(A) $\dfrac{5}{6}$
(B) $\dfrac{1}{6}$
(C) $\dfrac{5}{11}$
(D) $\dfrac{6}{11}$

Q70·MathematicsSingle correctJEE Main 2024

An integer is chosen at random from the integers

1,2,3,\ldots,50

. The probability that the chosen integer is a multiple of atleast one of

4,6

and

7

is:

(A) $\dfrac{8}{25}$
(B) $\dfrac{21}{50}$
(C) $\dfrac{9}{50}$
(D) $\dfrac{14}{50}$

Q71·MathematicsSingle correctJEE Main 2024

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 :

(A) $\dfrac{5}{256}$
(B) $\dfrac{5}{715}$
(C) $\dfrac{3}{715}$
(D) $\dfrac{3}{256}$

Q72·MathematicsNumericalJEE Main 2024

A fair die is tossed repeatedly until a six is obtained. Let

X

denote the number of tosses required and let

a=P(X=3)

,

b=P(X\ge 3)

and

c=P(X\ge 6\mid X>3)

. Then

\dfrac{b+c}{a}

is equal to __________.

Q73·MathematicsSingle correctJEE Advanced 2023

Let

X : \left\{(x, y) \in Z \times Z : \frac{x^2}{8} + \frac{y^2}{20} < 1 \text{ and } y^2 < 5x \right\}

. Three distinct points P, Q and R are randomly chosen from X . Then the probability that P, Q and R form a triangle whose area is a positive integer, is

(A) $\frac{71}{220}$
(B) $\frac{73}{220}$
(C) $\frac{79}{220}$
(D) $\frac{83}{220}$

Q74·MathematicsSingle correctJEE Advanced 2023

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is

\frac{1}{3}

, then the probability that the experiment stops with head is

(A) $\frac{1}{3}$
(B) $\frac{5}{21}$
(C) $\frac{4}{21}$
(D) $\frac{2}{7}$

Q75·MathematicsNumericalJEE Advanced 2023

Consider the

6 \times 6

square in the figure. Let

A_{1}

,

A_{2}

, .....,

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}

, .....,

A_{49}

. Let

p

be the probability that they are friends. Then the value of

7p

is

Q76·MathematicsNumericalJEE Advanced 2023

Consider the

6 \times 6

square in the figure. Let

A_{1}

,

A_{2}

, .....,

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. Let

p_{i}

be the probability that a randomly chosen point has

i

many friends,

i = 0, 1, 2, 3, 4

. Let

X

be a random variable such that for

i = 0, 1, 2, 3, 4

, the probability

P(X = i) = p_{i}

. Then the value of

7E(X)

is

Q77·MathematicsSingle correctJEE Main 2023

A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is:

(A) $\frac{1}{4}$
(B) $\frac{9}{50}$
(C) $\frac{1}{5}$
(D) $\frac{11}{50}$

Q78·MathematicsSingle correctJEE Main 2023

A coin is biased so that a head is

3

times as likely to occur as a tail. This coin is tossed until a head or three tails occur. If

X

denotes the number of tosses of the coin, then the mean of

X

is:

(A) $\dfrac{21}{16}$
(B) $\dfrac{81}{64}$
(C) $\dfrac{15}{16}$
(D) $\dfrac{37}{16}$

Q79·MathematicsSingle correctJEE Main 2023

The random variable X follows binomial distribution

B(n, p)

for which the difference of the mean and the variance is 1. If

2P(X = 2) = 3P(X = 1)

, then

n^2 P(X > 1)

is equal to

(A)12
(B)15
(C)11
(D)16

Q80·MathematicsNumericalJEE Main 2023

A fair die with faces marked

1

to

n

(

n>1

) is tossed repeatedly until a number less than n appears. If the mean of the number of tosses required is

\dfrac{n}{9}

, then n is equal to

Q81·MathematicsSingle correctJEE Main 2023

Let R be a rectangle given by the lines

x = 0

,

x = 2

,

y = 0

and

y = 5

. Let

A(\alpha, 0)

and

B(0, \beta)

,

\alpha \in [0, 2]

and

\beta \in [0, 5]

, be such that the line segment AB divides the area of the rectangle R in the ratio

4:1

. Then, the mid-point of AB lies on a

(A)parabola
(B)hyberbola
(C)straight line
(D)circle

Q82·MathematicsSingle correctJEE Main 2023

Let

S = \{ M = [a_{ij}],\ a_{ij} \in \{0, 1, 2\},\ 1 \le i, j \le 2 \}

be a sample space and

A = \{ M \in S : M \text{ is invertible} \}

be an event. Then

P(A)

is equal to

(A) $\frac{50}{81}$
(B) $\frac{47}{81}$
(C) $\frac{49}{81}$
(D) $\frac{16}{27}$

Q83·MathematicsNumericalJEE Main 2023

The probability of getting a head for a biased coin is

\dfrac{1}{4}

. It is tossed repeatedly until a head appears. Let

N

be the number of tosses required. If the probability that the equation

64x^2+5Nx+1=0

has no real root is

\dfrac{p}{q}

, where

p

and

q

are co-prime, then

q-p

is equal to

Q84·MathematicsSingle correctJEE Main 2023

Let N denotes the sum of the numbers obtained when two dice are rolled. If the probability that

2^N<N!

is

\dfrac{m}{n}

, where m and n are coprime, then

4m-3n

is equal to

(A) $8$
(B) $16$
(C) $10$
(D) $12$

Q85·MathematicsSingle correctJEE Main 2023

Let a die be rolled

n

times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is

\frac{k}{2^{15}}

, then

k

is equal to:

(A) $30$
(B) $90$
(C) $15$
(D) $60$

Q86·MathematicsSingle correctJEE Main 2023

In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine C is

(A) $\frac{2}{7}$
(B) $\frac{9}{28}$
(C) $\frac{5}{14}$
(D) $\frac{3}{7}$

Q87·MathematicsSingle correctJEE Main 2023

If the probability that the random variable X takes values x is given by

P(X=x)=k(x+1)3^{-x}

,

x=0,1,2,3,\ldots

, where k is a constant, then

P(X\geq2)

is equal to

(A) $\dfrac{1}{27}$
(B) $\dfrac{11}{18}$
(C) $\dfrac{7}{18}$
(D) $\dfrac{20}{27}$

Q88·MathematicsSingle correctJEE Main 2023

A pair of dice are thrown

5

times. For each throw, a total of

5

is considered a success. If the probability of at least

4

successes is

\dfrac{k}{3^{11}}

, then

k

is equal to:

(A) $82$
(B) $123$
(C) $164$
(D) $75$

Q89·MathematicsSingle correctJEE Main 2023

Three dice are rolled. If the probability of getting different numbers on the three dice is

\frac{p}{q}

, where

p

and

q

are co-prime, then

q-p

is equal to

(A)4
(B)3
(C)1
(D)2

Q90·MathematicsSingle correctJEE Main 2023

Two dice are thrown independently. Let A be the event that the number appeared on the

1^{st}

die is less than the number appeared on the

2^{nd}

die, B be the event that the number appeared on the

1^{st}

die is even and that on the second die is odd, and C be the event that the number appeared on the

1^{st}

die is odd and that on the

2^{nd}

die is even. Then:

(A)the number of favourable cases of the events A, B and C are 15, 6 and 6 respectively
(B)the number of favourable cases of the event $(A\cup B)\cap C$ is 6
(C)B and C are independent
(D)A and B are mutually exclusive

Q91·MathematicsSingle correctJEE Main 2023

In a binomial distribution

B(n,p)

, the sum and the product of the mean and the variance are

5

and

6

respectively, then

6(n+p-q)

is equal to

(A) $52$
(B) $50$
(C) $51$
(D) $53$

Q92·MathematicsNumericalJEE Main 2023

Let

A

be the event that the absolute difference between two randomly chosen real numbers in the sample space

[0,60]

is less than or equal to

a

. If

P(A)=\dfrac{11}{36}

, then

a

is equal to

Q93·MathematicsSingle correctJEE Main 2023

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 balls is

(A) $\frac{3}{7}$
(B) $\frac{5}{7}$
(C) $\frac{5}{6}$
(D) $\frac{2}{7}$

Q94·MathematicsSingle correctJEE Main 2023

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 positive, is:

(A) $\dfrac{881}{2592}$
(B) $\dfrac{27}{288}$
(C) $\dfrac{440}{2592}$
(D) $\dfrac{521}{2592}$

Q95·MathematicsNumericalJEE Main 2023

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 colour is

q

. If

p:q=m:n

, where

m

and

n

are coprime, then

m + n

is equal to _______.

Q96·MathematicsSingle correctJEE Main 2023

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

(A) $\dfrac{5}{24}$
(B) $\dfrac{1}{6}$
(C) $\dfrac{5}{36}$
(D) $\dfrac{2}{15}$

Q97·MathematicsSingle correctJEE Main 2023

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable

X

denote the number of rotten apples. If

\mu

and

\sigma^2

represent mean and variance of

X

, respectively, then

10(\mu^2 + \sigma^2)

is equal to

(A)250
(B)25
(C)30
(D)20

Q98·MathematicsSingle correctJEE Main 2023

Let

S=\{w_1,w_2,\dots\}

be the sample space associated to a random experiment. Let

P(w_n)=\dfrac{P(w_{n-1})}{2},\,n\ge 2

. Let

A=\{2k+3l:\,k,l\in\mathbb{N}\}

and

B=\{w_n:\,n\in A\}

. Then

P(B)

is equal to:

(A) $\dfrac{3}{64}$
(B) $\dfrac{1}{16}$
(C) $\dfrac{1}{32}$
(D) $\dfrac{3}{32}$

Q99·MathematicsNumericalJEE Main 2023

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 and the probability that this person is a smoker is

\dfrac{k}{10}

. Then the value of

k

is.

Q100·MathematicsSingle correctJEE Main 2023

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 be

\dfrac{k}{48}

. Then the value of

k

is

(A)8
(B)16
(C)2
(D)4

Q101·MathematicsSingle correctJEE Main 2023

Let

M

be the maximum value of the product of two positive integers when their sum is

66

. Let the sample space

S=\Big\{x\in\mathbb{Z}:x(66-x)\ge\dfrac{5}{9}M\Big\}

and the event

A=\{x\in S:x\text{ is a multiple of }3\}

. Then

P(A)

is equal to:

(A) $\dfrac{15}{22}$
(B) $\dfrac{1}{5}$
(C) $\dfrac{15}{44}$
(D) $\dfrac{1}{3}$

Q102·MathematicsSingle correctJEE Main 2023

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+Ny+2z=2

,

3x+3y+Nz=3

has unique solution is

\dfrac{k}{6}

, then the sum of value of

k

and all possible values of

N

is

(A) $21$
(B) $18$
(C) $20$
(D) $19$

Q103·MathematicsNumericalJEE Main 2023

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 and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn C is

0.4

, then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola

y^2=\lambda x

with one vertex at the vertex of the parabola, is

Q104·MathematicsSingle correctJEE Main 2023

Let

\Omega

be the sample space and

A\subseteq\Omega

be an event. Given below are two statements: (S1): If

P(A)=0

, then

A=\varnothing

. (S2): If

P(A)=1

, then

A=\Omega

. Then

(A)both (S1) and (S2) are true
(B)only (S1) is true
(C)only (S2) is true
(D)both (S1) and (S2) are false

Q105·MathematicsSingle correctJEE Advanced 2022

Suppose that Box-I contains 8 red, 3 blue and 5 green balls, Box-II contains 24 red, 9 blue and 15 green balls, Box-III contains 1 blue, 12 green and 3 yellow balls, Box-IV contains 10 green, 16 orange and 6 white balls, A ball is chosen randomly for Box-I; call that ball b. If b is red then a ball is chosen randomly from Box-II, if b is blue then a ball is chosen randomly from Box-III, and if b is green then a ball is chosen randomly from Box-IV. The conditional probability of the event 'one of the chosen balls is white' given that the event 'at least one of the chosen ball is green' has happened, is equal to

(A) $\frac{15}{256}$
(B) $\frac{3}{16}$
(C) $\frac{5}{52}$
(D) $\frac{1}{8}$ .

Q106·MathematicsSingle correctJEE Advanced 2022

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 points. 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. The correct option is:

List-I	List-II
I.Probability of $\left(X_{2} \geq Y_{2}\right)$ is	P. $\frac{3}{8}$
II.Probability of $\left(X_{2} > Y_{2}\right)$ is	Q. $\frac{11}{16}$
III.Probability of $\left(X_{3} = Y_{3}\right)$ is	R. $\frac{5}{16}$
IV.Probability of $\left(X_{3} > Y_{3}\right)$ is	S. $\frac{355}{864}$
	T. $\frac{77}{432}$

(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)

Q107·MathematicsSingle correctJEE Main 2022

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 then a ball is draw from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is:

(A) $\frac{4}{9}$
(B) $\frac{5}{18}$
(C) $\frac{1}{6}$
(D) $\frac{3}{10}$

Q108·MathematicsSingle correctJEE Main 2022

Let

S = \{1,\,2,\,3,\,\ldots,\,2022\}

. Then the probability, that a randomly chosen number n from the set S such that HCF

(n,\,2022) = 1

, is :

(A) $\frac{128}{1011}$
(B) $\frac{166}{1011}$
(C) $\frac{127}{337}$
(D) $\frac{112}{337}$

Q109·MathematicsNumericalJEE Main 2022

The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. They the number of trials in the binomial distribution is:

Q110·MathematicsNumericalJEE Main 2022

A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let X be the number of white balls, among the drawn balls. If

\sigma^{2}

is the variance of X, then

100\,\sigma^{2}

is equal to

Q111·MathematicsSingle correctJEE Main 2022

Let A and B be two events such that

P(B|A)=\frac{2}{5}

,

P(A|B)=\frac{1}{7}

and

P(A \cap B)=\frac{1}{9}

. Consider (S1)

P(A' \cup B)=\frac{5}{6}

, (S2)

P(A' \cap B')=\frac{1}{18}

. Then

(A)Both (S1) and (S2) are true
(B)Both (S1) and (S2) are false
(C)Only (S1) is true
(D)Only (S2) is true

Q112·MathematicsSingle correctJEE Main 2022

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

9p

is equal to

(A)1.0146
(B)1.2085
(C)1.0285
(D)1.1521

Q113·MathematicsSingle correctJEE Main 2022

A six faced die is biased such that

3 \times P(\text{a prime number}) = 6 \times P(\text{a composite number}) = 2 \times P(1)

. Let X be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of X is :

(A) $\frac{3}{11}$
(B) $\frac{5}{11}$
(C) $\frac{7}{11}$
(D) $\frac{8}{11}$

Q114·MathematicsSingle correctJEE Main 2022

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) = \frac{k}{2^{n}}

, then k is equal to

(A)528
(B)529
(C)629
(D)630

Q115·MathematicsSingle correctJEE Main 2022

Let E

_{1}

, E

_{2}

, E

_{3}

be three mutually exclusive events such that

P(E_{1}) = \frac{2+3p}{6}

,

P(E_{2}) = \frac{2-p}{8}

and

P(E_{3}) = \frac{1-p}{2}

. If the maximum and minimum values of p are p

_{1}

and p

_{2}

, then (p

_{1}

+ p

_{2}

) is equal to :

(A) $\frac{2}{3}$
(B) $\frac{5}{3}$
(C) $\frac{5}{4}$
(D)1

Q116·MathematicsSingle correctJEE Main 2022

Let X be a binomially distributed random variable with mean 4 and variance

\frac{4}{3}

. Then

54\,P(X\le 2)

is equal to

(A) $\frac{73}{27}$
(B) $\frac{146}{27}$
(C) $\frac{146}{81}$
(D) $\frac{126}{81}$

Q117·MathematicsSingle correctJEE Main 2022

The mean and variance of a binomial distribution are α and

\frac{\alpha}{3}

respectively. If

P\left(X = 1\right) = \frac{4}{243}

, then P(X = 4 or 5) is equal to :

(A) $\frac{5}{9}$
(B) $\frac{64}{81}$
(C) $\frac{16}{27}$
(D) $\frac{145}{243}$

Q118·MathematicsSingle correctJEE Main 2022

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

, for all

x \in R

, is :

(A) $\frac{17}{36}$
(B) $\frac{4}{9}$
(C) $\frac{1}{2}$
(D) $\frac{19}{36}$

Q119·MathematicsSingle correctJEE Main 2022

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 is :

(A) $\frac{33}{2^{32}}$
(B) $\frac{33}{2^{29}}$
(C) $\frac{33}{2^{28}}$
(D) $\frac{33}{2^{27}}$

Q120·MathematicsSingle correctJEE Main 2022

The probability that a randomly chosen 2 × 2 matrix with all the entries from the set of first 10 primes, is singular, is equal to :

(A) $\frac{133}{10^{4}}$
(B) $\frac{18}{10^{3}}$
(C) $\frac{19}{10^{3}}$
(D) $\frac{271}{10^{4}}$

Q121·MathematicsSingle correctJEE Main 2022

The probability that a relation R from

\{x,y\}

to

\{x,y\}

is both symmetric and transitive, is equal to:

(A) $\frac{5}{16}$
(B) $\frac{9}{16}$
(C) $\frac{11}{16}$
(D) $\frac{13}{16}$

Q122·MathematicsSingle correctJEE Main 2022

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)

is :

(A) $\frac{1}{24}$
(B) $\frac{1}{40}$
(C) $\frac{1}{30}$
(D) $\frac{1}{20}$

Q123·MathematicsSingle correctJEE Main 2022

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

(A) $\frac{19}{36}$
(B) $\frac{15}{36}$
(C) $\frac{13}{36}$
(D) $\frac{23}{36}$

Q124·MathematicsSingle correctJEE Main 2022

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<1

is :

(A) $\dfrac{1}{6}$
(B) $\dfrac{5}{6}$
(C) $\dfrac{2}{3}$
(D) $\dfrac{6}{7}$

Q125·MathematicsSingle correctJEE Main 2022

Let X be a random variable having binomial distribution B(7, p). If P(X = 3) = 5P(X = 4), then the sum of the mean and the variance of X is :

(A) $\frac{105}{16}$
(B) $\frac{7}{16}$
(C) $\frac{77}{36}$
(D) $\frac{49}{16}$

Q126·MathematicsNumericalJEE Main 2022

Let

S=\{E, E_{2}....E_{8}\}

be a sample space of random experiment such that

P(E_{n})=\dfrac{n}{36}

for every n = 1, 2....8. Then the number of elements in the set

\left\{A\subset S:P(A)\ge\dfrac{4}{5}\right\}

is ______

Q127·MathematicsSingle correctJEE Main 2022

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 (

x_1 < x_2 < x_3 < x_4 < x_5

). The probability that

x_2 = 7

and

x_4 = 11

is :

(A) $\frac{1}{136}$
(B) $\frac{1}{72}$
(C) $\frac{1}{68}$
(D) $\frac{1}{34}$

Q128·MathematicsSingle correctJEE Main 2022

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 getting atmost two heads is:

(A) $\frac{275}{6^{5}}$
(B) $\frac{36}{5^{4}}$
(C) $\frac{181}{5^{5}}$
(D) $\frac{46}{6^{4}}$

Q129·MathematicsNumericalJEE Main 2022

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 ______ .

Q130·MathematicsSingle correctJEE Main 2022

Let

E_1

and

E_2

be two events such that the conditional probabilities

P(E_1 | E_2) = \frac{1}{2}

,

P(E_2 | E_1) = \frac{3}{4}

and

P(E_1 \cap E_2) = \frac{1}{8}

. Then:

(A) $P(E_1 \cap E_2) = P(E_1)\cdot P(E_2)$
(B) $P(E'_1 \cap E'_2) = P(E'_1)\cdot P(E_2)$
(C) $P(E_1 \cap E'_2) = P(E_1)\cdot P(E_2)$
(D) $P(E'_1 \cap E_2) = P(E_1)\cdot P(E_2)$

Q131·MathematicsSingle correctJEE Main 2022

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

\frac{1}{n}

. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48, is

(A) $\frac{7}{2^{11}}$
(B) $\frac{7}{2^{12}}$
(C) $\frac{3}{2^{10}}$
(D) $\frac{13}{2^{12}}$

Q132·MathematicsSingle correctJEE Main 2022

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 from it at random, are found to be 1 red and 1 black. If the probability that both balls come from Bag A is

\dfrac{6}{11}

, then n is equal to ______ .

(A)13
(B)6
(C)4
(D)3

Q133·MathematicsNumericalJEE Main 2022

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

\dfrac{3}{4}

and the remaining 6 questions correctly with probability

\dfrac{1}{4}

. If the probability that the student guesses the answers of exactly 8 questions correctly out of 10 is

\dfrac{27k}{4^{10}}

, then k is equal to ______.

Q134·MathematicsSingle correctJEE Main 2022

If a random variable X follows the Binomial distribution B (33, p) such that

3P(X=0)=P(X=1)

, then the value of

\dfrac{P(X=15)}{P(X=18)}-\dfrac{P(X=16)}{P(X=17)}

is equal to

(A)1320
(B)1088
(C) $\dfrac{120}{1331}$
(D) $\dfrac{1088}{1089}$

Q135·MathematicsNumericalJEE Advanced 2021

Three numbers are chosen at random, one after another with replacement, from the set

S = \{1, 2, 3, ..., 100\}

. Let

p_1

be the probability that the maximum of chosen numbers is at least 81 and

p_2

be the probability that the minimum of chosen numbers is at most 40. The value of

\frac{125}{4}p_2

is ________.

Q136·MathematicsIntegerJEE Advanced 2021

A number is chosen at random from the set {1, 2, 3, ... , 2000}. Let p be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500p is ___.

Q137·MathematicsNumericalJEE Advanced 2021

Three numbers are chosen at random, one after another with replacement, from the set

S = \{1, 2, 3, ..., 100\}

. Let

p_1

be the probability that the maximum of chosen numbers is at least 81 and

p_2

be the probability that the minimum of chosen numbers is at most 40. The value of

\frac{625}{4}p_1

is ________.

Q138·MathematicsMultiple correctJEE Advanced 2021

Let E,F and G be three events having probabilities

P(E) = \frac{1}{8}, P(F) = \frac{1}{6}

and

P(G) = \frac{1}{4}

, and let

P(E \cap F \cap G) = \frac{1}{10}

. For any event H, if

H^C

denotes its complement, then which of the following statements is(are) TRUE ?

(A) $P\left(E \cap F \cap G^C\right) \le \frac{1}{40}$
(B) $P\left(E^C \cap F \cap G\right) \le \frac{1}{15}$
(C) $P\left(E \cup F \cup G\right) \le \frac{13}{24}$
(D) $P\left(E^C \cap F^C \cap G^C\right) \le \frac{5}{12}$

Q139·MathematicsSingle correctJEE Advanced 2021

Consider three sets

E_1 = \{1, 2, 3\}

,

F_1 = \{1, 3, 4\}

and

G_1 = \{2, 3, 4, 5\}

. Two elements are chosen at random, without replacement, from the set

E_1

, and let

S_1

denote the set of these chosen elements. Let

E_2 = E_1 - S_1

and

F_2 = F_1 \cup S_1

. Now two elements are chosen at random, without replacement, from the set

F_2

and let

S_2

denote the set of these chosen elements. Let

G_2 = G_1 \cup S_2

. Finally, two elements are chosen at random, without replacement, from the set

G_2

and let

S_3

denote the set of these chosen elements. Let

E_3 = E_2 \cup S_3

. Given that

E_1 = E_3

, let p be the conditional probability of the event

S_1 = \{1, 2\}

. Then the value of p is

(A) $\frac{1}{5}$
(B) $\frac{3}{5}$
(C) $\frac{1}{2}$
(D) $\frac{2}{5}$

Q140·MathematicsSingle correctJEE Main 2021

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

(A) $\frac{2}{7}$
(B) $\frac{1}{18}$
(C) $\frac{1}{7}$
(D) $\frac{1}{9}$

Q141·MathematicsNumericalJEE Main 2021

Let X be a random variable with distribution. If the mean of X is 2.3 and variance of X is

\sigma^{2}

, then

100\,\sigma^{2}

is equal to ________ .

x	−2	−1	3	4	6
P(X = x)	$\frac{1}{5}$	a	$\frac{1}{3}$	$\frac{1}{5}$	b

Q142·MathematicsNumericalJEE Main 2021

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 98 p is equal to ______.

Q143·MathematicsSingle correctJEE Main 2021

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 :

(A) $\frac{1}{10}$
(B) $\frac{1}{15}$
(C) $\frac{1}{5}$
(D) $\frac{1}{30}$

Q144·MathematicsSingle correctJEE Main 2021

When a certain biased die is rolled, a particular face occurs with probability

\frac{1}{6} - x

and its opposite face occurs with probability

\frac{1}{6} + x

. All other faces occur with probability

\frac{1}{6}

. Note that opposite faces sum to 7 in any die. If

0 < x < \frac{1}{6}

, and the probability of obtaining total sum

= 7

, when such a die is rolled twice, is

\frac{13}{96}

, then the value of x is:

(A) $\frac{1}{16}$
(B) $\frac{1}{8}$
(C) $\frac{1}{9}$
(D) $\frac{1}{12}$

Q145·MathematicsNumericalJEE Main 2021

The probability distribution of random variable X is given by: X | 1 | 2 | 3 | 4 | 5 P(X) | K | 2K | 2K | 3K | K Let p = P(1 < X < 4 | X < 3). If 5p = λK, then λ equal to _________ .

Q146·MathematicsSingle correctJEE Main 2021

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 :

(A) $\frac{1}{8}$
(B) $\frac{5}{8}$
(C) $\frac{5}{16}$
(D)1

Q147·MathematicsSingle correctJEE Main 2021

A fair die is tossed until six is obtained on it. Let X be the number of required tosses, then the conditional probability

P\left(X \geq 5 \mid X > 2\right)

is :

(A) $\frac{125}{216}$
(B) $\frac{11}{36}$
(C) $\frac{5}{6}$
(D) $\frac{25}{36}$

Q148·MathematicsSingle correctJEE Main 2021

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)

= \frac{5}{9}

, is :

(A) $\frac{1}{3}$
(B) $\frac{2}{9}$
(C) $\frac{4}{9}$
(D) $\frac{5}{12}$

Q149·MathematicsSingle correctJEE Main 2021

The probability that a randomly selected 2-digit number belongs to the set

\left\{ n \in N : \left(2^n - 2\right) \text{ is a multiple of } 3 \right\}

is equal to :

(A) $\frac{2}{3}$
(B) $\frac{1}{2}$
(C) $\frac{1}{6}$
(D) $\frac{1}{3}$

Q150·MathematicsSingle correctJEE Main 2021

A student appeared in an examination consisting of 8 true − false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least 'n' correct answers is less than

\frac{1}{2}

, is :

(A)5
(B)6
(C)3
(D)4

Q151·MathematicsSingle correctJEE Main 2021

Let X be a random variable such that the probability function of a distribution is given by

P\left(X = 0\right) = \frac{1}{2}, P\left(X = j\right) = \frac{1}{3^j}\left(j = 1,2,3,.....,\infty\right)

. Then the mean of the distribution and P(X is positive and even) respectively are :

(A) $\frac{3}{4}$ and $\frac{1}{9}$
(B) $\frac{3}{4}$ and $\frac{1}{16}$
(C) $\frac{3}{4}$ and $\frac{1}{8}$
(D) $\frac{3}{8}$ and $\frac{1}{8}$

Q152·MathematicsSingle correctJEE Main 2021

Let 9 distinct balls be distributed among 4 boxes,

B_1, B_2, B_3

and

B_4

. If the probability that

B_3

contains exactly 3 balls is

k\left(\frac{3}{4}\right)^{9}

then k lies in the set :

(A) $\left\{x \in R : \left|x - 2\right| \leq 1\right\}$
(B) $\left\{x \in R : \left|x - 5\right| \leq 1\right\}$
(C) $\left\{x \in R : \left|x - 3\right| < 1\right\}$
(D) $\left\{x \in R : \left|x - 1\right| < 1\right\}$

Q153·MathematicsNumericalJEE Main 2021

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...........

Q154·MathematicsSingle correctJEE Main 2021

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 x 2 matrices. The probability that such formed matrices have all different entries and are non-singular, is :

(A) $\dfrac{22}{81}$
(B) $\dfrac{45}{162}$
(C) $\dfrac{43}{162}$
(D) $\dfrac{23}{81}$

Q155·MathematicsSingle correctJEE Main 2021

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 position in any such word is :

(A) $\frac{1}{9}$
(B) $\frac{1}{11}$
(C) $\frac{2}{11}$
(D) $\frac{1}{66}$

Q156·MathematicsSingle correctJEE Main 2021

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 :

(A)greater than $\frac{1}{2}$
(B)greater than $\frac{1}{4}$ but less than $\frac{1}{2}$
(C)exactly equal to $\frac{1}{2}$
(D)greater than $\frac{1}{8}$ but less than $\frac{1}{4}$

Q157·MathematicsSingle correctJEE Main 2021

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 the probability of getting exactly 3 successes is equal to :

(A) $\frac{32}{625}$
(B) $\frac{80}{243}$
(C) $\frac{40}{243}$
(D) $\frac{128}{625}$

Q158·MathematicsSingle correctJEE Main 2021

Two dices 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 less than or equal to 8 is :

(A) $\frac{4}{9}$
(B) $\frac{17}{36}$
(C) $\frac{5}{12}$
(D) $\frac{1}{2}$

Q159·MathematicsSingle correctJEE Main 2021

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 :

(A) $\frac{1}{18}$
(B) $\frac{1}{3}$
(C) $\frac{1}{6}$
(D) $\frac{1}{9}$

Q160·MathematicsNumericalJEE Main 2021

Let there be three independent events

E_1

,

E_2

and

E_3

. The probability that only

E_1

occurs is

\alpha

, only

E_2

occurs is

\beta

and only

E_3

occurs is

\gamma

. Let 'p' denote the probability of none of events occurs that satisfies the equations

(\alpha - 2\beta)p = \alpha\beta

and

(\beta - 3\gamma)p = 2\beta\gamma

. All the given probabilities are assumed to lie in the interval (0, 1). Then,

\frac{\text{Probability of occurrence of } E_1}{\text{Probability of occurrence of } E_3}

is equal to _____.

Q161·MathematicsSingle correctJEE Main 2021

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 equal to :

(A) $\frac{9}{56}$
(B) $\frac{4}{9}$
(C) $\frac{3}{7}$
(D) $\frac{11}{27}$

Q162·MathematicsSingle correctJEE Main 2021

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, is :

(A) $\frac{3}{4}$
(B) $\frac{52}{867}$
(C) $\frac{39}{50}$
(D) $\frac{22}{425}$

Q163·MathematicsSingle correctJEE Main 2021

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

(A) $\frac{6}{7}$
(B) $\frac{4}{7}$
(C) $\frac{3}{7}$
(D) $\frac{1}{7}$

Q164·MathematicsSingle correctJEE Main 2021

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 of getting 2 heads is :

(A) $\frac{15}{2^{12}}$
(B) $\frac{15}{2^{13}}$
(C) $\frac{15}{2^{14}}$
(D) $\frac{15}{2^{8}}$

Q165·MathematicsSingle correctJEE Main 2021

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. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is:

(A)457
(B)458
(C)1445
(D)2845

Q166·MathematicsSingle correctJEE Main 2021

The coefficients a, b and c of the quadratic equation,

ax^{2} + bx + c = 0

are obtained by throwing a dice three times. The probability that this equation has equal roots is :

(A) $\frac{1}{54}$
(B) $\frac{1}{72}$
(C) $\frac{1}{36}$
(D) $\frac{5}{216}$

Q167·MathematicsSingle correctJEE Main 2021

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:

(A)15
(B)29
(C)29797
(D)122297

Q168·MathematicsSingle correctJEE Main 2021

When a missile is fired from a ship, the probability that it is intercepted is

\frac{1}{3}

and the probability that the missile hits the target, given that it is not intercepted, is

\frac{3}{4}

. If three missiles are fired independently from the ship, then the probability that all three hit the target, is:

(A) $\frac{1}{8}$
(B) $\frac{1}{27}$
(C) $\frac{3}{4}$
(D) $\frac{3}{8}$

Q169·MathematicsSingle correctJEE Main 2021

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 an even number 3 times, then the probability of getting an odd number for odd number of times is :

(A) $\frac{3}{16}$
(B) $\frac{1}{2}$
(C) $\frac{5}{16}$
(D) $\frac{1}{32}$

Q170·MathematicsSingle correctJEE Main 2021

The probability that two randomly selected subsets of the set

\{1,2,3,4,5\}

have exactly two elements in their intersection, is:

(A) $\frac{65}{2^{7}}$
(B) $\frac{135}{2^{9}}$
(C) $\frac{65}{2^{8}}$
(D) $\frac{35}{2^{7}}$

Q171·MathematicsNumericalJEE Main 2021

Let

B_i(i = 1, 2, 3)

be three independent events in a sample space. The probability that only

B_1

occur is

\alpha

, only

B_2

occurs is

\beta

and only

B_3

occurs is

\gamma

. Let p be the probability that none of the events

B_i

occurs and these 4 probabilities satisfy the equations

(\alpha - 2\beta)p = \alpha\beta

and

(\beta - 3\gamma)p = 2\beta\gamma

(All the probabilities are assumed to lie in the interval (0, 1)). Then

\frac{P(B_1)}{P(B_3)}

is equal to _____

Q172·MathematicsSingle correctJEE Advanced 2020

Let

C_{1}

and

C_{2}

be two biased coins such that the probabilities of getting head in a single toss are

\frac{2}{3}

and

\frac{1}{3}

, respectively. Suppose

\alpha

is the number of heads that appear when

C_{1}

is tossed twice, independently, and suppose

\beta

is the number of heads that appear when

C_{2}

is tossed twice, independently, Then probability that the roots of the quadratic polynomial

x^{2} - \alpha x + \beta

are real and equal, is

(A) $\frac{40}{81}$
(B) $\frac{20}{81}$
(C) $\frac{1}{2}$
(D) $\frac{1}{4}$

Q173·MathematicsNumericalJEE Advanced 2020

Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If

p

is the probability that this perfect square is an odd number, then the value of

14p

is ________

Q174·MathematicsIntegerJEE Advanced 2020

The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is ________

Q175·MathematicsSingle correctJEE Main 2020

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 \cap B \cap C) = 0.2

,

P(B \cap C) = \beta

and

P(A \cup B \cup C) = \alpha

, where

0.85 \leq \alpha \leq 0.95

, then

\beta

lies in the interval:

(A) $[0.35, 0.36]$
(B) $[0.25, 0.35]$
(C) $[0.20, 0.25]$
(D) $[0.36, 0.40]$

Q176·MathematicsSingle correctJEE Main 2020

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 positive common difference, is:

(A) $\dfrac{15}{101}$
(B) $\dfrac{5}{101}$
(C) $\dfrac{5}{33}$
(D) $\dfrac{10}{99}$

Q177·MathematicsNumericalJEE Main 2020

In a bombing attack, there is 50% chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least 99% chance of completely destroying the target, is __________.

Q178·MathematicsNumericalJEE Main 2020

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, a ____.

Q179·MathematicsSingle correctJEE Main 2020

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:

(A) $\frac{5}{6}$
(B) $\frac{31}{61}$
(C) $\frac{30}{61}$
(D) $\frac{5}{31}$

Q180·MathematicsNumericalJEE Main 2020

The probability of a man hitting a target is

\frac{1}{10}

. The least number of shots required, so that the probability of his hitting the target at least once is greater than

\frac{1}{4}

, is __________.

Q181·MathematicsSingle correctJEE Main 2020

A die is thrown two times and the sum of the scorers appearing on the die is observed to be a multiple of 4. Then the conditional probability that the score 4 has appeared atleast once is

(A) $\frac{1}{9}$
(B) $\frac{1}{3}$
(C) $\frac{1}{8}$
(D) $\frac{1}{4}$

Q182·MathematicsSingle correctJEE Main 2020

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

(A) $\frac{134}{10^4}$
(B) $\frac{121}{10^4}$
(C) $\frac{150}{10^4}$
(D) $\frac{135}{10^4}$

Q183·MathematicsSingle correctJEE Main 2020

In a box, there are 20 cards, out of which 10 are labeled as A and the remaining 10 are labelled as B. Cards are drawn at random, one after the other and with replacement, till a second A – card is obtained. The probability that the second A – card appears before the third B – card is:

(A) $\dfrac{9}{16}$
(B) $\dfrac{15}{16}$
(C) $\dfrac{13}{16}$
(D) $\dfrac{11}{16}$

Q184·MathematicsSingle correctJEE Main 2020

A random variable X has the following probability distribution: Then

P(X > 2)

is equal to:

X	1	2	3	4	5
P(X)	$K^{2}$	2K	K	2K	$5K^{2}$

(A) $\dfrac{1}{36}$
(B) $\dfrac{7}{12}$
(C) $\dfrac{23}{36}$
(D) $\dfrac{1}{6}$

Q185·MathematicsSingle correctJEE Main 2020

Let A and B be two independent events such that

P(A)=\dfrac{1}{3}

and

P(B)=\dfrac{1}{6}

. Then, which of the following is TRUE?

(A) $P\left(\dfrac{A}{B}\right)=\dfrac{2}{3}$
(B) $P\left(\dfrac{A'}{B'}\right)=\dfrac{1}{3}$
(C) $P\left(\dfrac{A}{B'}\right)=\dfrac{1}{3}$
(D) $P\left(\dfrac{A}{(A\cup B)}\right)=\dfrac{1}{4}$

Q186·MathematicsSingle correctJEE Main 2020

Let A and B be two events such that the probability that exactly one of them occurs is

\frac{2}{5}

and the probability that A or B occurs is

\frac{1}{2}

, then the probability of both of them occur together is:

(A)0.01
(B)0.10
(C)0.20
(D)0.02

Q187·MathematicsSingle correctJEE Main 2020

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is

\dfrac{1}{4}

. If the probability that at most two machines will be out of service on the same day is

\left(\dfrac{3}{4}\right)^{3}k

, then k is equal to:

(A) $4$
(B) $\dfrac{17}{8}$
(C) $\dfrac{17}{2}$
(D) $\dfrac{17}{4}$

Q188·MathematicsSingle correctJEE Main 2020

An unbiased coin is tossed 5 time. Suppose that a variable X is assigned the value k when k consecutive heads are obtained for k = 3, 4, 5, otherwise X takes the value −1. Then the expected value of X, is

(A) $-\frac{3}{16}$
(B) $-\frac{1}{8}$
(C) $\frac{1}{8}$
(D) $\frac{3}{16}$

Q189·MathematicsNumericalJEE Advanced 2019

Let

|x|

denote the number of elements in a set X. Let

S = \{1, 2, 3, 4, 5, 6\}

be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such that

1 \le |B| < |A|

, equals ______

Q190·MathematicsMultiple correctJEE Advanced 2019

There are three bags

B_1

,

B_2

and

B_3

. The bag

B_1

contains 5 red and 5 green balls,

B_2

contains 3 red and 5 green balls and

B_3

contains 5 red and 3 green balls. Bags

B_1

,

B_2

and

B_3

have probabilities

\frac{3}{10}

,

\frac{3}{10}

and

\frac{4}{10}

respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

(A)Probability that the chosen ball is green equals $\frac{39}{80}$
(B)Probability that the chosen ball is green, given that the selected bag is $B_3$ , equals $\frac{3}{8}$
(C)Probability that the selected bag is $B_3$ and the chosen ball is green equals $\frac{3}{10}$
(D)Probability that the selected bag is $B_3$ , given that the chosen ball is green, equals $\frac{5}{13}$

Q191·MathematicsNumericalJEE Advanced 2019

Let S be the sample space of all

3 \times 3

matrices with entries from the set

\{0, 1\}

. Let the events

E_1

and

E_2

be given by

E_1 = \{A \in S : \det A = 0\}

and

E_2 = \{A \in S :

sum of entries of A is

7\}

If a matrix is chosen at random from S, then the conditional probability

P(E_1/E_2)

equals ____

Q192·MathematicsSingle correctJEE Main 2019

Let a random variable X have a binomial distribution with mean 8 and variance 4. If

P(X \le 2) = \frac{k}{2^{16}}

, then k is equal to :

(A)17
(B)137
(C)1
(D)121

Q193·MathematicsSingle correctJEE Main 2019

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 equilateral is :

(A) $\frac{3}{10}$
(B) $\frac{1}{5}$
(C) $\frac{1}{10}$
(D) $\frac{3}{20}$

Q194·MathematicsSingle correctJEE Main 2019

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 9, and loses Rs. 6 for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is :

(A) $\dfrac{1}{4}$ loss
(B)2 gain
(C) $\dfrac{1}{2}$ gain
(D) $\dfrac{1}{2}$ loss

Q195·MathematicsSingle correctJEE Main 2019

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

\dfrac{4}{5}

, then the probability that he is unable to solve less than two problems is :

(A) $\dfrac{164}{25}\left(\dfrac{1}{5}\right)^{48}$
(B) $\dfrac{201}{5}\left(\dfrac{1}{5}\right)^{49}$
(C) $\dfrac{54}{5}\left(\dfrac{4}{5}\right)^{49}$
(D) $\dfrac{316}{25}\left(\dfrac{4}{5}\right)^{48}$

Q196·MathematicsSingle correctJEE Main 2019

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

(A)8
(B)6
(C)7
(D)5

Q197·MathematicsSingle correctJEE Main 2019

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 children are girls given that at least two are girls is:

(A) $\dfrac{1}{10}$
(B) $\dfrac{1}{17}$
(C) $\dfrac{1}{12}$
(D) $\dfrac{1}{11}$

Q198·MathematicsSingle correctJEE Main 2019

Four persons can hit a target correctly with probabilities

\dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}

and

\dfrac{1}{8}

respectively. If all hit at the target independently, then the probability that the target would be hit, is:

(A) $\dfrac{25}{32}$
(B) $\dfrac{25}{192}$
(C) $\dfrac{7}{32}$
(D) $\dfrac{1}{192}$

Q199·MathematicsSingle correctJEE Main 2019

Two newspaper A and B are published in a city. It is known that 25% of the city populations 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 advertisement is:

(A)12.8
(B)13.5
(C)13.9
(D)13

Q200·MathematicsSingle correctJEE Main 2019

Let A and b be two non-null events such that

A\subset B

. Then, which of the following statements is always correct?

(A) $P(A|B)=1$
(B) $P(A|B)\le P(A)$
(C) $P(A|B)=P(B)-P(A)$
(D) $P(A|B)\ge P(A)$

Q201·MathematicsSingle correctJEE Main 2019

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:

(A)2
(B)3
(C)4
(D)5

Q202·MathematicsSingle correctJEE Main 2019

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 the probability that the student selected has opted neither for NCC nor for NSS is :

(A) $\frac{1}{6}$
(B) $\frac{1}{3}$
(C) $\frac{2}{3}$
(D) $\frac{5}{6}$

Q203·MathematicsSingle correctJEE Main 2019

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 throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :

(A) $\dfrac{400}{9}$ loss
(B) $0$
(C) $\dfrac{400}{3}$ gain
(D) $\dfrac{400}{3}$ loss

Q204·MathematicsSingle correctJEE Main 2019

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 throw of the die is equal to:

(A) $\dfrac{200}{6^{5}}$
(B) $\dfrac{150}{6^{5}}$
(C) $\dfrac{225}{6^{5}}$
(D) $\dfrac{175}{6^{5}}$

Q205·MathematicsSingle correctJEE Main 2019

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 both the numbers are even is:

(A) $\frac{7}{10}$
(B) $\frac{1}{2}$
(C) $\frac{2}{5}$
(D) $\frac{3}{5}$

Q206·MathematicsSingle correctJEE Main 2019

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 balls drawn, then

\left(\dfrac{\text{mean of }X}{\text{standard deviation of }X}\right)

is equal to:

(A)4
(B) $4\sqrt{3}$
(C) $3\sqrt{2}$
(D) $\dfrac{4\sqrt{3}}{3}$

Q207·MathematicsSingle correctJEE Main 2019

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 randomly chosen subset of S is ‘nice’ is:

(A) $\frac{7}{2^{20}}$
(B) $\frac{5}{2^{20}}$
(C) $\frac{4}{2^{20}}$
(D) $\frac{6}{2^{20}}$

Q208·MathematicsSingle correctJEE Main 2019

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 if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is:

(A) $\dfrac{26}{49}$
(B) $\dfrac{32}{49}$
(C) $\dfrac{27}{49}$
(D) $\dfrac{21}{49}$

Q209·MathematicsSingle correctJEE Main 2019

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 in the two drawn cards. Then

P(X=1)+P(X=2)

equals:

(A) $\dfrac{49}{169}$
(B) $\dfrac{52}{169}$
(C) $\dfrac{24}{169}$
(D) $\dfrac{25}{169}$

Q210·MathematicsSingle correctJEE Advanced 2018

PARAGRAPH "A" There are five students

S_{1}

,

S_{2}

,

S_{3}

,

S_{4}

and

S_{5}

in a music class and for them there are five seats

R_{1}

,

R_{2}

,

R_{3}

,

R_{4}

and

R_{5}

arranged in a row, where initially the seat

R_{i}

is allotted to the student

S_{i}

,

i = 1, 2, 3, 4, 5

. But, on the examination day, the five students are randomly allotted the five seats. (There are two questions based on PARAGRAPH "A", the question given below is one of them) For

i = 1, 2, 3, 4

, let

T_{i}

denote the event that the students

S_{i}

and

S_{i+1}

do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event

T_{1} \cap T_{2} \cap T_{3} \cap T_{4}

is

(A) $\frac{1}{15}$
(B) $\frac{1}{10}$
(C) $\frac{7}{60}$
(D) $\frac{1}{5}$

Q211·MathematicsSingle correctJEE Advanced 2018

PARAGRAPH "A" There are five students

S_{1}

,

S_{2}

,

S_{3}

,

S_{4}

and

S_{5}

in a music class and for them there are five seats

R_{1}

,

R_{2}

,

R_{3}

,

R_{4}

and

R_{5}

arranged in a row, where initially the seat

R_{i}

is allotted to the student

S_{i}

,

i = 1, 2, 3, 4, 5

. But, on the examination day, the five students are randomly allotted the five seats. (There are two questions based on PARAGRAPH "A", the question given below is one of them) The probability that, on the examination day, the student

S_{1}

gets the previously allotted seat

R_{1}

, and NONE of the remaining students gets the seat previously allotted to him/her is

(A) $\frac{3}{40}$
(B) $\frac{1}{8}$
(C) $\frac{7}{40}$
(D) $\frac{1}{5}$

Q212·MathematicsMultiple correctJEE Advanced 2017

Let X and Y be two events such that

P(X) = \frac{1}{3}

,

P(X|Y) = \frac{1}{2}

and

P(Y|X) = \frac{2}{5}

. Then

(A) $P(X'|Y) = \frac{1}{2}$
(B) $P(X \cap Y) = \frac{1}{5}$
(C) $P(X \cup Y) = \frac{2}{5}$
(D) $P(Y) = \frac{4}{15}$

Q213·MathematicsSingle correctJEE Advanced 2017

Three randomly chosen nonnegative integers

x

,

y

and

z

are found to satisfy the equation

x + y + z = 10

. Then the probability that

z

is even, is

(A) $\dfrac{36}{55}$
(B) $\dfrac{6}{11}$
(C) $\dfrac{1}{2}$
(D) $\dfrac{5}{11}$

Q214·MathematicsSingle correctJEE Advanced 2016

Football teams

T_{1}

and

T_{2}

have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of

T_{1}

winning, drawing and losing a game against

T_{2}

are

\frac{1}{2}

,

\frac{1}{6}

and

\frac{1}{3}

, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams

T_{1}

and

T_{2}

, respectively, after two games.

P(X > Y)

is

(A) $\frac{1}{4}$
(B) $\frac{5}{12}$
(C) $\frac{1}{2}$
(D) $\frac{7}{12}$

Q215·MathematicsSingle correctJEE Advanced 2016

Football teams

T_{1}

and

T_{2}

have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of

T_{1}

winning, drawing and losing a game against

T_{2}

are

\frac{1}{2}

,

\frac{1}{6}

and

\frac{1}{3}

, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams

T_{1}

and

T_{2}

, respectively, after two games.

P(X = Y)

is

(A) $\frac{11}{36}$
(B) $\frac{1}{3}$
(C) $\frac{13}{36}$
(D) $\frac{1}{2}$

Q216·MathematicsSingle correctJEE Advanced 2016

A computer producing factory has only two plants

T_1

and

T_2

. Plant

T_1

produces 20% and plant

T_2

produces 80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that P(computer turns out to be defective given that it is produced in plant

T_1

) = 10 P(computer turns out to be defective given that it is produced in plant

T_2

), where P(E) denotes the probability of an event E. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant

T_2

is

(A) $\frac{36}{73}$
(B) $\frac{47}{79}$
(C) $\frac{78}{93}$
(D) $\frac{75}{83}$

Q217·MathematicsIntegerJEE Advanced 2015

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96 is

Q218·MathematicsMultiple correctJEE Advanced 2015

Let

n_{1}

and

n_{2}

be the number of red and black balls, respectively, in box I. Let

n_{3}

and

n_{4}

be the number of red and black balls, respectively, in box II. A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is

\dfrac{1}{3}

, then the correct option(s) with the possible values of

n_{1}

and

n_{2}

is(are)

(A) $n_{1} = 4$ , $n_{2} = 6$
(B) $n_{1} = 2$ , $n_{2} = 3$
(C) $n_{1} = 10$ , $n_{2} = 20$
(D) $n_{1} = 3$ , $n_{2} = 6$

Q219·MathematicsMultiple correctJEE Advanced 2015

Let

n_{1}

and

n_{2}

be the number of red and black balls, respectively, in box I. Let

n_{3}

and

n_{4}

be the number of red and black balls, respectively, in box II. One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is

\dfrac{1}{3}

, then the correct option(s) with the possible values of

n_{1}

,

n_{2}

,

n_{3}

and

n_{4}

is(are)

(A) $n_{1} = 3$ , $n_{2} = 3$ , $n_{3} = 5$ , $n_{4} = 15$
(B) $n_{1} = 3$ , $n_{2} = 6$ , $n_{3} = 10$ , $n_{4} = 50$
(C) $n_{1} = 8$ , $n_{2} = 6$ , $n_{3} = 5$ , $n_{4} = 20$
(D) $n_{1} = 6$ , $n_{2} = 12$ , $n_{3} = 5$ , $n_{4} = 20$

Q220·MathematicsSingle correctJEE Advanced 2014

Box 1 contains three cards bearing numbers 1, 2, 3 ; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5 ; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let

x_{i}

be the number on the card drawn from the

i^{th}

box,

i = 1, 2, 3

. The probability that

x_{1} + x_{2} + x_{3}

is odd, is

(A) $\frac{29}{105}$
(B) $\frac{53}{105}$
(C) $\frac{57}{105}$
(D) $\frac{1}{2}$

Q221·MathematicsSingle correctJEE Advanced 2014

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) $\frac{3}{4}$

Q222·MathematicsSingle correctJEE Advanced 2014

Box 1 contains three cards bearing numbers 1, 2, 3 ; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5 ; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let

x_{i}

be the number on the card drawn from the

i^{th}

box,

i = 1, 2, 3

. The probability that

x_{1}, x_{2}, x_{3}

are in an arithmetic progression, is

(A) $\frac{9}{105}$
(B) $\frac{10}{105}$
(C) $\frac{11}{105}$
(D) $\frac{7}{105}$

Q223·MathematicsSingle correctJEE Advanced 2013

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 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box

B_{2}

is

(A) $\frac{116}{181}$
(B) $\frac{126}{181}$
(C) $\frac{65}{181}$
(D) $\frac{55}{181}$

Q224·MathematicsIntegerJEE Advanced 2013

Of the three independent events

E_1

,

E_2

, and

E_3

, the probability that only

E_1

occurs is

\alpha

, only

E_2

occurs is

\beta

and only

E_3

occurs is

\gamma

. Let the probability

p

that none of events

E_1

,

E_2

or

E_3

occurs satisfy the equations

(\alpha-2\beta)p=\alpha\beta

and

(\beta-3\gamma)p=2\beta\gamma

. All the given probabilities are assumed to lie in the interval

(0,1)

. Then

\dfrac{\text{Probability of occurrence of }E_1}{\text{Probability of occurrence of }E_3}=

___________

Q225·MathematicsSingle correctJEE Advanced 2013

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) $\frac{566}{648}$

Q226·MathematicsSingle correctJEE Advanced 2013

Four persons independently solve a certain problem correctly with probabilities

\frac{1}{2},\frac{3}{4},\frac{1}{4},\frac{1}{8}

. Then the probability that the problem is solved correctly by at least one of them is

(A) $\frac{235}{256}$
(B) $\frac{21}{256}$
(C) $\frac{3}{256}$
(D) $\frac{253}{256}$

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