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Probability — JEE Main & Advanced Mathematics PYQs

561 previous year questions from Probability with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

561 questionsMathematicsSolutions on every page
1

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 ch

JEE Advanced 2026 Solution View → 2

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

JEE Main 2026 Solution View → 3

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 gree

JEE Main 2026 Solution View → 4

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 9 16 , then N is

JEE Main 2026 Solution View → 5

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

JEE Main 2026 Solution View → 6

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 fr

JEE Main 2026 Solution View → 7

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 a b , where a,b N and (a,b)=1 , then a+

JEE Main 2026 Solution View → 8

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 ____

JEE Main 2026 Solution View → 9

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 m n , (m,

JEE Main 2026 Solution View → 10

The probability distribution of a random variable X is given below : ( array |c|c|c|c|c|c|c|c|c| X & 4k & 30 7 k & 32 7 k & 34 7 k & 36 7 k & 38 7 k & 40 7 k & 6k P(X) & 2 15 & 1 1

JEE Main 2026 Solution View → 11

A bag contains 10 balls out of which k are red and ( 10-k ) are black, where 0 k 10 . If three balls are drawn at random without replacement and all of them are found to be black,

JEE Main 2026 Solution View → 12

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 ) P ( S ) such that A B = . If the pr

JEE Main 2026 Solution View → 13

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

JEE Main 2026 Solution View → 14

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. Th

JEE Main 2026 Solution View → 15

From the first 100 natural numbers, two numbers first a and then b are selected randomly without replacement. If the probability that a-b 10 is m n , gcd (m, n)=1 , then m+n is equ

JEE Main 2026 Solution View → 16

If a random variable x has the probability distribution array |c|c|c|c|c|c|c|c|c| x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 P(x) & 0 & 2k & k & 3k & 2k² & 2k & k²+k & 7k² array then P(3 <

JEE Main 2026 Solution View → 17

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

JEE Main 2026 Solution View → 18

A random variable X takes values 0,1,2,3 with probabilities 2 a+1 30 , 8 a-1 30 , 4 a+1 30 , b respectively, where a , b R . Let and respectively be the mean and standard deviation

JEE Main 2026 Solution View → 19

Three students S₁, S₂ and S₁ are given a problem to solve. Consider the following events : U: At least one of S₁, S₂ and S₃ can solve the problem, V: S₁ can solve the problem, give

JEE Advanced 2025 Solution View → 20

A factory has a total of three manufacturing units, M₁, M₂ , and M₃ , which produce bulbs independent of each other. The units M₁, M₂ , and M₃ produce bulbs in the proportions of 2

JEE Advanced 2025 Solution View → 21

If A and B are two events such that P(A)=0.7 , P ( B )=0.4 and P ( A B )=0.5 , where B denotes the complement of B , then P(B (A B )) is equal:-

JEE Main 2025 Solution View → 22

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

JEE Main 2025 Solution View → 23

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 )=2 E ( X ) . Then the value of 8 p -1 is :

JEE Main 2025 Solution View → 24

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 11 50 , the

JEE Main 2025 Solution View → 25

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:

JEE Main 2025 Solution View → 26

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

JEE Main 2025 Solution View → 27

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 , where k is a constant, then P ( X 3) is equal to

JEE Main 2025 Solution View → 28

All five letter words are made using all the letters A , B , C , D , E and arranged as in an English dictionary with serial numbers. Let the word at serial number n be denoted by W

JEE Main 2025 Solution View → 29

Given three indentical bags each containing 10 balls, whose colours are as follows : array cccc & Red & Blue & Green Bag I & 3 & 2 & 5 Bag II & 4 & 3 & 3 Bag III & 5 & 1 & 4 array

JEE Main 2025 Solution View → 30

Three distinct numbers are selected randomly from the set 1,2,3, , 40 . If the probability, that the selected numbers are in an increasing G.P. is m n , gcd (m, n)=1 , then m+n is

JEE Main 2025 Solution View → 31

Let A = [ a _ i j ] be a 2 2 matrix such that a _ i j 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 vari

JEE Main 2025 Solution View → 32

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 d

JEE Main 2025 Solution View → 33

Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selec

JEE Main 2025 Solution View → 34

Bag B₁ contains 6 white and 4 blue balls, Bag B₂ contains 4 white and 6 blue balls, and Bag B₃ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball i

JEE Main 2025 Solution View → 35

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 = -1 ) is non-zero, equals

JEE Main 2025 Solution View → 36

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

JEE Main 2025 Solution View → 37

Let A= [a_ i j ] be a square matrix of order 2 with entries either 0 or 1 . Let E be the event that A is an invertible matrix. Then the probability P ( E ) is :

JEE Main 2025 Solution View → 38

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 probabilit

JEE Main 2025 Solution View → 39

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 :

JEE Main 2025 Solution View → 40

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

JEE Main 2025 Solution View → 41

If A and B are two events such that P(A B)=0.1 , and P(A B) and P(B A) are the roots of the equation 12 x^2-7 x+1=0 , then the value of P ( A B ) P ( A B ) is :

JEE Main 2025 Solution View → 42

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given tha

JEE Main 2025 Solution View → 43

A coin is tossed three times. Let X denote the number of times a tail follows a head. If and ^2 denote the mean and variance of X , then the value of 64 ( + ^2 ) is :

JEE Main 2025 Solution View → 44

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 ran

JEE Advanced 2024 Solution View → 45

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

JEE Advanced 2024 Solution View → 46

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

JEE Main 2024 Solution View → 47

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 a x^2+b x+c=0 has all real ro

JEE Main 2024 Solution View → 48

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 co

JEE Main 2024 Solution View → 49

Let the sum of two positive integers be 24 . If the probability, that their product is not less than 3 4 times their greatest possible product, is m n , where gcd (m, n)=1 , then n

JEE Main 2024 Solution View → 50

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 X and

JEE Main 2024 Solution View → 51

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:

JEE Main 2024 Solution View → 52

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 t

JEE Main 2024 Solution View → 53

A company has two plants A and B to manufacture motorcycles. 60 % motorcycles are manufactured at plant A and the remaining are manufactured at plant B .80 % of the motorcycles man

JEE Main 2024 Solution View → 54

The coefficients a, b, c in the quadratic equation a x^2+b x+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

JEE Main 2024 Solution View → 55

The coefficients a, b, c in the quadratic equation a x^2+b x+c=0 are chosen from the set 1,2,3,4,5,6,7,8 . The probability of this equation having repeated roots is :

JEE Main 2024 Solution View → 56

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as 1 3 and 2 3 respectively. Let x be the number of matches that the team wins, and y b

JEE Main 2024 Solution View → 57

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 d

JEE Main 2024 Solution View → 58

Let Ajay will not appear in JEE exam with probability p = 2 7 , while both Ajay and Vijay will appear in the exam with probability q = 1 5 . Then the probability, that Ajay will ap

JEE Main 2024 Solution View → 59

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 other 2 balls are black

JEE Main 2024 Solution View → 60

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-

JEE Main 2024 Solution View → 61

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, tha

JEE Main 2024 Solution View → 62

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

JEE Main 2024 Solution View → 63

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 the

JEE Main 2024 Solution View → 64

Two integers x and y are chosen with replacement from the set 0 , 1 , 2 , 3 , … . . , 10 . Then the probability that | x - y | > 5 is :

JEE Main 2024 Solution View → 65

An integer is chosen at random from the integers 1 , 2 , 3 , . . . . . , 50 . The probability that the chosen integer is a multiple of atleast one of 4 , 6 and 7 is

JEE Main 2024 Solution View → 66

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

JEE Main 2024 Solution View → 67

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 d

JEE Main 2024 Solution View → 68

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 ≥ 3 ) and c = P ( X ≥ 6 ∣ X > 3 ) . Then b +

JEE Main 2024 Solution View → 69

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 1 3 , then the p

JEE Advanced 2023 Solution View → 70

Consider the 6 &#215; 6 square in the figure. Let A 1 , A 2 , &#8230; , A 49 be the points of intersections (dots in the picture) in some order. We say that A i and A j are friends

JEE Advanced 2023 Solution View → 71

Let X = x , y &#8712; &#8484; &#215; &#8484; : x 2 8 + y 2 20 &#60; 1 and y 2 &#60; 5 x . Three distinct point P , Q and R are randomly chosen from X . Then the probability that P

JEE Advanced 2023 Solution View → 72

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 tha

JEE Main 2023 Solution View → 73

The random variable X follows binomial distribution B ( n , p ) , for which the difference of the mean and the variance is 1 . If 2 P ( X = 2 ) = 3 P ( X = 1 ) , then n 2 P ( X &#6

JEE Main 2023 Solution View → 74

A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If X denotes the number of tosses of the coin, then

JEE Main 2023 Solution View → 75

Two dice A and B are rolled. Let the numbers obtained on A and B be &#945; and &#946; respectively. If the variance of &#945; - &#946; is p q , where p and q are co-prime, then the

JEE Main 2023 Solution View → 76

A fair n n &#62; 1 faces die is rolled repeatedly until a number less than n appears. If the mean of the number of tosses required is n 9 , then n is equal to

JEE Main 2023 Solution View → 77

Let the probability of getting head for a biased coin be 1 4 . It is tossed repeatedly until a head appears. Let N be the number of tosses required. If the probability that the equ

JEE Main 2023 Solution View → 78

Let S = M = a i j , a i j &#8712; 0 , 1 , 2 , 1 &#8804; i , j &#8804; 2 be a sample space and A M &#8712; S : M is invertible be an even. Then P A is equal to

JEE Main 2023 Solution View → 79

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 eve

JEE Main 2023 Solution View → 80

Let N denote the sum of the numbers obtained when two dice are rolled. If the probability that 2 N &#60; N ! is m n where m and n are coprime, then 4 m - 3 n is equal to

JEE Main 2023 Solution View → 81

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 , &#8230; &#8230; , where k is a constant, then P ( X &

JEE Main 2023 Solution View → 82

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 bo

JEE Main 2023 Solution View → 83

Three dice are rolled. If the probability of getting different numbers on the three dice is p q , where p and q are co-prime, then q - p is equal to

JEE Main 2023 Solution View → 84

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is k 3 11 , then k is equal to

JEE Main 2023 Solution View → 85

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 appe

JEE Main 2023 Solution View → 86

In a binomial distribution B ( n , p ) , the sum and product of the mean & variance are 5 and 6 respectively, then find 6 ( n + p - q ) is equal to :-

JEE Main 2023 Solution View → 87

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 = 11 36 , then a is equal to

JEE Main 2023 Solution View → 88

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

JEE Main 2023 Solution View → 89

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

JEE Main 2023 Solution View → 90

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 :

JEE Main 2023 Solution View → 91

Let S = w 1 , w 2 , &#8230; . be the sample space associated to a random experiment. Let P w n = P w n - 1 2 , n &#8805; 2 . Let A = 2 k + 3 l ; k , l &#8712; &#8469; and B = w n ;

JEE Main 2023 Solution View → 92

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 l

JEE Main 2023 Solution View → 93

There 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 appl

JEE Main 2023 Solution View → 94

Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that N - 2 , 3 N , N + 2 are in geometric progression be k 48 . Then the value of k i

JEE Main 2023 Solution View → 95

25 % of the population are smokers. A smoker has 27 times more chances to develop lung cancer then a non-smoker. A person is diagnosed with lung cancer and the probability that thi

JEE Main 2023 Solution View → 96

Let M be the maximum value of the product of two positive integers when their sum is 66 . Let the sample space S = x &#8712; Z : x ( 66 - x ) &#8805; 5 9 M and the event A = x &#87

JEE Main 2023 Solution View → 97

The urns A , B and C contains 4 red, 6 black; 5 red, 5 black and &#955; red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball draw

JEE Main 2023 Solution View → 98

Let &#937; be the sample space and A &#8838; &#937; be an event. Given below are two statements: (S1): If P ( A ) = 0 , then A = &#981; (S2): If P ( A ) = , then A = &#937; Then

JEE Main 2023 Solution View → 99

Suppose that Box-I contains 8 red, 3 blue and 5 green balls, Box-ll contains 24 red, 9 blue and 15 green balls, Box-III contains 1 blue, 12 green and 3 yellow balls, Box-IV contain

JEE Advanced 2022 Solution View → 100

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.

JEE Advanced 2022 Solution View → 101

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

JEE Main 2022 Solution View → 102

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

JEE Main 2022 Solution View → 103

Let S = 1 , 2 , 3 , &#8230; , 2022 . Then the probability, that a randomly chosen number n from the set S such that H C F n , 2022 = 1 , is

JEE Main 2022 Solution View → 104

Let A and B be two events such that P B &#8739; A = 2 5 , P A &#8739; B = 1 7 and P A &#8745; B = 1 9 . Consider S 1 P A &#39; &#8746; B = 5 6 , S 2 P A &#39; &#8745; B &#39; = 1 1

JEE Main 2022 Solution View → 105

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 &#963; 2 is the variance of X

JEE Main 2022 Solution View → 106

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

JEE Main 2022 Solution View → 107

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 &#62; n - 3 = k 2 n , then k is equal

JEE Main 2022 Solution View → 108

A six faced die is biased such that 3 &#215; P (a prime number) = 6 &#215; P (a composite number) = 2 &#215; P 1 . Let X be a random variable that counts the number of times one ge

JEE Main 2022 Solution View → 109

Let S be the sample space of all five digit numbers. If p is the probability that a randomly selected number from S , is a multiple of 7 but not divisible by 5 , then 9 p is equal

JEE Main 2022 Solution View → 110

Let X be a binomially distributed random variable with mean 4 and variance 4 3 . Then 54 P X &#8804; 2 is equal to

JEE Main 2022 Solution View → 111

The mean and variance of a binomial distribution are &#945; and &#945; 3 respectively. If P X = 1 = 4 243 , then P X = 4 or 5 is equal to:

JEE Main 2022 Solution View → 112

Let E 1 , E 2 , E 3 be three mutually exclusive events such that P E 1 = 2 + 3 p 6 , P E 2 = 2 - p 8 and P E 3 = 1 - p 2 . If the maximum and minimum values of p are p 1 and p 2 th

JEE Main 2022 Solution View → 113

If A and B are two events such that P A = 1 3 , P B = 1 5 and P A &#8746; B = 1 2 , then P A B ' + P B A ' is equal to

JEE Main 2022 Solution View → 114

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 :

JEE Main 2022 Solution View → 115

If the numbers appeared on the two throws of a fair six faced die are &#945; and &#946; , then the probability that x 2 + &#945; x + &#946; &#62; 0 , for all x &#8712; R , is

JEE Main 2022 Solution View → 116

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

JEE Main 2022 Solution View → 117

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

JEE Main 2022 Solution View → 118

If a point A x , y lies in the region bounded by the y -axis, straight lines 2 y + x = 6 and 5 x - 6 y = 30 , then the probability that y &#60; 1 is

JEE Main 2022 Solution View → 119

Let S = E , E 2 &#8230; E 8 be a sample space of raddom experiment such that P E n = n 36 for every n = 1 , 2 &#8230; . 8 . Then the number of elements in the set A &#8834; S : P A

JEE Main 2022 Solution View → 120

Five numbers x 1 , x 2 , x 3 , x 4 , x 5 are randomly selected from the numbers 1 , 2 , 3 , &#8230; &#8230; , 18 and are arranged in the increasing order x 1 &#60; x 2 &#60; x 1 &#

JEE Main 2022 Solution View →
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