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Probability

Get chapter-wise JEE Main & Advanced questions with solutions

Q Jee Main 2024
Let the area of the region $\left\{(x, y): 0 \leq x \leq 3,0 \leq y \leq \min \left\{\mathrm{x}^2+2,2 \mathrm{x}+2\right\}\right\}$ be...
JEE Main Mathematics Medium
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Q Jee Main 2024
An integer is chosen at random from the integers $1,2,3, \ldots, 50$. The probability that the chosen integer is a...
JEE Main Mathematics Medium
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Q JEE MAIN 2024
Let Ajay will not appear in JEE exam with probability $\mathrm{p}=\frac{2}{7}$, while both Ajay and Vijay will appear in the...
JEE Main Mathematics Easy
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Q JEE-Main 2024
An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The...
JEE Main Mathematics Medium
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Q Jee Main 2024
A coin is based so that a head is twice as likely to occur as a tail. If the coin...
JEE Main Mathematics Easy
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Q JEE MAIN 2024
Bag A contains 3 white, 7 red balls and bag $B$ contains 3 white, 2 red balls. One bag is...
JEE Main Mathematics Easy
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Q JEE Main 2026
Let S be a set of 5 elements and $\mathrm{P}(\mathrm{S})$ denote the power set of S . Let E be...
JEE Main Mathematics Hard
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Q JEE MAIN 2026
A bag contains 10 balls out of which $k$ are red and $(10-k)$ are black, where $...
JEE Main Mathematics Hard
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Q JEE MAIN 2026
From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then...
JEE Main Mathematics Easy
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Q JEE MAIN_2026_
From the first 100 natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If the...
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Q JEE MAIN 2026
Let $n$ be the number obtained on rolling a fair die. If the probability that the system $$ \begin{aligned} &...
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Q JEE MAIN_2026
If a random variable X has the probability distribution then $\mathrm{P}(3
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Q JEE MAIN_2026_
Two distinct numbers $a$ and $b$ are selected at random from $1,2,3, \ldots, 50$. The probability, that their product ab...
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Q JEE MAIN 2026
A random variable X takes values $0,1,2,3$ with probabilities $\frac{2 a+1}{30}, \frac{8 a-1}{30}, \frac{4 a+1}{30}, b$ respectively, where $...
JEE Main Mathematics Medium
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Q JEE MAIN_2019_
Assume that each born child is equally likely to be a boy or a girl. If two families have two...
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Q JEE MAIN 2019
Four persons can hit a target correctly with probabilities $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and $\frac{1}{8}$ respectively. If all hit at the...
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Q JEE MAIN 2019
A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice),...
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Q JEE Main 2020
Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A...
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Q JEE-MAIN 2019
In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the...
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Q JEE MAIN 2019
An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the...
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Q JEE MAIN 2019
For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that...
JEE Main Mathematics Hard
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Q JEE MAIN 2020
Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up...
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Q JEE MAIN 2020
Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they...
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Q JEE MAIN 2020
The probabilities of three events $\mathrm{A}, \mathrm{B}$ and C are given by $\mathrm{P}(\mathrm{A})=0.6, \mathrm{P}(\mathrm{B})=0.4$ and $\mathrm{P}(\mathrm{C})=0.5$. If $...
JEE Main Mathematics Hard
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Q JEE MAIN 2019
An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the...
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Q JEE MAIN 2019
Two integers are selected at random from the set {1, 2, ..., 11}. Given that the sum of selected numbers...
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Q JEE MAIN 2020
In a box, there are 20 cards, out of which 10 are labelled as A and the remaining 10 are...
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Q JEE MAIN 2019
Minimum number of times a fair coin must be tossed so that the probability of getting at least one head...
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Q JEE MAIN 2020
Let $A$ and $B$ be two independent events such that $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}$. Then, which of the following is TRUE...
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Q JEE MAIN 2020
In a game two players A and B take turns in throwing a pair of fair dice starting with player...
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Q JEE Main 2019
An unbiased coin is tossed. If the outcome is a head, then a pair of unbiased dice is rolled and...
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Q JEE MAIN 2020
The probability of a man hitting a target is $\frac{1}{10}$. The least number of shots required, so that the probability...
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Q JEE MAIN_2021_
Each of the persons A and B independently tosses three fair coins. The probability that both of them get the...
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Q JEE MAIN 2020
A random variable X has the following probability distribution X : 1 2 3 4 5 $...
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Q JEE MAIN 2020
If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of...
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Q JEE MAIN 2019
Let S = {1, 2, ...., 20}. A subset B of S is said to be “nice”, if the sum...
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Q JEE-MAIN 2020
The probability that a randomly chosen 5-digit number is made from exactly two digits is :
JEE Main Mathematics Easy
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Q JEE MAIN 2020
Let A and B be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and...
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Q JEE MAIN 2021
An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that...
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Q JEE MAIN 2020
An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value k when k consecutive...
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Q JEE MAIN 2019
If the probability of hitting a target by a shooter, in any shot, is $\frac{1}{3}$, then the minimum number of...
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Q JEE 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 $g(3) =2 g(1)$...
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Q JEE 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...
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Q JEE MAIN_2021_
The probability distribution of random variable X is given by: Let $p=P(1
JEE Main Mathematics Easy
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Q JEE Main 2021
Let $A$ and $B$ be independent events such that $P(A)=p, P(B)=2 p$. The largest value of $p$, for which $P$...
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Q JEE MAIN 2019
Let $A$ and $B$ be two non-null events such that $A \subset$ B. Then, which of the following statements is...
JEE Main Mathematics Easy
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Q JEE MAIN 2021
Two fair dice are thrown. The numbers on them are taken as l and m, and a system of linear...
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Q JEE Main 2019
Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable...
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Q JEE MAIN 2021
Let X be a random variable with distribution. If the mean of $X$ is 2.3 and variance of $X$ is...
JEE Main Mathematics Hard
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Q JEE MAIN 2021
Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common...
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Q JEE MAIN_2021_
Let 9 distinct balls be distributed among 4 boxes, $B_1, B_2, B_3$ and $B_4$. If the probability than $B_3$ contains...
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Q JEE MAIN 2021 S2
A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability....
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Q JEE MAIN 2021 S2
A fair coin is tossed $n$-times such that the probability of getting at least one head is at least 0.9...
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Q JEE MAIN 2019
The minimum number of times one has to toss a fair coin so that the probability of observing at least...
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Q JEE MAIN 2021
The probability that a randomly selected 2 -digit number belongs to the set $\left\{n \in N:\left(2^n-2\right)\right.$ is a multiple of...
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Q JEE MAIN 2021
Let $x$ be a random variable such that the probability function of a distribution is given by $...
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Q JEE MAIN 2021
The probability of selecting integers $a \in[-5,30]$ such that $x^2+2(a+4) x-5 a+64>0$, for all $x \in R$, is :
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Q JEE MAIN 2021
Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that...
JEE Main Mathematics Easy
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Q JEE MAIN_2021_
Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 × 2 matrices. The...
JEE Main Mathematics Medium
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Q JEE MAIN 2021
Let A be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that...
JEE Main Mathematics Easy
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Q JEE MAIN 2021
Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and...
JEE Main Mathematics Medium
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Q JEE-MAIN 2021
Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability...
JEE Main Mathematics Easy
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Q JEE MAIN 2021
In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140...
JEE Main Mathematics Hard
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Q JEE MAIN_2021_
Let there be three independent events $\mathrm{E}_1, \mathrm{E}_2$ and $E_3$. The probability that only $E_1$ occurs is $\alpha$, only $E_2$...
JEE Main Mathematics Hard
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Q JEE MAIN 2021
Let $\mathrm{B}_i(\mathrm{i}=1,2,3)$ be three independent events in a sample space. The probability that only $\mathrm{B}_1$ occur is $\alpha$, only $\mathrm{B}_2$...
JEE Main Mathematics Medium
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Q JEE MAIN 2021
Let $A, B$ and $C$ be three events such that the probability that exactly one of $A$ and $B$ occurs...
JEE Main Mathematics Hard
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Q JEE MAIN 2021
An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2...
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Q JEE MAIN 2021
Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions,...
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Q _JEE 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...
JEE Main Mathematics Easy
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Q JEE MAIN 2021
The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in...
JEE Main Mathematics Easy
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Q JEE MAIN 2021
A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The...
JEE Main Mathematics Easy
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Q JEE MAIN 2021
A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to...
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Q JEE MAIN 2021
The coefficients $a, b$ and $c$ of the quadratic equation, $a x^2+b x+c=0$ are obtained by throwing a dice three...
JEE Main Mathematics Medium
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Q JEE MAIN 2021
When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}$ and the probability that...
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Q JEE MAIN_2022
Let $E_1, E_2, E_3$ be three mutually exclusive events such that $P\left(E_1\right)=\frac{2+3 p}{6}, P\left(E_2\right)=\frac{2-p}{8}$ and $P\left(E_3\right)=\frac{1-p}{2}$. If the maximum and...
JEE Main Mathematics Medium
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Q JEE MAIN_2022
The mean and variance of a binomial distribution are $\alpha$ and $\frac{\alpha}{3}$ respectively. If $P(X=1)=\frac{4}{243}$, then $P(X=4$ or 5) is...
JEE Main Mathematics Medium
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Q JEE MAIN 2022
If the numbers appeared on the two throws of a fair six faced die are $a$ and $b$, then the...
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Q JEE MAIN 2022
If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then...
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Q JEE MAIN 2022
Let $S=\left\{E, E_2 \ldots E_8\right\}$ be a sample space of random experiment such that $P\left(E_n\right)=\frac{n}{36}$ for every $...
JEE Main Mathematics Hard
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Q JEE MAIN 2022
If a point $A(x, y)$ lies in the region bounded by the $y$-axis, straight lines $2 y+x=6$ and $...
JEE Main Mathematics Medium
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Q JEE MAIN 2022
If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple...
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Q JEE MAIN 2022
In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4...
JEE Main Mathematics Hard
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Q JEE MAIN 2022
The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to:
JEE Main Mathematics Medium
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Q JEE MAIN 2022
The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2,...
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Q JEE MAIN 2022
JEE Main Mathematics Medium
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Q JEE MAIN 2022
A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of...
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Q JEE MAIN 2022
The probability, that in a randomly selected 3-digit number at least two digits are odd, is
JEE Main Mathematics Easy
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Q JEE MAIN 2022
Let X be a random variable having binomial distribution B(7, p). If P(X = 3) = 5P(X = 4), then...
JEE Main Mathematics Easy
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Q JEE MAIN 2022
Five numbers $x_1, x_2, x_3, x_4, x_5$ are randomly selected from the numbers $1,2,3, \ldots \ldots, 18$ and are arranged...
JEE Main Mathematics Easy
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Q JEE MAIN 2022
The probability that a randomly chosen 2 × 2 matrix with all the entries from the set of first 10...
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Q JEE MAIN 2022
Let the normal at the point P on the parabola $\mathrm{y}^2=6 \mathrm{x}$ pass through the point $(5,-8)$. If the tangent...
JEE Main Mathematics Easy
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Q JEE MAIN 2022
If a random variable $X$ follows the Binomial distribution $B(33, p)$ such that $3 P(X=0)=P(X=1)$, then the value of $\frac{P(X=15)}{P(X=18)}-\frac{P(X=16)}{P(X=17)}$...
JEE Main Mathematics Easy
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Q JEE MAIN 2022
Let $E_1$ and $E_2$ be two events such that the conditional probabilities $P\left(E_1 \mid E_2\right)=\frac{1}{2}, P\left(E_2 \mid E_1\right)=\frac{3}{4}$ and $...
JEE Main Mathematics Medium
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Q JEE MAIN 2022
Let $x^2+y^2+A x+B y+C=0$ be a circle passing through $(0,6)$ and touching the parabola $y=x^2$ at $(2,4)$. Then $\mathrm{A}+\mathrm{C}$ is...
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Q JEE MAIN 2022
Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and...
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Q JEE MAIN 2023
The ordinates of the points $P$ and $Q$ on the parabola with focus $(3,0)$ and directrix $x=-3$ are in the...
JEE Main Mathematics Hard
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Q JEE MAIN
Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2-12 x+[x]+31=0$ and $...
JEE Main Mathematics Medium
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Q JEE-Main 2023
Let $\mathrm{A}(0,1), \mathrm{B}(1,1)$ and $\mathrm{C}(1,0)$ be the mid - points of the sides of a triangle with in Centre at...
JEE Main Mathematics Hard
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Q JEE-Main 2023
If the probability that the random variable $X$ takes values $x$ is given by $P(X=x)=k(x+1) 3^{-x}, x=0,1$, $2,3 \ldots \ldots$,...
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Q JEE MAIN 2023
The random valuable X follows binomial distribution B(n, p) for which the difference of the mean and the variance is...
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Q JEE MAIN_2025
A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is...
JEE Main Physics Easy
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Q JEE 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 $E\left(X^2\right)=2 E(X)$. Then the value of $8 p-1$...
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Q JEE MAIN 2023
Three dice are rolled. If the probability of getting different numbers on the three dice is , where p and...
JEE Main Mathematics Easy
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Q JEE 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...
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Q JEE MAIN 2025
A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and...
JEE Main Mathematics Hard
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Q JEE MAIN 2025
Given three identical bags each containing 10 balls, whose colours are as follows : A person chooses a bag at...
JEE Main Mathematics Easy
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Q JEE 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...
JEE Main Mathematics Medium
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Q JEE-Main 2023
A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If...
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Q JEE MAIN 2023
A fair n (n > 1) faces die is rolled repeatedly until a number less than n appears. If the...
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Q JEE MAIN 2023
Two dice A and B are rolled, Let the numbers obtained on A and B be $\alpha$ and $\beta$ respectively....
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Q JEE MAIN 2023
Let ij $\mathrm{S}=\left\{\mathrm{M}=\left[a_{i j}\right], a_{i j} \in\{0,1,2\}, 1 \leq \mathrm{i}_k \mathrm{j} \leq 2\right\}$ be a sample space and $...
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Q JEE-Main 2023
Let the tangent to the curve $x^2+2 x-4 y+9=0$ at the point P(1, 3) on it meet the y-axis at...
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Q JEE MAIN 2023
In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. of...
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Q JEE-Main 2023
A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If...
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Q JEE MAIN
Let a,b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are...
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Q JEE MAIN 2024
Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables...
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Q JEE-Main 2024
A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the...
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Q JEE MAIN 2024
Let the sum of two positive integers be 24 . If the probability, that their product is not less than...
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Q JEE MAIN 2024
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...
JEE Main Mathematics Medium
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Q JEE MAIN 2024
Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black...
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Q JEE MAIN 2024
Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable x to be the number of...
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Q JEE MAIN 2024
Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles,...
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Q JEE MAIN 2024
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 Mathematics Medium
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Q JEE-Main 2024
A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement...
JEE Main Mathematics Medium
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Q JEE MAIN 2024
A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and...
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Q JEE MAIN 2025
Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black...
JEE Main Mathematics Easy
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Q JEE MAIN 2025
Let $\mathrm{A}=\left[\mathrm{a}_{i j}\right]$ be a $2 \times 2$ matrix such that $\mathrm{a}_{i j} \in\{0,1\}$ for all $i$ and $j$. Let...
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Q
Bag $B_{1}$ contains 6 white and 4 blue balls, Bag $B_{2}$ contains 4 white and 6 blue balls, and Bag...
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Q JEE MAIN 2025
Let $S$ be the set of all the words that can be formed by arranging all the letters of the...
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Q JEE MAIN 2025
A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and...
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Q JEE MAIN 2025
The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3...
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Q JEE MAIN 2025
All five letter words are made using all the letters A, B, C, D, E and arranged as in an...
JEE Main Mathematics Hard
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Q JEE MAIN 2025
Three distinct numbers are selected randomly from the set $\{ 1,2,3, \ldots ,40\}$. If the probability, that the selected numbers...
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Q JEE MAIN 2025
Let $\mathrm{A}=\left[a_{i j}\right]$ be a square matrix of order 2 with entries either 0 or 1 . Let E be...
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Q JEE MAIN 2025
A board has 16 square as shown in the figures : Out of these 16 squares, two squares are chosen...
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Q JEE MAIN 2025
Two number ${k_1}$ and ${k_2}$ are randomly chosen from the set of natural numbers. Then, the probability that the value...
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Q JEE MAIN 2025
A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B...
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Q JEE MAIN 2025
One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4....
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Q JEE MAIN 2025
Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black...
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Q JEE MAIN 2025
A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$...
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Q JEE MAIN 2025
If $A$ and $B$ are two events such that $P(A \cap B)=0.1$, and $P(A \mid B)$ and $P(B \mid A)$...
JEE Main Mathematics Easy
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Q JEE Advanced 2021
A number is chosen at random from the set {1, 2, 3, ... , 2000}. Let p be the probability...
JEE Advance Mathematics Medium
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Q JEE 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 $...
JEE Advance Mathematics Medium
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Q JEE Advanced 2021
Three numbers are chosen at random, one after another with replacement, from the set $S=\{1,2,3, \ldots, 100\}$. Let $p_1$ be...
JEE Advance Mathematics Easy
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Q JEE Advanced 2021
Three numbers are chosen at random, one after another with replacement, from the set $S=\{1,2,3, \ldots, 100\}$. Let $p_1$ be...
JEE Advance Mathematics Easy
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Q JEE 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 $\mathrm{E}_1$, and...
JEE Advance Mathematics Hard
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Q JEE Advanced 2023
Let $P$ be a point on the parabola $y^2=4 a x$, where $a>0$. The normal to the parabola at $P$...
JEE Advance Mathematics Easy
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Q JEE Advanced 2023
Let $\mathrm{X}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}: \frac{\mathrm{x}^2}{8}+\frac{\mathrm{y}^2}{20}
JEE Advance Mathematics Medium
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Q JEE Advanced 2024
Let $X$ be a random variable, and let $P(X=x)$ denote the probability that $X$ takes the value $x$. Suppose that...
JEE Advance Mathematics Medium
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Q JEE Advanced 2024
A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows...
JEE Advance Mathematics Easy
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Q JEE 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....
JEE Advance Mathematics Easy
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Q JEE 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...
JEE Advance Mathematics Easy
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