JEE MAIN Probability Previous Year Question Paper and Solution with PDF

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JEE MAIN

JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Probability PYQ

Qus : 1 JEE MAIN PYQ

Let $E_1, E_2, E_3$ be three mutually exclusive events such that $P(E_1)=\dfrac{2+3p}{6}$, $P(E_2)=\dfrac{2-p}{8}$ and $P(E_3)=\dfrac{1-p}{2}$. If the maximum and minimum values of $p$ are $p_1$ and $p_2$, then $(p_1+p_2)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 July Morning Shift) PYQ

Solution

Qus : 2 JEE MAIN PYQ

Let $A$ and $B$ be two events such that $P(\overline{A\cup B})=\dfrac{1}{6}$, $P(A\cap B)=\dfrac{1}{4}$ and $P(\overline{A})=\dfrac{1}{4}$, where $\overline{A}$ stands for the complement of the event $A$. Then the events $A$ and $B$ are :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2014 (Offline) PYQ

Solution

Qus : 3 JEE MAIN PYQ

Two dice are thrown independently. Let $A$ be the event that the number on the $1^{\text{st}}$ die is less than the number on the $2^{\text{nd}}$ die; $B$ be the event that the number on the $1^{\text{st}}$ die is even and that on the $2^{\text{nd}}$ die is odd; and $C$ be the event that the number on the $1^{\text{st}}$ die is odd and that on the $2^{\text{nd}}$ die is even. Then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Evening Shift) PYQ

Solution

Qus : 4 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Morning Shift) PYQ

Solution

Qus : 5 JEE MAIN PYQ

A box $A$ contains $2$ white, $3$ red and $2$ black balls. Another box $B$ contains $4$ white, $2$ red and $3$ black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box $B$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Morning Shift) PYQ

Solution

Qus : 6 JEE MAIN PYQ

The coefficients $a,b,c$ in the quadratic $ax^{2}+bx+c=0$ are chosen from the set ${1,2,3,4,5,6,7,8}$. The probability that the equation has repeated roots is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (5 April Morning Shift) PYQ

Solution

Qus : 7 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Evening Shift) PYQ

Solution

Qus : 8 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Morning Shift) PYQ

Solution

Qus : 9 JEE MAIN PYQ

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 ${6 \over {11}}$, then n is equal to __________.

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (24 June Morning Shift) PYQ

Solution

Qus : 10 JEE MAIN PYQ

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 August Morning Shift) PYQ

Solution

Qus : 11 JEE MAIN PYQ

$A$ 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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Morning Shift) PYQ

Solution

Qus : 12 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Evening Shift) PYQ

Solution

Qus : 13 JEE MAIN PYQ

A bag contains $19$ unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and a head turns up. If the probability that the drawn coin was unbiased is $\dfrac{m}{n}$ with $\gcd(m,n)=1$, then $n^2-m^2$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Evening Shift) PYQ

Solution

Qus : 14 JEE MAIN PYQ

A player $X$ has a biased coin whose probability of showing heads is $p$ and a player $Y$ has a fair coin. They start playing a game with their own coins and play alternately. The player who throws a head first is a winner. If $X$ starts the game, and the probability of winning the game by both the players is equal, then the value of $p$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Evening Shift) PYQ

Solution

Qus : 15 JEE MAIN PYQ

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. A candidate is randomly chosen from the qualified candidates. The probability that the chosen candidate is a female, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 July Morning Shift) PYQ

Solution

Qus : 16 JEE MAIN PYQ

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 is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Morning Shift) PYQ

Solution

Qus : 17 JEE MAIN PYQ

A bag contains $4$ red and $6$ black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (Offline) PYQ

Solution

Qus : 18 JEE MAIN PYQ

$ \text{Let A and B be two events such that } P(B|A)=\frac{2}{5}, P(A|B)=\frac{1}{7},; \text{and } P(A\cap B)=\frac{1}{9}. $ Consider:(S1) $P(A' \cup B)=\frac{5}{6}$ (S2) $P(A' \cap B')=\frac{1}{18}$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 July Evening Shift) PYQ

Solution

Qus : 19 JEE MAIN PYQ

Consider three boxes, each containing $10$ balls labelled $1,2,\ldots,10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$ the label of the ball drawn from the $i^{\text{th}}$ box ($i=1,2,3$). Then, the number of ways in which the balls can be chosen such that $n_1
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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Morning Shift) PYQ

Solution

Qus : 20 JEE MAIN PYQ

Two numbers $k_{1}$ 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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Morning Shift) PYQ

Solution

Qus : 21 JEE MAIN PYQ

Let $N$ denote 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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (10 April Morning Shift) PYQ

Solution

Qus : 22 JEE MAIN PYQ

$.$ 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. 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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Evening Shift) PYQ

Solution

Qus : 23 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

Qus : 24 JEE MAIN PYQ

Two different families $A$ and $B$ are blessed with equal number of children. There are $3$ tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of the family $B$ is $\dfrac{1}{12}$, then the number of children in each family is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (16 April Morning Shift) PYQ

Solution

Qus : 25 JEE MAIN PYQ

Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Morning) PYQ

Solution

Given:
Box I → cards numbered 1 to 30 (30 cards)
Box II → cards numbered 31 to 50 (20 cards)
A box is selected at random → probability of each box = $\dfrac{1}{2}$

Non-prime numbers in each box:
Box I (1–30): Prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 → 10 primes.
Non-prime numbers = 30 − 10 = 20
(Including 1 as non-prime)

Box II (31–50): Prime numbers are 31, 37, 41, 43, 47 → 5 primes.
Non-prime numbers = 20 − 5 = 15

Let
A = “card drawn from Box I”
B = “card drawn from Box II”
N = “number on the card is non-prime”

We need $P(A|N)$.

Using Bayes’ theorem:
P(A|N) = \frac{P(A)P(N|A)}{P(A)P(N|A) + P(B)P(N|B)} \]
Now substitute:
\[ P(A) = P(B) = \frac{1}{2}, \quad \] \[ P(N|A) \frac{20}{30} = \frac{2}{3}, \quad \] \[ P(N|B) = \frac{15}{20} = \frac{3}{4} \]
\[ P(A|N) = \frac{\frac{1}{2}\cdot\frac{2}{3}}{\frac{1}{2}\cdot\frac{2}{3} + \frac{1}{2}\cdot\frac{3}{4}} \] \[= \frac{\frac{1}{3}}{\frac{1}{3} + \frac{3}{8}} \] \[= \frac{\frac{1}{3}}{\frac{17}{24}}\] \[= \frac{8}{17} \]

Final Answer: $\boxed{\dfrac{8}{17}}$

Qus : 26 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Evening Shift) PYQ

Solution

Qus : 27 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Evening Shift) PYQ

Solution

Qus : 28 JEE MAIN PYQ

Let A, B and C be three events, which are pair-wise independent and $\overrightarrow E $ denotes the completement of an event E. If $P\left( {A \cap B \cap C} \right) = 0$ and $P\left( C \right) > 0,$ then $P\left[ {\left( {\overline A \cap \overline B } \right)\left| C \right.} \right]$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (16 April Morning Shift) PYQ

Solution

Qus : 29 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Evening Shift) PYQ

Solution

Qus : 30 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Evening Shift) PYQ

Solution

Qus : 31 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Morning Shift) PYQ

Solution

Qus : 32 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (24 January Morning Shift) PYQ

Solution

Qus : 33 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Morning Shift) PYQ

Solution

Qus : 34 JEE MAIN PYQ

If two different numbers are taken from the set {0,1,2,3,...,10} then the probability that their sum as well as absolute difference are both multiple of 4, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (Offline) PYQ

Solution

Qus : 35 JEE MAIN PYQ

For three events A, B and C, P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P(Exactly one of C or A occurs) = $\dfrac{1}{4}$ and P(All the three events occur simultaneously) =$ \dfrac{1}{16}$. Then the probability that at least one of the events occurs, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (Offline) PYQ

Solution

Qus : 36 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Morning Shift) PYQ

Solution

Qus : 37 JEE MAIN PYQ

If $A$ and $B$ are two events such that $P(A)=0.7,\ P(B)=0.4$ and $P(A\cap \overline{B})=0.5$, where $\overline{B}$ denotes the complement of $B$, then $P!\left(B,\middle|,(A\cup \overline{B})\right)$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (8 April Evening Shift) PYQ

Solution

Qus : 38 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Morning Shift) PYQ

Solution

Qus : 39 JEE MAIN PYQ

Let E^C denote the complement of an event E.Let E₁, E₂ and E₃ be any pairwise independentevents with P(E₁) > 0

and P(E₁ $\ \cap $ E₂ $ \cap $ E₃) = 0.

Then P($E_2^C \cap E_3^C/{E_1}$) is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Evening) PYQ

Solution

Qus : 40 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 June Morning Shift) PYQ

Solution

Qus : 41 JEE MAIN PYQ

Two dice $A$ and $B$ are rolled. Let the numbers obtained on $A$ and $B$ be $\alpha$ and $\beta$ respectively. If the variance of $\alpha-\beta$ is $\dfrac{p}{q}$, where $p$ and $q$ are co-prime, then the sum of the positive divisors of $p$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (12 April Morning Shift) PYQ

Solution

Qus : 42 JEE MAIN PYQ

There are three bags $X,Y,Z$. 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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Evening Shift) PYQ

Solution

Qus : 43 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

Qus : 44 JEE MAIN PYQ

Let $M$ be the maximum value of the product of two positive integers when their sum is $66$. Let the sample space $S=\{\,x\in\mathbb{Z}: x(66-x)\ge \tfrac{5}{9}M\,\}$ and the event $A=\{\,x\in S:\ x\ \text{is a multiple of }3\,\}$. Then $P(A)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Morning Shift) PYQ

Solution

Qus : 45 JEE MAIN PYQ

Three persons P, Q and R independently try to hit a target. If the probabilities of their hitting the target are $\dfrac{3}{4},\ \dfrac{1}{2}$ and $\dfrac{5}{8}$ respectively, then the probability that the target is hit by $P$ or $Q$ but not by $R$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

Qus : 46 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 June Evening Shift) PYQ

Solution

Qus : 47 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Morning) PYQ

Solution

Qus : 48 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Evening Shift) PYQ

Solution

Qus : 49 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Evening Shift) PYQ

Solution

Qus : 50 JEE MAIN PYQ

Bag A contains $3$ white and $7$ red balls; Bag B contains $3$ white and $2$ red balls. One bag is selected at random and a ball is drawn. If the ball drawn is white, the probability that it was drawn from Bag A is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Evening Shift) PYQ

Solution

Qus : 51 JEE MAIN PYQ

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 the mean of $X$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Morning Shift) PYQ

Solution

Qus : 52 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Evening Shift) PYQ

Solution

Qus : 53 JEE MAIN PYQ

Let $E$ and $F$ be two independent events. The probability that both $E$ and $F$ happen is $\dfrac{1}{12}$ and the probability that neither $E$ nor $F$ happens is $\dfrac{1}{2}$. Then a value of $\dfrac{P(E)}{P(F)}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

Qus : 54 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 June Morning Shift) PYQ

Solution

Qus : 55 JEE MAIN PYQ

From a group of $10$ men and $5$ women, four-member committees are to be formed, each of which must contain at least one woman. Then the probability for these committees to have more women than men is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

Qus : 56 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Morning Shift) PYQ

Solution

Qus : 57 JEE MAIN PYQ

The distance of the point $Q(0,2,-2)$ from the line passing through the point $P(5,-4,3)$ and perpendicular to the lines $\ \vec r = (-3\hat i + 2\hat k) + \lambda(2\hat i + 3\hat j + 5\hat k),\ \lambda\in\mathbb R,$ and $\ \vec r = (\hat i - 2\hat j + \hat k) + \mu(-\hat i + 3\hat j + 2\hat k),\ \mu\in\mathbb R,$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Morning Shift) PYQ

Solution

Qus : 58 JEE MAIN PYQ

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_{1}$ is the event that die $A$ shows up four, $E_{2}$ is the event that die $B$ shows up two, and $E_{3}$ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (Offline) PYQ

Solution

Qus : 59 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Evening Shift) PYQ

Solution

Qus : 60 JEE MAIN PYQ

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{\text{th}}$ roll than the number obtained in the $(i-1)^{\text{th}}$ roll, $i=2,3$, is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Evening Shift) PYQ

Solution

Qus : 61 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Evening) PYQ

Solution

Qus : 62 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (15 April Morning Shift) PYQ

Solution

Qus : 63 JEE MAIN PYQ

Let $ S=\{w_1,w_2,\ldots\} $ be the sample space of a random experiment. Let the probabilities satisfy \[ P(w_n)=\frac{P(w_{n-1})}{2},\qquad n\ge 2. \] Let \[ A=\{\,2k+3\ell : k,\ell\in\mathbb{N}\,\},\qquad B=\{\,w_n : n\in A\,\}. \] Then $P(B)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Evening Shift) PYQ

Solution

Qus : 64 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Morning Shift) PYQ

Solution

Qus : 65 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Morning Shift) PYQ

Solution

Qus : 66 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (1 September Evening Shift) PYQ

Solution

Qus : 67 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (30 January Morning Shift) PYQ

Solution

Qus : 68 JEE MAIN PYQ

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 total a 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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Evening) PYQ

Solution

Qus : 69 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Evening Shift) PYQ

Solution

Qus : 70 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Evening Shift) PYQ

Solution

Qus : 71 JEE MAIN PYQ

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 $x \in \mathbb{R}$, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 July Morning Shift) PYQ

Solution

Qus : 72 JEE MAIN PYQ

If $A$ and $B$ are two events such that $P(A) = \tfrac{1}{3},\ P(B) = \tfrac{1}{5}$ and $P(A \cup B) = \tfrac{1}{2}$, then

$P(A \mid B') + P(B \mid A')$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 July Evening Shift) PYQ

Solution

Qus : 73 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2015 (Offline) PYQ

Solution

Qus : 74 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Morning Shift) PYQ

Solution

Qus : 75 JEE MAIN PYQ

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 that the sum of the numbers is 4 or 5 when both dice are thrown together is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Morning Shift) PYQ

Solution

Qus : 76 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

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