Question

Let $E^C$ denote the complement of an event $E$. Let $E_1,E_2$ and  $E_3$ be any pairwise independent events with $P\left(E_1\right)>0$ and $P\left(E_1\cap E_2\cap E_3\right)=0$ then $P\left(\left(E_2^C\cap E_3^C\right)/E_1\right)$ is equal to

JEE Main 2020 (02 Sep Shift 2)

Options

• A: $P\left(E_2^C\right)+P\left(E_3\right)$
• B: $P\left(E_3^C\right)-P\left(E_2^C\right)$
• C: $P\left(E_3\right)-P\left(E_2^C\right)$
• D: $P\left(E_3^C\right)-P\left(E_2\right)$

Question

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 :

JEE Main 2020 (04 Sep Shift 2)

Options

• A: $\frac{5}{31}$
• B: $\frac{31}{61}$
• C: $\frac{5}{6}$
• D: $\frac{30}{61}$

Question

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

JEE Main 2021 (01 Sep Shift 2)

Options

• A: $\frac{1}{9}$
• B: $\frac{1}{7}$
• C: $\frac{2}{7}$
• D: $\frac{1}{18}$

Question

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:

JEE Main 2021 (20 Jul Shift 2)

Options

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

Question

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

JEE Main 2021 (22 Jul Shift 1)

Options

• A: $\frac{45}{162}$
• B: $\frac{23}{81}$
• C: $\frac{22}{81}$
• D: $\frac{43}{162}$

Question

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 :

JEE Main 2021 (25 Feb Shift 2)

Options

• A: $\frac{14}{45}$
• B: $\frac{7}{45}$
• C: $\frac{8}{45}$
• D: $\frac{28}{45}$

Question

Let $X$ be a random variable such that the probability function of a distribution is given by $P\left(X=0\right)=\frac{1}{2},P\left(X=j\right)=\frac{1}{3^j}\left(j=1,2,3,\dots ,\infty \right).$ Then the mean of the distribution and $P(X$ is positive and even$)$ respectively, are:

JEE Main 2021 (25 Jul Shift 2)

Options

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

Question

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

JEE Main 2021 (27 Aug Shift 1)

Options

• A: $\frac{1}{16}$
• B: $\frac{1}{12}$
• C: $\frac{1}{8}$
• D: $\frac{1}{9}$

Question

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:

JEE Main 2021 (25 Feb Shift 2)

Options

• A: $\frac{1}{5}$
• B: $\frac{122}{297}$
• C: $\frac{97}{297}$
• D: $\frac{2}{9}$

Question

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 :

JEE Main 2021 (01 Sep Shift 2)

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Question

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 $\frac{6}{11}$, then $n$ is equal to _____

JEE Main 2022 (24 Jun Shift 1)

Options

• A: $13$
• B: $6$
• C: $4$
• D: $3$

Question

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

JEE Main 2022 (25 Jun Shift 2)

Options

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

Question

Let $S=\left\{M=\left[a_{ij}\right], a_{ij}\in \left\{0,1,2\right\}, \left\{1\le i,j\le 2\right\}\right\}$ be a sample space and $A\left\{M\in S:M \text{is invertible}\right\}$ be an even. Then $P\left(A\right)$ is equal to

JEE Main 2023 (11 Apr Shift 1)

Options

• A: $\frac{16}{27}$
• B: $\frac{47}{81}$
• C: $\frac{49}{81}$
• D: $\frac{50}{81}$

Question

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 $\frac{p}{q},$ where $p$ and $q$ are co-prime, then the sum of the positive divisors of $p$ is equal to

JEE Main 2023 (12 Apr Shift 1)

Options

• A: $72$
• B: $36$
• C: $48$
• D: $31$

Question

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 $\mathrm{A}$ is :

JEE Main 2024 (04 Apr Shift 1)

Options

• A: $\frac{5}{18}$
• B: $\frac{5}{16}$
• C: $\frac{4}{17}$
• D: $\frac{7}{18}$

Question

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac{1}{3}$ and $\frac{2}{3}$ 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 $\mathrm{P}(|x-y| \leq$ 2) is $p$, then $3^9 p$ equals ______

JEE Main 2024 (04 Apr Shift 2)

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Question

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 (05 Apr Shift 1)

Options

• A: $\frac{1}{128}$
• B: $\frac{1}{64}$
• C: $\frac{3}{256}$
• D: $\frac{3}{128}$

Question

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 manufactured at plant $A$ are rated of the standard quality, while $90 \%$ of the motorcycles manufactured at plant $B$ are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant $B$, then $126 p$ is

JEE Main 2024 (06 Apr Shift 1)

Options

• A: 54
• B: 66
• C: 64
• D: 56

Question

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 (06 Apr Shift 2)

Options

• A: $\frac{18}{25}$
• B: $\frac{12}{25}$
• C: $\frac{6}{25}$
• D: $\frac{4}{25}$

Question

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $\mathrm{X}$ 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 $\operatorname{gcd}(m, n)=1$, then $n-m$ is equal to _________

JEE Main 2024 (06 Apr Shift 2)

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Question

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

JEE Main 2024 (08 Apr Shift 1)

Options

• A: 10
• B: 9
• C: 11
• D: 8

Question

Let $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ 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 roots is $\frac{m}{n}$, $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to ________

JEE Main 2024 (09 Apr Shift 1)

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Question

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 (31 Jan Shift 2)

Options

• A: $\frac{2}{9}$
• B: $\frac{1}{9}$
• C: $\frac{2}{27}$
• D: $\frac{1}{27}$

Question

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

JEE Main 2024 (30 Jan Shift 1)

Options

• A: $\frac{30}{121}$
• B: $\frac{62}{121}$
• C: $\frac{60}{121}$
• D: $\frac{31}{121}$

Question

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $\mathrm{a}=\mathrm{P}(\mathrm{X}=3),\mathrm{b}=\mathrm{P}(\mathrm{X}\ge 3)$ and $\mathrm{c}=$ $\mathrm{P}(\mathrm{X}\ge 6\mid \mathrm{X}>3)$. Then $\frac{\mathrm{b}+\mathrm{c}}{\mathrm{a}}$ is equal to

JEE Main 2024 (27 Jan Shift 1)

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