Probability

182 questions · Mathematics · JEE Main

Probability

182 questions · Mathematics · JEE Main

Three distinct numbers are selected randomly from the set ${1, 2, 3, \dots, 40}$ . If the probability, that the selected numbers are in an increasing G.P., is $\frac{m}{n}, gcd (m, n) = 1$ , then $m + n$ is equal to …2025 · 2 Apr · Shift 1 · Q46 · Numerical
$Given three indentical bags each containing 10 balls, whose colours are as follows : Bag I Bag II Bag III Red 345 Blue 231 Green 534$ …2025 · 2 Apr · Shift 2 · Q45 · MCQ
If the probability that the random variable $X$ takes the value $x$ is given by $P (X = x) = k (x + 1) 3^{- x}, x = 0, 1, 2, 3 \dots$ , where $k$ is a constant, then $P (X \geq 3)$ is equal to2025 · 3 Apr · Shift 2 · Q40 · MCQ
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, is2025 · 4 Apr · Shift 1 · Q36 · MCQ
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 $X$ is2025 · 4 Apr · Shift 1 · Q37 · MCQ
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 $\frac{11}{50}$ , then n is equal to .2025 · 4 Apr · Shift 2 · Q47 · Numerical
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 $\frac{m}{n}$ , $g cd (m, n) = 1$ , then $n^{2} - m^{2}$ is equal…2025 · 7 Apr · Shift 2 · Q30 · MCQ
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(X2)=2E(X). Then the value of 8p−1 is :2025 · 7 Apr · Shift 2 · Q43 · MCQ
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 $P (B ∣ (A \cup \overline{B}))$ is equal to2025 · 8 Apr · Shift 2 · Q42 · MCQ
A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $μ$ and $σ^{2}$ denote the mean and variance of $X$ , then the value of $64 (μ + σ^{2})$ is:2025 · 22 Jan · Shift 1 · Q35 · MCQ
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 that the second selected ball is also black, is $\frac{m}{n}$ ,…2025 · 22 Jan · Shift 1 · Q37 · MCQ
If $A$ and $B$ are two events such that $P (A \cap B) = 0.1$ , and $P (A ∣ B)$ and $P (B ∣ A)$ are the roots of the equation $12 x^{2} - 7 x + 1 = 0$ , then the value of $\frac{P ( A ˉ \cup B ˉ )}{P ( A ˉ \cap B ˉ )}$ is :2025 · 22 Jan · Shift 2 · Q33 · MCQ
One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4. The probability of getting the sum of…2025 · 23 Jan · Shift 1 · Q32 · MCQ
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 : Includes diagram2025 · 23 Jan · Shift 2 · Q30 · MCQ
$A$ 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, is2025 · 24 Jan · Shift 1 · Q37 · MCQ
Let $A = [a_{ij}]$ 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 :2025 · 24 Jan · Shift 2 · Q27 · MCQ
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, equals2025 · 28 Jan · Shift 1 · Q35 · MCQ
Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denote the number of defective oranges, then…2025 · 28 Jan · Shift 1 · Q36 · MCQ
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…2025 · 28 Jan · Shift 2 · Q27 · MCQ
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 selected word will NOT have vowels in alphabetical order is:2025 · 28 Jan · Shift 2 · Q36 · MCQ
Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball…2025 · 29 Jan · Shift 2 · Q34 · MCQ
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. The probability that the bag contains equal number of…2024 · 1 Feb · Shift 1 · Q31 · MCQ
Let Ajay will not appear in JEE exam with probability $p = \frac{2}{7}$ , while both Ajay and Vijay will appear in the exam with probability $q = \frac{1}{5}$ . Then the probability, that Ajay will appear in the exam and Vijay…2024 · 1 Feb · Shift 2 · Q38 · MCQ
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…2024 · 4 Apr · Shift 1 · Q42 · MCQ
If the mean of the following probability distribution of a radam variable $X$ : is $\frac{46}{9}$ , then the variance of the distribution is Includes table2024 · 4 Apr · Shift 2 · Q42 · MCQ
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…2024 · 4 Apr · Shift 2 · Q55 · Numerical
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 :2024 · 5 Apr · Shift 1 · Q39 · MCQ
From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $X$ denote the number of defective items in the sample. If the variance of $X$ is $σ^{2}$ , then $96 σ^{2}$ is…2024 · 5 Apr · Shift 1 · Q52 · Numerical
The coefficients $a, b, c$ in the quadratic equation $a x^{2} + bx + c = 0$ are from the set ${1, 2, 3, 4, 5, 6}$ . If the probability of this equation having one real root bigger than the other is…2024 · 5 Apr · Shift 2 · Q44 · MCQ
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…2024 · 6 Apr · Shift 1 · Q50 · MCQ
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 :2024 · 6 Apr · Shift 2 · Q36 · MCQ
From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $X$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If…2024 · 6 Apr · Shift 2 · Q59 · Numerical
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 $gcd (m, n) = 1$ , then $n$ - $m$ equals2024 · 8 Apr · Shift 1 · Q40 · MCQ
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 $\overset{ˉ}{X}$ and $\overset{ˉ}{Y}$ are the means of $X$ and $Y$ …2024 · 8 Apr · Shift 1 · Q56 · Numerical
There are three bags $X, Y$ and $Z$ . Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag $Z$ contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at…2024 · 8 Apr · Shift 2 · Q33 · MCQ
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 roots is $\frac{m}{n}, gcd (m, n) = 1$ …2024 · 9 Apr · Shift 1 · Q60 · Numerical
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 to2024 · 9 Apr · Shift 2 · Q39 · MCQ
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 $\frac{b + c}{a}$ is equal to .2024 · 27 Jan · Shift 1 · Q60 · Numerical
An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :2024 · 27 Jan · Shift 2 · Q34 · MCQ
A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is2024 · 29 Jan · Shift 1 · Q36 · MCQ
An integer is chosen at random from the integers $1, 2, 3, \dots, 50$ . The probability that the chosen integer is a multiple of atleast one of 4, 6 and 7 is2024 · 29 Jan · Shift 2 · Q43 · MCQ
Two integers $x$ and $y$ are chosen with replacement from the set ${0, 1, 2, 3, \dots, 10}$ . Then the probability that $∣ x - y ∣ > 5$ , is :2024 · 30 Jan · Shift 1 · Q43 · MCQ
A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics and Chemistry. It was found that all students passed in atleast one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics,…2024 · 30 Jan · Shift 1 · Q51 · Numerical
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 bag A, if the ball drawn is white, is2024 · 30 Jan · Shift 2 · Q49 · MCQ
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 $x$ is2024 · 31 Jan · Shift 1 · Q38 · MCQ
Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is…2024 · 31 Jan · Shift 1 · Q41 · MCQ
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 is2024 · 31 Jan · Shift 2 · Q32 · MCQ
Two dice are thrown independently. Let $A$ be the event that the number appeared on the $1^{st}$ die is less than the number appeared on the $2^{nd}$ die, $B$ be the event that the number appeared on the…2023 · 1 Feb · Shift 2 · Q35 · MCQ
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 :2023 · 6 Apr · Shift 2 · Q34 · MCQ
In a bolt factory, machines $A, B$ and $C$ manufacture respectively $20%, 30%$ and $50%$ of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt…2023 · 8 Apr · Shift 1 · Q26 · MCQ
If the probability that the random variable $X$ takes values $x$ is given by $P (X = x) = k (x + 1) 3^{- x}, x = 0, 1, 2, 3, \dots$ , where $k$ is a constant, then $P (X \geq 2)$ is equal…2023 · 8 Apr · Shift 2 · Q25 · MCQ
Let N denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^{N} < N!$ is $\frac{m}{n}$ , where m and n are coprime, then $4 m - 3 n$ is equal to :2023 · 10 Apr · Shift 1 · Q31 · MCQ
Let $S = {M = [a_{ij}], a_{ij} \in {0, 1, 2}, 1 \leq i, j \leq 2}$ be a sample space and $A = {M \in S : M$ is invertible $}$ be an event. Then $P (A)$ is equal to :2023 · 11 Apr · Shift 1 · Q26 · MCQ
Let the probability of getting head for a biased coin be $\frac{1}{4}$ . It is tossed repeatedly until a head appears. Let $N$ be the number of tosses required. If the probability that the equation $64 x^{2} + 5 Nx + 1 = 0$ …2023 · 11 Apr · Shift 2 · Q39 · Numerical
Two dice A and B are rolled. Let the numbers obtained on A and B be $α$ and $β$ respectively. If the variance of $α - β$ is $\frac{p}{q}$ , where $p$ and $q$ are co-prime, then the sum of the positive divisors of $p$ is…2023 · 12 Apr · Shift 1 · Q23 · MCQ
A fair $n (n > 1)$ faces die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9}$ , then $n$ is equal to .2023 · 12 Apr · Shift 1 · Q38 · Numerical
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 :2023 · 13 Apr · Shift 1 · Q28 · MCQ
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 :2023 · 15 Apr · Shift 1 · Q32 · MCQ
Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x + y + z = 12 x + N y + 2 z = 23 x + 3 y + N z = 3$ has unique solution is $\frac{k}{6}$ , then the sum of value…2023 · 24 Jan · Shift 1 · Q25 · MCQ
Let $Ω$ be the sample space and $A \subseteq Ω$ be an event. Given below are two statements : (S1) : If P(A) = 0, then A = $ϕ$ (S2) : If P(A) = 1, then A = $Ω$ Then :2023 · 24 Jan · Shift 1 · Q30 · MCQ
Three urns A, B and C contain 4 red, 6 black; 5 red, 5 black; and $λ$ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn…2023 · 24 Jan · Shift 2 · Q44 · Numerical
Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $S = {x \in Z : x (66 - x) \geq \frac{5}{9} M}$ and the event $A = {x \in S : x is a multiple of 3}$ …2023 · 25 Jan · Shift 1 · Q29 · MCQ
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 $\frac{k}{48}$ . Then the value of k is :2023 · 25 Jan · Shift 2 · Q25 · MCQ
25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$ . Then the value of k…2023 · 25 Jan · Shift 2 · Q39 · Numerical
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 :2023 · 29 Jan · Shift 1 · Q26 · MCQ
Let $S = {w_{1}, w_{2}, ......}$ be the sample space associated to a random experiment. Let $P (w_{n}) = \frac{P ( w _{n - 1} )}{2}, n \geq 2$ . Let $A = {2 k + 3 l : k, l \in N}$ and $B = {w_{n} : n \in A}$ . Then P(B) is equal to :2023 · 29 Jan · Shift 2 · Q29 · MCQ
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 :2023 · 30 Jan · Shift 1 · Q32 · MCQ
A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the…2023 · 30 Jan · Shift 2 · Q42 · Numerical
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 :2023 · 31 Jan · Shift 1 · Q26 · MCQ
Let A be the event that the absolute difference between two randomly choosen real numbers in the sample space $[0, 60]$ is less than or equal to a . If $P (A) = \frac{11}{36}$ , then $a$ is equal to …2023 · 31 Jan · Shift 2 · Q43 · Numerical
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…2022 · 24 Jun · Shift 1 · Q25 · MCQ
A random variable X has the following probability distribution : The value of P(1 < X < 4 | X $\leq$ 2) is equal to : Includes table2022 · 24 Jun · Shift 2 · Q32 · MCQ
If the numbers appeared on the two throws of a fair six faced die are $α$ and $β$ , then the probability that $x^{2} + α x + β > 0$ , for all $x \in R$ , is :2022 · 25 Jul · Shift 1 · Q35 · MCQ
If $A$ and $B$ are two events such that $P (A) = \frac{1}{3}, P (B) = \frac{1}{5}$ and $P (A \cup B) = \frac{1}{2}$ , then $P (A ∣ B^{'}) + P (B ∣ A^{'})$ is equal to :2022 · 25 Jul · Shift 2 · Q28 · MCQ
Let E1 and E2 be two events such that the conditional probabilities $P (E_{1} ∣ E_{2}) = \frac{1}{2}$ , $P (E_{2} ∣ E_{1}) = \frac{3}{4}$ and $P (E_{1} \cap E_{2}) = \frac{1}{8}$ . Then :2022 · 25 Jun · Shift 1 · Q30 · MCQ
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 :2022 · 25 Jun · Shift 2 · Q34 · MCQ
Let $E_{1}, E_{2}, E_{3}$ be three mutually exclusive events such that $P (E_{1}) = \frac{2 + 3 p}{6}, P (E_{2}) = \frac{2 - p}{8}$ and $P (E_{3}) = \frac{1 - p}{2}$ …2022 · 26 Jul · Shift 1 · Q37 · MCQ
If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is p, then 96 p is equal to .2022 · 26 Jun · Shift 2 · Q46 · Numerical
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 $9 p$ is equal to :2022 · 27 Jul · Shift 1 · Q35 · MCQ
A six faced die is biased such that $3 \times P ($ a prime number $) = 6 \times P ($ a composite number $) = 2 \times P (1)$ . Let X be a random variable that counts the number of times one gets a perfect square on…2022 · 27 Jul · Shift 2 · Q33 · MCQ
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 $x_{4} = 11$ …2022 · 27 Jun · Shift 1 · Q33 · MCQ
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 :2022 · 27 Jun · Shift 2 · Q34 · MCQ
Let S = {E1, E2, ........., E8} be a sample space of a random experiment such that $P (E_{n}) = \frac{n}{36}$ for every n = 1, 2, ........, 8. Then the number of elements in the set $$\left\{ {A \subseteq S:P(A) \ge {4 \over 5}}…2022 · 27 Jun · Shift 2 · Q44 · Numerical
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…2022 · 28 Jul · Shift 1 · Q30 · MCQ
Let $A$ and $B$ be two events such that $P (B ∣ A) = \frac{2}{5}, P (A ∣ B) = \frac{1}{7}$ and $P (A \cap B) = \frac{1}{9} \cdot$ Consider (S1) $P (A^{'} \cup B) = \frac{5}{6}$ , (S2) $P (A^{'} \cap B^{'}) = \frac{1}{18}$ …2022 · 28 Jul · Shift 2 · Q34 · MCQ
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 $σ^{2}$ is the variance of $X$ , then $100 σ^{2}$ is equal…2022 · 28 Jul · Shift 2 · Q38 · Numerical
The probability, that in a randomly selected 3-digit number at least two digits are odd, is :2022 · 28 Jun · Shift 1 · Q34 · MCQ
The probability that a randomly chosen one-one function from the set {a, b, c, d} to the set {1, 2, 3, 4, 5} satisfies f(a) + 2f(b) $-$ f(c) = f(d) is :2022 · 28 Jun · Shift 2 · Q37 · MCQ
Let $S = {1, 2, 3, \dots, 2022}$ . Then the probability, that a randomly chosen number n from the set S such that $HCF (n, 2022) = 1$ , is :2022 · 29 Jul · Shift 1 · Q39 · MCQ
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…2022 · 29 Jul · Shift 2 · Q33 · MCQ
The probability that a randomly chosen 2 $\times$ 2 matrix with all the entries from the set of first 10 primes, is singular, is equal to :2022 · 29 Jun · Shift 1 · Q20 · MCQ
The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to :2022 · 29 Jun · Shift 2 · Q33 · MCQ
The probability distribution of X is : For the minimum possible value of d, sixty times the mean of X is equal to . Includes table2022 · 30 Jun · Shift 1 · Q41 · Numerical
Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is : Includes diagram2021 · 1 Sep · Shift 2 · Q27 · MCQ
Let X be a random variable with distribution. If the mean of X is 2.3 and variance of X is $σ$ 2, then 100 $σ$ 2 is equal to : Includes table2021 · 1 Sep · Shift 2 · Q38 · Numerical
A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :2021 · 16 Mar · Shift 1 · Q35 · MCQ
Let A denote the event that a 6-digit integer formed by 0, 1, 2, 3, 4, 5, 6 without repetitions, be divisible by 3. Then probability of event A is equal to :2021 · 16 Mar · Shift 2 · Q39 · MCQ
Two dies are rolled. If both dices have six faces numbered 1, 2, 3, 5, 7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is :2021 · 17 Mar · Shift 1 · Q30 · MCQ
Let there be three independent events E1, E2 and E3. The probability that only E1 occurs is $α$ , only E2 occurs is $β$ and only E3 occurs is $γ$ . Let 'p' denote the probability of none of events occurs that satisfies the…2021 · 17 Mar · Shift 1 · Q38 · Numerical
Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be $\frac{1}{2}$ and probability of occurrence of 0 at the odd place be $\frac{1}{3}$ . Then the…2021 · 17 Mar · Shift 2 · Q22 · MCQ
Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is :2021 · 20 Jul · Shift 1 · Q35 · MCQ
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 :2021 · 20 Jul · Shift 1 · Q36 · MCQ
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…2021 · 20 Jul · Shift 2 · Q31 · MCQ
Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 $\times$ 2 matrices. The probability that such formed matrix have all different entries and are non-singular, is :2021 · 22 Jul · Shift 2 · Q27 · MCQ
Let Bi (i = 1, 2, 3) be three independent events in a sample space. The probability that only B1 occur is $α$ , only B2 occurs is $β$ and only B3 occurs is $γ$ . Let p be the probability that none of the events Bi occurs and…2021 · 24 Feb · Shift 1 · Q44 · Numerical
The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is :2021 · 24 Feb · Shift 2 · Q31 · MCQ
When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}$ and the probability that the missile hits the target, given that it is not intercepted, is $\frac{3}{4}$ . If three missiles are fired…2021 · 25 Feb · Shift 1 · Q24 · MCQ
The coefficients a, b and c of the quadratic equation, ax2 + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is :2021 · 25 Feb · Shift 1 · Q35 · MCQ
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 :2021 · 25 Feb · Shift 2 · Q28 · MCQ
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…2021 · 25 Feb · Shift 2 · Q36 · MCQ
Let 9 distinct balls be distributed among 4 boxes, B1, B2, B3 and B4. If the probability than B3 contains exactly 3 balls is $k (\frac{3}{4})^{9}$ then k lies in the set :2021 · 25 Jul · Shift 1 · Q35 · MCQ
Let X be a random variable such that the probability function of a distribution is given by $P (X = 0) = \frac{1}{2}, P (X = j) = \frac{1}{3 ^{j}} (j = 1, 2, 3, ..., \infty)$ . Then the mean of the distribution and P(X is positive and even)…2021 · 25 Jul · Shift 2 · Q38 · MCQ
A fair coin is tossed n-times such that the probability of getting at least one head is at least 0.9. Then the minimum value of n is .2021 · 25 Jul · Shift 2 · Q45 · Numerical
Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P (exactly one of A, B occurs) = $\frac{5}{9}$ , is :2021 · 26 Aug · Shift 1 · Q25 · MCQ
A fair die is tossed until six is obtained on it. Let x be the number of required tosses, then the conditional probability P(x $\geq$ 5 | x > 2) is :2021 · 26 Aug · Shift 2 · Q29 · MCQ
Two fair dice are thrown. The numbers on them are taken as $λ$ and $μ$ , and a system of linear equations x + y + z = 5 x + 2y + 3z = $μ$ x + 3y + $λ$ z = 1 is constructed. If p is the probability that the system has a unique…2021 · 26 Aug · Shift 2 · Q31 · MCQ
A seven digit number is formed using digits 3, 3, 4, 4, 4, 5, 5. The probability, that number so formed is divisible by 2, is :2021 · 26 Feb · Shift 2 · Q39 · MCQ
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…2021 · 27 Aug · Shift 1 · Q30 · MCQ
The probability distribution of random variable X is given by : Let p = P(1 < X < 4 | X < 3). If 5p = $λ$ K, then $λ$ equal to . Includes table2021 · 27 Aug · Shift 2 · Q36 · Numerical
The probability that a randomly selected 2-digit number belongs to the set {n $\in$ N : (2n $-$ 2) is a multiple of 3} is equal to :2021 · 27 Jul · Shift 1 · Q37 · MCQ
An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8. The instrument is switched on and…2021 · 31 Aug · Shift 1 · Q45 · Numerical
Let S = {1, 2, 3, 4, 5, 6}. Then the probability that a randomly chosen onto function g from S to S satisfies g(3) = 2g(1) is :2021 · 31 Aug · Shift 2 · Q24 · MCQ
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…2020 · 2 Sep · Shift 1 · Q22 · MCQ
Let EC denote the complement of an event E. Let E1 , E2 and E3 be any pairwise independent events 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 :2020 · 2 Sep · Shift 2 · Q38 · MCQ
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 :2020 · 3 Sep · Shift 1 · Q36 · MCQ
The probability that a randomly chosen 5-digit number is made from exactly two digits is :2020 · 3 Sep · Shift 2 · Q31 · MCQ
The probability of a man hitting a target is $\frac{1}{10}$ . The least number of shots required, so that the probability of his hitting the target at least once is greater than $\frac{1}{4}$ , is .2020 · 4 Sep · Shift 1 · Q23 · Numerical
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…2020 · 4 Sep · Shift 2 · Q26 · MCQ
Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is :2020 · 6 Sep · Shift 1 · Q34 · MCQ
The probabilities of three events A, B and C are given by P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5. If P(A $\cup$ B) = 0.8, P(A $\cap$ C) = 0.3, P(A $\cap$ B $\cap$ C) = 0.2, P(B $\cap$ C) = $β$ and P(A $\cup$ B $\cup$ C) = $α$ ,…2020 · 6 Sep · Shift 2 · Q33 · MCQ
An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value of k when k consecutive heads are obtained for k = 3, 4, 5, otherwise X takes the value -1. Then the expected value of X, is :2020 · 7 Jan · Shift 1 · Q40 · MCQ
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?2020 · 8 Jan · Shift 1 · Q28 · MCQ
Let A and B be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that A or B occurs is $\frac{1}{2}$ , then the probability of both of them occur together is :2020 · 8 Jan · Shift 2 · Q23 · MCQ
In a box, there are 20 cards, out of which 10 are lebelled as A and the remaining 10 are labelled as B. Cards are drawn at random, one after the other and with replacement, till a second A-card is obtained. The probability that the second…2020 · 9 Jan · Shift 1 · Q25 · MCQ
If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is :2020 · 9 Jan · Shift 2 · Q21 · MCQ
A random variable X has the following probability distribution : Then P(X > 2) is equal to : Includes table2020 · 9 Jan · Shift 2 · Q34 · MCQ
Let A and B be two non-null events such that A $\subset$ B . Then, which of the following statements is always correct?2019 · 8 Apr · Shift 1 · Q26 · MCQ
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 :2019 · 8 Apr · Shift 2 · Q40 · MCQ
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 target independently, then the probability that the target would be hit, is :2019 · 9 Apr · Shift 1 · Q36 · MCQ
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…2019 · 9 Jan · Shift 2 · Q45 · MCQ
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 :2019 · 10 Apr · Shift 1 · Q23 · MCQ
Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is :2019 · 10 Apr · Shift 2 · Q27 · MCQ
An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards…2019 · 10 Jan · Shift 1 · Q29 · MCQ
Two integers are selected at random from the set {1, 2, ...., 11}. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :2019 · 11 Jan · Shift 1 · Q37 · MCQ
Let S = {1, 2, . . . . . ., 20}. A subset B of S is said to be "nice", if the sum of the elements of B is 203. Then the probability that a randonly chosen subset of S is "nice" is :2019 · 11 Jan · Shift 2 · Q23 · MCQ
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 :2019 · 12 Apr · Shift 1 · Q31 · MCQ
A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice), wins Rs. 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw. Then the expected gain/loss (in…2019 · 12 Apr · Shift 2 · Q40 · MCQ
In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to :2019 · 12 Jan · Shift 1 · Q28 · MCQ
In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his…2019 · 12 Jan · Shift 2 · Q25 · MCQ
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 students selected has opted neither for NCC nor for NSS is :2019 · 12 Jan · Shift 2 · Q39 · MCQ
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 eplacement, from a randomly selected box and one ball turns out to be…2018 · 15 Apr · Shift 1 · Q36 · MCQ
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,…2018 · 15 Apr · Shift 2 · Q32 · MCQ
Two different families A and B are blessed with equal numbe 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…2018 · 16 Apr · Shift 1 · Q39 · MCQ
Let A, B and C be three events, which are pair-wise independent and $E$ denotes the completement of an event E. If $P (A \cap B \cap C) = 0$ and $P (C) > 0,$ then $P [(\overline{A} \cap \overline{B}) ∣ C]$ …2018 · 16 Apr · Shift 1 · Q42 · MCQ
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,…2018 · Shift 0 · Q34 · MCQ
Three persons P, Q and R independently try to hit a target. I the probabilities of their hitting the target are $\frac{3}{4}, \frac{1}{2}$ and $\frac{5}{8}$ respectively, then the probability that the target is hit by P or Q but not by R is…2017 · 8 Apr · Shift 1 · Q37 · MCQ
An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :2017 · 8 Apr · Shift 1 · Q38 · MCQ
Let E and F be two independent events. The probability that both E and F happen is $\frac{1}{12}$ and the probability that neither E nor F happens is $\frac{1}{2}$ , then a value of $\frac{P ( E )}{P ( F )}$ is…2017 · 9 Apr · Shift 1 · Q33 · MCQ
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 :2017 · 9 Apr · Shift 1 · Q36 · MCQ
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 :2017 · Shift 0 · Q33 · MCQ
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) = $\frac{1}{4}$ and P(All the three events occur simultaneously) = $\frac{1}{16}$ . Then the probability that…2017 · Shift 0 · Q34 · MCQ
If A and B are any two events such that P(A) = $\frac{2}{5}$ and P (A $\cap$ B) = $\frac{3}{20}$ , hen the conditional probability, P(A $∣$ (A' $\cup$ B')), where A' denotes the complement of A, is equal to :2016 · 9 Apr · Shift 1 · Q37 · MCQ
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…2016 · Shift 0 · Q24 · MCQ
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 :2015 · Shift 0 · Q27 · MCQ
Let $A$ and $B$ be two events such that $P (\overline{A \cup B}) = \frac{1}{6}, P (A \cap B) = \frac{1}{4}$ and $P (\overline{A}) = \frac{1}{4},$ where $\overline{A}$ stands for the…2014 · Shift 0 · Q35 · MCQ
Three numbers are chosen at random without replacement from ${1, 2, 3, ..8} .$ The probability that their minimum is $3,$ given that their maximum is $6,$ is :2012 · Shift 0 · Q29 · MCQ
If $C$ and $D$ are two events such that $C \subset D$ and $P (D) e 0,$ then the correct statement among the following is :2011 · Shift 0 · Q36 · MCQ
An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is :2010 · Shift 0 · Q31 · MCQ
Four numbers are chosen at random (without replacement) from the set ${1, 2, 3, ....20} .$ Statement - 1: The probability that the chosen numbers when arranged in some order will form an AP is $\frac{1}{85} .$ Statement -…2010 · Shift 0 · Q32 · MCQ
One ticket is selected at random from $50$ tickets numbered $00, 01, 02, ...., 49.$ Then the probability that the sum of the digits on the selected ticket is $8$ , given that the product of these digits is zer, equals :2009 · Shift 0 · Q32 · MCQ
A die is thrown. Let $A$ be the event that the number obtained is greater than $3.$ Let $B$ be the event that the number obtained is less than $5.$ Then $P (A \cup B)$ is :2008 · Shift 0 · Q33 · MCQ
It is given that the events $A$ and $B$ are such that $P (A) = \frac{1}{4}, P (A ∣ B) = \frac{1}{2}$ and $P (B ∣ A) = \frac{2}{3} .$ Then $P (B)$ is :2008 · Shift 0 · Q34 · MCQ
Two aeroplanes $I$ and $I I$ bomb a target in succession. The probabilities of $I$ and $I I$ scoring a hit correctly are $0.3$ and $0.2,$ respectively. The second plane will bomb only if the first misses…2007 · Shift 0 · Q45 · MCQ
At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of $5$ phone calls during $10$ minute time intervals. The probability that there is at the most one phone call…2006 · Shift 0 · Q64 · MCQ
Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is :2005 · Shift 0 · Q87 · MCQ
A random variable $X$ has Poisson distribution with mean $2$ . Then $P (X > 1.5)$ equals :2005 · Shift 0 · Q88 · MCQ
Let $A$ and $B$ two events such that $P (\overline{A \cup B}) = \frac{1}{6}, P (A \cap B) = \frac{1}{4}$ and $P (\overline{A}) = \frac{1}{4},$ where $\overline{A}$ stands for complement of…2005 · Shift 0 · Q89 · MCQ
The probability that $A$ speaks truth is $\frac{4}{5},$ while the probability for $B$ is $\frac{3}{4} .$ The probability that they contradict each other when asked to speak on a fact is :2004 · Shift 0 · Q90 · MCQ
Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is :2003 · Shift 0 · Q76 · MCQ
Events $A, B, C$ are mutually exclusive events such that $P (A) = \frac{3 x + 1}{3}, P (B) = \frac{1 - x}{4}$ and $P (C) = \frac{1 - 2 x}{2}$ The set of possible values of $x$ are in the interval.2003 · Shift 0 · Q88 · MCQ
A problem in mathematics is given to three students $A, B, C$ and their respective probability of solving the problem is $\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{4} .$ Probability that the problem is solved is :2002 · Shift 0 · Q87 · MCQ
$A$ and $B$ are events such that $P (A \cup B) = 3/4$ , $P (A \cap B) = 1/4, P (\overline{A}) = 2/3$ then $P (\overline{A} \cap B)$ is :2002 · Shift 0 · Q88 · MCQ