A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required and let a=P(X=3), b=P(X ≥ 3) and c=P(X ≥ 6 ∣ X>3). Then b+ca is equal to

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 :

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

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 in white, is:

Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable x to be the number of rotten apples in a draw of two apples, the variance of x is

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 white, is

A coin is based 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-

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 white and black balls is:

Let Ajay will not appear in JEE exam with probability p=27, while both Ajay and Vijay will appear in the exam with probability q=15. Then the probability, that Ajay will appear in the exam and Vijay will not appear is :

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 A is :

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as 13 and 23 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 P(|x−y| ≤ 2) is p , then 3⁹ p equals.....

If the mean of the following probability distribution of a random variable X; is 469, then the variance of the distribution is The probability distribution is: X: 0, 2, 4, 6, 8; P(X): a, 2a, a+b, 2b, 3b.

The coefficients a, b, c in the quadratic equation a x²+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:

Let the mean and the standard deviation of the probability distribution be μ and σ, respectively. If σ−μ=2, then σ+μ is equal to ____ . The probability distribution is: X: α, 1, 0, -3; P(X): 13, K, 16, 14.

The coefficients a, b, c in the quadratic equation a x²+b x+c=0 are from the set {1,2,3,4,5,6}. If the probability of this equation having one real root bigger than the other is p, then 216 p equals :

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 _____.

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 X̅ and Y̅ are the means of X and Y respectively, then 7 X̅+4 Y̅ is equal to

Let the sum of two positive integers be 24 . If the probability, that their product is not less than 34 times their greatest positive product, is mn, where gcd(m, n)=1, then n−m equals :

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 x²+b x+c=0 has all real roots is mn, gcd( m, n)=1 , then m+n is equal to

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the i^th roll than the number obtained in the (i−1)^th roll, i=2,3, is equal to:

A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ² denote the mean and variance of X, then the value of 64(μ+σ²) is :

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 mn, where gcd(m, n)=1, then m+n is equal to :

If A and B are two events such that P(A ∩ B)=0.1, and P(A ∣ B) and P(B ∣ A) are the roots of the equation 12 x²−7 x+1=0, then the value of P(A̅ ∪ B̅)P(A̅ ∩ B̅) is :

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 numbers to be 4 or 5 , when both the dice are thrown together, is

A board has 16 square as shown in the figures : Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: The board is a 4 by 4 grid of 16 squares.

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

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

Two number k₁ and k₂ are randomly chosen from the set of natural numbers. Then, the probability that the value of i^k₁ + i^k₂,(i = √(−1) ) is non-zero, equals

Bag B₁ contains 6 white and 4 blue balls, Bag B₂ contains 4 white and 6 blue balls, and Bag B₃ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag B₂, is

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 :

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 drawn is white, is 29 / 45, then n is equal to:

Let A=[a_{i j}] be a 2 × 2 matrix such that a_{i j} ∈{0,1} for all i and j. Let the random variable X denote the possible values of the determinant of the matrix A. Then, the variance of X is :

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

Given three identical bags each containing 10 balls, whose colours are as follows : A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is p and if the ball is Green, the probability that it is from bag III is q, then the value of (1p+1q) is: The bags contain: Bag I: Red 3, Blue 2, Green 5; Bag II: Red 4, Blue 3, Green 3; Bag III: Red 5, Blue 1, Green 4.

All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number n be denoted by W_n. Let the probability P( W_n ) of choosing the word W_n satisfy P( W_n ) = 2P( W_{n − 1} ),n > 1. If P(CDBEA) = 2^α 2^β − 1,α ,β ∈ ℕ, then α + β is equal to : ______

If the probability that the random variable X takes the value x is given by P(X=x)=k(x+1) 3^−x, x=0,1,2,3 …, where k is a constant, then P(X ≥ 3) is equal to

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let X denote the number of defective pens. Then the variance of X is

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

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 1150, then n is equal to ____ .

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 mn, gcd(m, n)=1. then n²−m² is equal to:

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)=2 E(X). Then the value of 8 p−1 is:

If A and B are two events such that P(A)=0.7, P(B)=0.4 and P(A ∩ B̅)=0.5, where B̅ denotes the complement of B, then P(B ∣(A ∪ B̅)) is equal to

A random variable X takes values 0,1,2,3 with probabilities 2 a+130, 8 a−130, 4 a+130, b respectively, where a, b ∈ ℝ. Let μ and σ respectively be the mean and standard deviation of X such that σ²⁺μ²=2. Then ab is equal to :

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

Let n be the number obtained on rolling a fair die. If the probability that the system x−n y+z=6
x+(n−2) y+(n+1) z=8
(n−1) y+z=1 has a unique solution is k6, then the sum of k and all possible values of n is :

From the first 100 natural numbers, two numbers first a and then b are selected randomly without replacement. If the probability that a−b ≥ 10 is mn, gcd(m, n)=1, then m+n is equal to ____ .

From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is

Let S be a set of 5 elements and P(S) denote the power set of S . Let E be an event of choosing an ordered pair ( A, B ) from the set P(S) × P(S) such that A ∩ B=φ. If the probability of the event E is 3^p2^q, where p, q ∈ N, then p+q is equal to ____ .

A bag contains 10 balls out of which k are red and (10−k) are black, where 0 ≤ k ≤ 10. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:

Let a, b, c ∈{1,2,3,4}. If the probability, that a x²+2 √(2) b x+c>0 for all x ∈ ℝ, is mn, gcd(m, n)=1, then m+n is equal to ____ .

A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is mn, gcd(m, n)=1, then m+n is equal to:

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is p, then 96 p is equal to ____ .

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to ab, where a, b ∈ ℕ and gcd(a, b)=1, then a+b is equal to

A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is:

The probabilities that players A and B of a team are selected for the captaincy for a tournament are 0.6 and 0.4 , respectively. If A is selected the captain, the probability that the team wins the tournament is 0.8 and if B is selected the captain, the probability that the team wins the tournament is 0.7 . Then the probability, that the team wins the tournament, is :

A bag contains ( ℕ+1 ) coins − N fair coins, and one coin with 'Head' on both sides. A coin is selected at random and tossed. If the probability of getting 'Head' is 916, then N is equal to:

A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is:

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are 25, 15 and 25. The probabilities that the candidate reaches late at the examination centre are 15, 13 and 14 if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is :