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GATE CSE
Probability
Discrete Mathematics
Previous Years Questions
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Marks 1
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Two numbers are chosen independently and uniformly at random from the set {1, 2, ...., 13}. The probability (rounded off...
GATE CSE 2019
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Let $$X$$ be a Gaussian random variable with mean $$0$$ and variance $${\sigma ^2}$$ . Let $$Y=max(X,0)$$ where $$max(a,...
GATE CSE 2017 Set 1
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A probability density function on the interval $$\left[ {a,1} \right]$$ is given by $$1/{x^2}$$ and outside this interva...
GATE CSE 2016 Set 1
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Each of the nine words in the sentence "The Quick brown fox jumps over the lazy dog" is written on a separate piece of p...
GATE CSE 2014 Set 2
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The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the ...
GATE CSE 2014 Set 2
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Suppose you break a stick of unit length at a point chosen uniformaly at random. Then the expected length of the shorter...
GATE CSE 2014 Set 1
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Suppose p is the number of cars per minute passing through a certain road junction between 5PM and 6PM and p has a poiss...
GATE CSE 2013
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Consider a random variable X that takes values + 1 and-1 with probability 0.5 each. The values of the cumulative distr...
GATE CSE 2012
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If the difference between the expectation of the square of a random variable $$\left( {E\left[ {{X^2}} \right]} \right)$...
GATE CSE 2011
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If two fair coins are flipped and at least one of the outcomes is known to be a head, what is the probability that both ...
GATE CSE 2011
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A sample space has two events A and B such that probabilities $$P\,(A\, \cap \,B)\, = \,1/2,\,\,P(\overline A )\, = \,1...
GATE CSE 2008
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Suppose there are two coins. The first coin gives heads with probability 5/8 when tossed, while the second coin gives he...
GATE CSE 2007
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In a certain town, the probability that it will rain in the afternoon is known to be 0.6. Moreover, meteorological data ...
GATE CSE 2006
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Let $$f(x)$$ be the continuous probability density function of a random variable X. The probability that $$a\, < \,X\...
GATE CSE 2005
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A bag contains 10 blue marbles, 20 green marbles and 30 red marbles. A marble is drawn from the bag, its colour recorded...
GATE CSE 2005
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If a fair coin is tossed four times, what is the probability that two heads and two tails will result?
GATE CSE 2004
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In a population of N families, 50% of the families have three children, 30% of the families have two children and the re...
GATE CSE 2004
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Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = $${\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em...
GATE CSE 2003
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Suppose that the expectation of a random variable X is 5. Which of the following statements is true?
GATE CSE 1999
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Suppose that the expectation of a random variable X is 5. Which of the following statements is true?
GATE CSE 1999
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A die is rolled three times. The probability that exactly one odd number turns up among the three outcomes is
GATE CSE 1998
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The probability that it will rain today is 0.5. The probability that it will rain tomorrow is 0.6. The probability that ...
GATE CSE 1997
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Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is
GATE CSE 1996
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The probability that a number selected at random between $$100$$ and $$999$$ (both inclusive ) will not contain the digi...
GATE CSE 1995
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Let A and B be any two arbitrary events, then, which one of the following is true?
GATE CSE 1994
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Marks 2
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A sender (S) transmits a signal, which can be one of the two kinds: H and L with probabilities 0.1 and 0.9 respectively,...
GATE CSE 2021 Set 1
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The lifetime of a component of a certain type is a random variable whose probability density function is exponentially d...
GATE CSE 2021 Set 1
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Consider the two statements. S1 : There exist random variables X and Y such that (E[X - E(X)) (Y - E(Y))])2 > ...
GATE CSE 2021 Set 1
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For n > 2, let a {0, 1}n be a non-zero vector. Suppose that x is chosen uniformly at random from {0, 1}n. Then, the p...
GATE CSE 2020
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Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial 3x2 + 6xY + 3Y + 6 ha...
GATE CSE 2019
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Consider Guwahati $$(G)$$ and Delhi $$(D)$$ whose temperatures can be classified as high $$(H),$$ medium $$(M)$$ and low...
GATE CSE 2018
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Two people, $$P$$ and $$Q,$$ decide to independently roll two identical dice, each with $$6$$ faces, numbered $$1$$ to $...
GATE CSE 2018
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For any discrete random variable $$X,$$ with probability mass function $$P\left( {X = j} \right) = {p_j},$$ $${p_j}\,\, ...
GATE CSE 2017 Set 2
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If a random variable $$X$$ has a Poisson distribution with mean $$5,$$ then the expectation $$E\left[ {{{\left( {X + 2} ...
GATE CSE 2017 Set 2
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$$P$$ and $$Q$$ are considering to apply for a job. The probability that $$P$$ applies for the job is $${1 \over 4},$$ t...
GATE CSE 2017 Set 2
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Consider the following experiment. Step1: Flip a fair coin twice. Step2: If the outcomes are (TAILS, HEADS) then output...
GATE CSE 2016 Set 1
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Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more...
GATE CSE 2016 Set 2
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Suppose $${X_i}$$ for $$i=1,2,3$$ are independent and identically distributed random variables whose probability mass fu...
GATE CSE 2015 Set 3
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The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c$$ respectively. Of th...
GATE CSE 2015 Set 1
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Given Set $$\,\,\,A = \left\{ {2,3,4,5} \right\}\,\,\,$$ and Set $$\,\,\,B = \left\{ {11,12,13,14,15} \right\},\,\,\,$$ ...
GATE CSE 2015 Set 1
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Let S be a sample space and two mutually exclusive events A and B be such that $$A\, \cup \,B = \,S$$. If P(.) denotes t...
GATE CSE 2014 Set 3
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The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is ...
GATE CSE 2014 Set 2
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Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is X/1296. The value of X is__...
GATE CSE 2014 Set 1
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Suppose a fair six-sided die is rolled once. If the value on the die is 1, 2 or 3 the die is rolled a second time. What ...
GATE CSE 2012
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Consider a finite sequence of random values $$X = \left\{ {{x_1},{x_2},{x_3}, - - - - - {x_n}} \right\}..$$ Let $${\...
GATE CSE 2011
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A deck of 5 cards (each carrying a distinct number from 1 to 5) is shuffled thoroughly. Two cards are then removed one ...
GATE CSE 2011
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Consider a company that assembles computers. The probability of a faulty assembly of any computer is P. The company ther...
GATE CSE 2010
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What is the probability that divisor of $${10^{99}}$$ is a multiple of $${10^{96}}$$ ?
GATE CSE 2010
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An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of ...
GATE CSE 2009
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What is the probability that in a randomly choosen group of r people at least three people have the same birthday?
GATE CSE 2008
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Let X be a random variable following normal distribution with mean + 1 and variance 4. Let Y be another normal variable ...
GATE CSE 2008
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Aishwarya studies either computer science or mathematics everyday. If she studies computer science on a day, then the pr...
GATE CSE 2008
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Suppose we uniformly and randomly select a permutation from the 20! permutations of 1, 2, 3,..., 20. What is the promuta...
GATE CSE 2007
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When a coin is tossed, the probability of getting a Head is p, 0 < p < 1. Let N be the random variable denoting th...
GATE CSE 2006
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A random bit string of length n is constructed by tossing a fair coin n times and setting a bit to 0 or 1 depending on o...
GATE CSE 2005
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An unbiased coin is tossed repeatedly until the outcome of two successive tosses is the same. Assuming that the tails ar...
GATE CSE 2005
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Box P has 2 red balls and 3 blue balls and box Q has 3 red balls and 1 blue ball. A ball is selected as follows: (i) sel...
GATE CSE 2005
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Two n bit binary stings, S1 and, are chosen randomly with uniform probability. The probability that the Hamming distance...
GATE CSE 2004
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An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each in...
GATE CSE 2004
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A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at (0, 0), (1...
GATE CSE 2004
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Four fair coins are tossed simultaneously. The probability that at least one head and one tail turn up is
GATE CSE 2002
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Seven (distinct) car accidents occurred in a week. What is the probability that they all occurred on the same day ?
GATE CSE 2001
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$${{E_1}}$$ and $${{E_2}}$$ are events in a probability space satisfying the following constraints: $$ \bullet $$ $$\Pr...
GATE CSE 2000
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Consider two events $${{E_1}}$$ and $${{E_2}}$$ such that probability of $${{E_1}}$$, Pr [$${{E_1}}$$] = 1/2, probabilit...
GATE CSE 1999
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Let X and Y be two exponentially distributed and independent random variables with mean $$\alpha $$ and $$\beta $$, resp...
GATE CSE 1999
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The probability that the top and bottom cards of a randomly shuffled deck are both access is
GATE CSE 1996
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A bag contains 10 white balls and 15 black balls. Two balls drawn in succession. The probability that one of them is bla...
GATE CSE 1995
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