# Probability · Discrete Mathematics · GATE CSE

Start Practice## Marks 1

GATE CSE 2024 Set 2

When six unbiased dice are rolled simultaneously, the probability of getting all distinct numbers (i.e., 1, 2, 3, 4, 5, and 6) is

GATE CSE 2024 Set 1

Consider a permutation sampled uniformly at random from the set of all permutations of {1, 2, 3, ..., n} for some n ≥ 4. Let X be the event that 1 occ...

GATE CSE 2024 Set 1

Let A and B be two events in a probability space with $P(A) = 0.3$, $P(B) = 0.5$, and $P(A \cap B) = 0.1$. Which of the following statements is/are TR...

GATE CSE 2019

Two numbers are chosen independently and uniformly at random from the set {1, 2, ...., 13}. The probability (rounded off to 3 decimal places) that the...

GATE CSE 2017 Set 1

Let $$X$$ be a Gaussian random variable with mean $$0$$ and variance $${\sigma ^2}$$ . Let $$Y=max(X,0)$$ where $$max(a, b)$$ is the maximum of $$a$$...

GATE CSE 2016 Set 1

A probability density function on the interval $$\left[ {a,1} \right]$$ is given by $$1/{x^2}$$ and outside this interval the value of the function is...

GATE CSE 2014 Set 2

Each of the nine words in the sentence "The Quick brown fox jumps over the lazy dog" is written on a separate piece of paper. These nine pieces of pap...

GATE CSE 2014 Set 2

The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the offi...

GATE CSE 2014 Set 1

Suppose you break a stick of unit length at a point chosen uniformaly at random. Then the expected length of the shorter stick is __________________.

GATE CSE 2013

Suppose p is the number of cars per minute passing through a certain road junction between 5PM and 6PM and p has a poisson distribution with mean 3. W...

GATE CSE 2012

Consider a random variable X that takes values + 1 and-1 with probability 0.5 each.
The values of the cumulative distribution function F(x) at x = -...

GATE CSE 2011

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 outcomes are heads?

GATE CSE 2011

If the difference between the expectation of the square of a random variable $$\left( {E\left[ {{X^2}} \right]} \right)$$ and the square of the expect...

GATE CSE 2008

A sample space has two events A and B such that probabilities
$$P\,(A\, \cap \,B)\, = \,1/2,\,\,P(\overline A )\, = \,1/3,\,\,P(\overline B )\, = \,1...

GATE CSE 2007

Suppose there are two coins. The first coin gives heads with probability 5/8 when tossed, while the second coin gives heads with probability 1/4. On e...

GATE CSE 2006

In a certain town, the probability that it will rain in the afternoon is known to be 0.6. Moreover, meteorological data indicates that if the temperat...

GATE CSE 2005

A bag contains 10 blue marbles, 20 green marbles and 30 red marbles. A marble is drawn from the bag, its colour recorded and it is put back in the bag...

GATE CSE 2005

Let $$f(x)$$ be the continuous probability density function of a random variable X. The probability that $$a\, < \,X\, \le \,b$$, is:

GATE CSE 2004

In a population of N families, 50% of the families have three children, 30% of the families have two children and the remaining families have one chil...

GATE CSE 2004

If a fair coin is tossed four times, what is the probability that two heads and two tails will result?

GATE CSE 2003

Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = $${\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbo...

GATE CSE 1999

Suppose that the expectation of a random variable X is 5. Which of the following statements is true?

GATE CSE 1999

Suppose that the expectation of a random variable X is 5. Which of the following statements is true?

GATE CSE 1998

A die is rolled three times. The probability that exactly one odd number turns up among the three outcomes is

GATE CSE 1997

The probability that it will rain today is 0.5. The probability that it will rain tomorrow is 0.6. The probability that it will rain either today or t...

GATE CSE 1996

Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is

GATE CSE 1995

The probability that a number selected at random between $$100$$ and $$999$$ (both inclusive ) will not contain the digit $$7$$ is

GATE CSE 1994

Let A and B be any two arbitrary events, then, which one of the following is true?

## Marks 2

GATE CSE 2024 Set 2

Let $ x $ and $ y $ be random variables, not necessarily independent, that take real values in the interval $[0,1]$. Let $ z = xy $ and let the mean v...

GATE CSE 2024 Set 1

A bag contains 10 red balls and 15 blue balls. Two balls are drawn randomly without replacement. Given that the first ball drawn is red, the probabili...

GATE CSE 2023

Consider a random experiment where two fair coins are tossed. Let A be the event that denotes HEAD on both the throws, B be the event that denotes HEA...

GATE CSE 2021 Set 1

Consider the two statements.
S1 : There exist random variables X and Y such that
(E[X - E(X)) (Y - E(Y))])2 > Var[X] Var[Y]
S2 : For all ra...

GATE CSE 2021 Set 1

The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter 2. F...

GATE CSE 2021 Set 1

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, to a receiver (R).
In the gr...

GATE CSE 2020

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 probability that $$\sum\limits_...

GATE CSE 2019

Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial 3x2 + 6xY + 3Y + 6 has only real roots is (rounded ...

GATE CSE 2018

Two people, $$P$$ and $$Q,$$ decide to independently roll two identical dice, each with $$6$$ faces, numbered $$1$$ to $$6.$$ The person with the lowe...

GATE CSE 2018

Consider Guwahati $$(G)$$ and Delhi $$(D)$$ whose temperatures can be classified as high $$(H),$$ medium $$(M)$$ and low $$(L).$$ Let $$P\left( {{H_G}...

GATE CSE 2017 Set 2

$$P$$ and $$Q$$ are considering to apply for a job. The probability that $$P$$ applies for the job is $${1 \over 4},$$ the probability that $$P$$ appl...

GATE CSE 2017 Set 2

If a random variable $$X$$ has a Poisson distribution with mean $$5,$$ then the expectation $$E\left[ {{{\left( {X + 2} \right)}^2}} \right]$$ equals ...

GATE CSE 2017 Set 2

For any discrete random variable $$X,$$ with probability mass function $$P\left( {X = j} \right) = {p_j},$$
$${p_j}\,\, \ge 0,\,j \in \left\{ {0,........

GATE CSE 2016 Set 2

Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than $$100$$ hours given that...

GATE CSE 2016 Set 1

Consider the following experiment.
Step1: Flip a fair coin twice.
Step2: If the outcomes are (TAILS, HEADS) then output $$Y$$ and stop.
Step3: If the...

GATE CSE 2015 Set 3

Suppose $${X_i}$$ for $$i=1,2,3$$ are independent and identically distributed random variables whose probability mass functions are $$\,\,\Pr \left[ {...

GATE CSE 2015 Set 1

Given Set $$\,\,\,A = \left\{ {2,3,4,5} \right\}\,\,\,$$ and Set $$\,\,\,B = \left\{ {11,12,13,14,15} \right\},\,\,\,$$ two numbers are randomly selec...

GATE CSE 2015 Set 1

The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c$$ respectively. Of these subjects, the student has ...

GATE CSE 2014 Set 2

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 3

Let S be a sample space and two mutually exclusive events A and B be such that $$A\, \cup \,B = \,S$$. If P(.) denotes the probability of the event, t...

GATE CSE 2014 Set 1

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 2012

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 is the probability that the su...

GATE CSE 2011

A deck of 5 cards (each carrying a distinct number from 1 to 5) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What ...

GATE CSE 2011

Consider a finite sequence of random values $$X = \left\{ {{x_1},{x_2},{x_3}, - - - - - {x_n}} \right\}..$$ Let $${\mu _x}$$ be the mean and $${\s...

GATE CSE 2010

What is the probability that divisor of $${10^{99}}$$ is a multiple of $${10^{96}}$$ ?

GATE CSE 2010

Consider a company that assembles computers. The probability of a faulty assembly of any computer is P. The company therefore subjects each computer t...

GATE CSE 2009

An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face ...

GATE CSE 2008

Aishwarya studies either computer science or mathematics everyday. If she studies computer science on a day, then the probability that the studies mat...

GATE CSE 2008

Let X be a random variable following normal distribution with mean + 1 and variance 4. Let Y be another normal variable with mean - 1 and variance unk...

GATE CSE 2008

What is the probability that in a randomly choosen group of r people at least three people have the same birthday?

GATE CSE 2007

Suppose we uniformly and randomly select a permutation from the 20! permutations of 1, 2, 3,..., 20. What is the promutations that 2 appears at an ear...

GATE CSE 2006

When a coin is tossed, the probability of getting a Head is p, 0 < p < 1. Let N be the random variable denoting the number of tosses till the fi...

GATE CSE 2005

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) select a box (ii) choose a ball f...

GATE CSE 2005

An unbiased coin is tossed repeatedly until the outcome of two successive tosses is the same. Assuming that the tails are independent, the expected nu...

GATE CSE 2005

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 outcomes head and tail, respect...

GATE CSE 2004

A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at
(0, 0), (1, 0), (1, 2) and (0, 2). If p ...

GATE CSE 2004

An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches-0.25 ma...

GATE CSE 2004

Two n bit binary stings, S1 and, are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the nu...

GATE CSE 2002

Four fair coins are tossed simultaneously. The probability that at least one head and one tail turn up is

GATE CSE 2001

Seven (distinct) car accidents occurred in a week. What is the probability that they all occurred on the same day ?

GATE CSE 2000

$${{E_1}}$$ and $${{E_2}}$$ are events in a probability space satisfying the following constraints:
$$ \bullet $$ $$\Pr \,\,({E_1}) = \Pr \,({E_2})$$...

GATE CSE 1999

Let X and Y be two exponentially distributed and independent random variables with mean $$\alpha $$ and $$\beta $$, respectively. If Z = min (X, Y), t...

GATE CSE 1999

Consider two events $${{E_1}}$$ and $${{E_2}}$$ such that probability of $${{E_1}}$$, Pr [$${{E_1}}$$] = 1/2, probability of $${{E_2}}$$, Pr[$${{E_2}}...

GATE CSE 1996

The probability that the top and bottom cards of a randomly shuffled deck are both access is

GATE CSE 1995

A bag contains 10 white balls and 15 black balls. Two balls drawn in succession. The probability that one of them is black the other is white is