Probability and Statistics: GATE CS previous year questions (Engineering Mathematics)

Hide the 1 question not in the current syllabus

GATE CS 2026 Set 1MCQ1 markEasy (estimate)
GATE CS 2026 Set 1, Q.11
An urn contains one red ball and one blue ball. At each step, a ball is picked uniformly at random from the urn, and this ball together with another ball of the same color is pu...
Probability and Statistics
GATE CS 2026 Set 1NAT2 marksEasy (estimate)
GATE CS 2026 Set 1, Q.58
Let $X$ be a random variable which takes values in the set ${1, 2, 3, 4, 5, 6, 7, 8}$ . Further, $Pr (X = 1) = Pr (X = 2) = Pr (X = 5) = Pr (X = 7) = \frac{1}{6}$ and…
Probability and Statistics
GATE CS 2026 Set 2MCQ1 markEasy (estimate)
GATE CS 2026 Set 2, Q.14
The probability density function $f (x)$ of a random variable $X$ which takes real values is…
Probability and Statistics
GATE CS 2026 Set 2NAT2 marksMedium (estimate)
GATE CS 2026 Set 2, Q.63
Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let $E_{1}$ be the event that among the second, fourth, and sixth coin tosse...
Probability and Statistics
GATE CS 2025 Set 1NAT1 markEasy (estimate)
GATE CS 2025 Set 1, Q.32
A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability $P (head) = 0.5$ and for a fake coin, $P (head) = 1$ . You pic...
Probability and Statistics
GATE CS 2025 Set 1NAT2 marksEasy (estimate)
GATE CS 2025 Set 1, Q.58
Consider a probability distribution given by the density function $P (x)$ .…
Probability and Statistics
GATE CS 2025 Set 2NAT2 marksMedium (estimate)
GATE CS 2025 Set 2, Q.64
A quadratic polynomial $(x - α) (x - β)$ over complex numbers is said to be square invariant if $(x - α) (x - β) = (x - α^{2}) (x - β^{2})$ . Suppose from the...
Probability and Statistics · Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2025 Set 2NAT2 marksMedium (estimate)
GATE CS 2025 Set 2, Q.65
The unit interval $(0, 1)$ is divided at a point chosen uniformly distributed over $(0, 1)$ in $R$ into two disjoint subintervals. The expected length of the subinterva...
Probability and Statistics · Calculus
GATE CS 2024 Set 1MCQ1 markEasy (estimate)
GATE CS 2024 Set 1, Q.14
Consider a permutation sampled uniformly at random from the set of all permutations of ${1, 2, 3, \dots, n}$ for some $n \geq 4$ . Let $X$ be the event that $1$ occurs before…
Probability and Statistics · Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2024 Set 1MSQ1 markEasy (estimate)
GATE CS 2024 Set 1, Q.27
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 TRUE?
Probability and Statistics
GATE CS 2024 Set 1NAT2 marksEasy (estimate)
GATE CS 2024 Set 1, Q.63
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 probability (rounded off to 3 decima...
Probability and Statistics
GATE CS 2024 Set 2MCQ1 markEasy (estimate)
GATE CS 2024 Set 2, Q.18
When six unbiased dice are rolled simultaneously, the probability of getting all distinct numbers (i.e., 1, 2, 3, 4, 5, and 6) is
Probability and Statistics · Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2024 Set 2MCQ2 marksMedium (estimate)
GATE CS 2024 Set 2, Q.44
Let $x$ and $y$ be random variables, not necessarily independent, that take real values in the interval $[0, 1]$ . Let $z = x y$ and let the mean values of $x, y, z$ be…
Probability and Statistics
GATE CS 2023MSQ2 marksEasy (estimate)
GATE CS 2023, Q.53
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 HEAD on the first throw, a...
Probability and Statistics
GATE CS 2021 Set 1NAT1 markEasy (estimate)
GATE CS 2021 Set 1, Q.28 (CS Q.18)
The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter $λ$ . For a randomly picke...
Probability and Statistics
GATE CS 2021 Set 1MCQ2 marksMedium (estimate)
GATE CS 2021 Set 1, Q.45 (CS Q.35)
Consider the two statements. $S_{1} : There exist random variables X and Y such that (E [(X - E [X]) (Y - E [Y]))^{2} > Var [X] Var [Y]$ ...
Probability and Statistics
GATE CS 2021 Set 1NAT2 marksMedium (estimate)
GATE CS 2021 Set 1, Q.64 (CS Q.54)
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 graph below, the weight of edg...
Probability and Statistics
GATE CS 2021 Set 2NAT1 markEasy (estimate)
GATE CS 2021 Set 2, Q.32 (CS Q.22)
For a given biased coin, the probability that the outcome of a toss is a head is 0.4. This coin is tossed 1,000 times. Let $X$ denote the random variable whose value is the numb...
Probability and Statistics
GATE CS 2021 Set 2MCQ2 marksEasy (estimate)
GATE CS 2021 Set 2, Q.39 (CS Q.29)
In an examination, a student can choose the order in which two questions (QuesA and QuesB) must be attempted. - If the first question is answered wrong, the student gets zer...
Probability and Statistics
GATE CS 2021 Set 2MCQ2 marksMedium (estimate)
GATE CS 2021 Set 2, Q.43 (CS Q.33)
A bag has $r$ red balls and $b$ black balls. All balls are identical except for their colours. In a trial, a ball is randomly drawn from the bag, its colour is noted and the bal...
Probability and Statistics
GATE CS 2020NAT1 markEasy (estimate)
GATE CS 2020, Q.27 (CS Q.17)
Let $R$ be the set of all binary relations on the set ${1, 2, 3}$ . Suppose a relation is chosen from $R$ at random. The probability that the chosen relation is reflexive (roun...
Sets, Relations, Functions, Partial Orders and Lattices · Probability and Statistics
GATE CS 2020NAT2 marksEasy (estimate)
GATE CS 2020, Q.55 (CS Q.45)
For $n \geq 2$ , let $a \in {0, 1}^{n}$ be a non-zero vector. Suppose that $x$ is chosen uniformly at random from ${0, 1}^{n}$ . Then, the probability that $\sum_{i = 1}^{n} a_{i} x_{i}$ ...
Probability and Statistics
GATE CS 2019NAT1 markMedium (estimate)
GATE CS 2019, Q.32 (CS Q.22)
Two numbers are chosen independently and uniformly at random from the set ${1, 2, \dots, 13}$ . The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) bi...
Probability and Statistics
GATE CS 2019NAT2 marksMedium (estimate)
GATE CS 2019, Q.57 (CS Q.47)
Suppose $Y$ is distributed uniformly in the open interval $(1, 6)$ . The probability that the polynomial $3 x^{2} + 6 x Y + 3 Y + 6$ has only real roots is (rounded off to 1 decimal pl...
Probability and Statistics
GATE CS 2018NAT1 markEasy (estimate)
GATE CS 2018, Q.25 (CS Q.15)
Two people, P and Q, decide to independently roll two identical dice, each with 6 faces, numbered 1 to 6. The person with the lower number wins. In case of a tie, they roll the...
Probability and Statistics
GATE CS 2018NAT2 marksMedium (estimate)
GATE CS 2018, Q.54 (CS Q.44)
Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high ( $H$ ), medium ( $M$ ) and low ( $L$ ). Let $P (H_{G})$ denote the probability that Guwahati has high te...
Probability and Statistics
GATE CS 2017 Set 1NAT1 markEasy (estimate)
GATE CS 2017 Set 1, Q.19 (CS Q.19)
Let $X$ be a Gaussian random variable with mean 0 and variance $σ^{2}$ . Let $Y = max (X, 0)$ where $max (a, b)$ is the maximum of $a$ and $b$ . The median of $Y$ is ______.
Probability and Statistics
GATE CS 2017 Set 2MCQ2 marksMedium (estimate)
GATE CS 2017 Set 2, Q.26 (CS Q.26)
P and Q are considering to apply for a job. The probability that P applies for the job is $\frac{1}{4}$ , the probability that P applies for the job given that Q applies for the...
Probability and Statistics
GATE CS 2017 Set 2MCQ2 marksMedium (estimate)
GATE CS 2017 Set 2, Q.31 (CS Q.31)
For any discrete random variable $X$ , with probability mass function…
Probability and Statistics
GATE CS 2017 Set 2NAT2 marksEasy (estimate)
GATE CS 2017 Set 2, Q.48 (CS Q.48)
If a random variable $X$ has a Poisson distribution with mean 5, then the expectation $E [(X + 2)^{2}]$ equals
Probability and Statistics
GATE CS 2016 Set 1NAT1 markEasy (estimate)
GATE CS 2016 Set 1, Q.14 (CS Q.4)
A probability density function on the interval $[a, 1]$ is given by $1/ x^{2}$ and outside this interval the value of the function is zero. The value of $a$ is _____.
Probability and Statistics
GATE CS 2016 Set 1NAT2 marksEasy (estimate)
GATE CS 2016 Set 1, Q.39 (CS Q.29)
Consider the following experiment. Step 1. Flip a fair coin twice. Step 2. If the outcomes are (TAILS, HEADS) then output Y and stop. Step 3. If the outcomes are either (HEADS,...
Probability and Statistics
GATE CS 2016 Set 2NAT1 markEasy (estimate)
GATE CS 2016 Set 2, Q.15 (CS Q.5)
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 it is of Type 1 is 0.7, and gi...
Probability and Statistics
GATE CS 2015 Set 2NAT2 marksEasy (estimate)
GATE CS 2015 Set 2, Q.64 (CS Q.54)
Let $X$ and $Y$ denote the sets containing 2 and 20 distinct objects respectively and $F$ denote the set of all possible functions defined from $X$ to $Y$ . Let $f$ be randomly c...
Probability and Statistics
GATE CS 2015 Set 3NAT1 markMedium (estimate)Not in current syllabus
GATE CS 2015 Set 3, Q.21 (CS Q.11)
Consider a software program that is artificially seeded with 100 faults. While testing this program, 159 faults are detected, out of which 75 faults are from those artificially...
Probability and Statistics
GATE CS 2015 Set 3NAT2 marksEasy (estimate)
GATE CS 2015 Set 3, Q.47 (CS Q.37)
Suppose $X_{i}$ for $i = 1, 2, 3$ are independent and identically distributed random variables whose probability mass functions are $Pr [X_{i} = 0] = Pr [X_{i} = 1] = 1/2$ for…
Probability and Statistics
GATE CS 2014 Set 1NAT1 markMedium (estimate)
GATE CS 2014 Set 1, Q.12 (CS Q.2)
Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .
Probability and Statistics
GATE CS 2014 Set 1NAT2 marksMedium (estimate)
GATE CS 2014 Set 1, Q.58 (CS Q.48)
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 ______________.
Probability and Statistics
GATE CS 2014 Set 2NAT1 markMedium (estimate)
GATE CS 2014 Set 2, Q.11 (CS Q.1)
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 officials inspect four of the c...
Probability and Statistics
GATE CS 2014 Set 2NAT1 markEasy (estimate)
GATE CS 2014 Set 2, Q.12 (CS Q.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 paper are kept in a box. One o...
Probability and Statistics
GATE CS 2014 Set 2NAT2 marksEasy (estimate)
GATE CS 2014 Set 2, Q.58 (CS Q.48)
The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is _____.
Probability and Statistics
GATE CS 2014 Set 3NAT2 marksEasy (estimate)
GATE CS 2014 Set 3, Q.58 (CS Q.48)
Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$ . If $P (\cdot)$ denotes the probability of the event, the maximum value of…
Probability and Statistics
GATE CS 2013MCQ1 markEasy (estimate)
GATE CS 2013, Q.2 (CS Q.2)
Suppose $p$ is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and $p$ has a Poisson distribution with mean 3. What is the probabili...
Probability and Statistics
GATE CS 2013MCQ1 markEasy (estimate)
GATE CS 2013, Q.24 (CS Q.24)
Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is $1/2$ . What is the expected number of unordered cycles...
Probability and Statistics
GATE CS 2012MCQ1 markEasy (estimate)
GATE CS 2012, Q.21 (CS Q.21)
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 $x = - 1$ and $+ 1$ are
Probability and Statistics
GATE CS 2012MCQ2 marksEasy (estimate)
GATE CS 2012, Q.33 (CS Q.33)
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 sum total of values that tu...
Probability and Statistics
GATE CS 2011MCQ1 markEasy (estimate)
GATE CS 2011, Q.3 (CS Q.3)
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?
Probability and Statistics
GATE CS 2011MCQ1 markEasy (estimate)
GATE CS 2011, Q.18 (CS Q.18)
If the difference between the expectation of the square of a random variable ( $E [X^{2}]$ ) and the square of the expectation of the random variable ( $(E [X])^{2}$ ) is denoted by $R$ ,...
Probability and Statistics
GATE CS 2011MCQ2 marksEasy (estimate)
GATE CS 2011, Q.33 (CS Q.33)
Consider a finite sequence of random values $X = [x_{1}, x_{2}, \dots, x_{n}]$ . Let $μ_{x}$ be the mean and $σ_{x}$ be the standard deviation of $X$ . Let another finite sequence…
Probability and Statistics
GATE CS 2011MCQ2 marksEasy (estimate)
GATE CS 2011, Q.34 (CS Q.34)
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 is the probability that the...
Probability and Statistics

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