What are common examiner traps in Probability and Statistics?

Assuming independence implies mutual exclusivity or vice versa (e.g., P(A ∩ B) = 0 vs P(A ∩ B) = P(A)P(B)). Forgetting that standard deviation is invariant under constant shifts: (aX + b) = |a| (X), omitting +b. Overlooking overlap/dependence in conditional coin-toss problems (e.g., treating shared coin tosses across two events as independent). Rounding errors in NAT questions when computing rational probabilities like 5/42 or powers of (0.99)5. In continuous interval questions (e.g., subinterval containing a point), failing to split the integral at the boundary point where the piecewise definition changes. Confusing covariance definition Cov(X, Y) = E[(X- X)(Y- Y)] (which can be negative) with the non-negative expectation of absolute differences. Assuming unconditional independence (e.g., assuming the second coin is independently 1/2 without accounting for the condition 'at least one outcome is a head') Confusing the conditioned event denominator with the total original sample space (e.g., obtaining 1/4 instead of 1/3) Failing to account for permutations in compound outcomes (e.g., counting HT and TH as a single event) Stopping at the first step and answering the single-step probability (e.g., (1)/(4) = 0.25) instead of solving the recurrence for the infinite trial sequence ((1)/(3) 0.33). Misidentifying the sample space of multiple independent coin flips (e.g., treating (TAILS, HEADS) and (HEADS, TAILS) as indistinguishable single outcomes). Rounding errors in NAT questions when converting fractions like (1)/(3) to decimal form. Confusing P(X < x) with P(X ≤ x), which causes an off-by-one probability calculation at discontinuity points. Assuming F(xmin) = 0 instead of F(xmin) = P(X = xmin) for discrete distributions. Applying standard stars-and-bars without enforcing the upper bound constraint (di ≤ 6), which is only valid when the transformed sum is less than 6. Off-by-one errors when shifting bounds (e.g., confusing yi = 6 - di with yi = di - 1). Mistaking the total sample space size (64 = 1296) when setting up the probability ratio.

GATE CSE Probability and Statistics PYQs

Q: What does GATE test in Probability and Statistics?

Probability and Statistics in GATE CS is consistently tested with 2 to 4 questions per year, spanning fundamental combinatorics, conditional probability, Bayes' theorem, discrete and continuous random variables, and expectation/variance properties. A notable proportion of questions require setting up a discrete probability model (urn/coin/dice experiments or bit strings) or computing expectations using linearity of expectation and integration over piece-wise uniform intervals. The question mix favors NAT and conceptual MCQ/MSQ formats with an emphasis on exact algebraic reasoning rather than heavy numerical statistical testing.

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Probability and Statistics in GATE CS is consistently tested with 2 to 4 questions per year, spanning fundamental combinatorics, conditional probability, Bayes'… Read the guide

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Questions: 46
Asked: since 2011
Total marks: 71
Marks / paper: 2.7

46 GATE CSE Probability and Statistics PYQs with solutions, from Engineering Mathematics. Browse all GATE CSE PYQs.

Questions

Every official paper that asked this topic, newest first.

What GATE tests in Probability and Statistics

Concepts, patterns, formulas and traps distilled from past papers.

Probability and Statistics in GATE CS is consistently tested with 2 to 4 questions per year, spanning fundamental combinatorics, conditional probability, Bayes' theorem, discrete and continuous random variables, and expectation/variance properties. A notable proportion of questions require setting up a discrete probability model (urn/coin/dice experiments or bit strings) or computing expectations using linearity of expectation and integration over piece-wise uniform intervals. The question mix favors NAT and conceptual MCQ/MSQ formats with an emphasis on exact algebraic reasoning rather than heavy numerical statistical testing.

Core concepts

Conditional probability, law of total probability, and Bayes' theorem
Independent vs. mutually exclusive events
Linearity of expectation and indicator random variables
Discrete distributions: Bernoulli, Binomial, Poisson, Geometric, Uniform
Continuous distributions: Uniform, Exponential, Normal (Gaussian)
Properties of Mean, Median, Mode, Variance, Standard Deviation, and Covariance
Probability density function normalization and continuous expectation integrals
Pólya urn and sequential sampling with/without replacement
Principle of Inclusion-Exclusion over finite sample spaces
Definition of conditional probability P(A B)
Sample space reduction under given conditions
Equally likely elementary events in coin tosses
First-step analysis
Law of total probability
Infinite geometric series in Bernoulli trials
Conditional probability with recursive reset states
Equally likely outcome spaces for independent trials
Discrete Random Variables
Probability Mass Function (PMF)
Cumulative Distribution Function (CDF)
Sum of independent discrete uniform random variables
Stars and bars / generating non-negative integer solutions
Variable substitution and symmetry in dice roll outcomes
Sample space calculation for multi-stage independent trials (6n)

Recurring question patterns

Bayes' Theorem and Conditional Probability Updates
CommonNAT2 marks2025, 2021, 2017, 2016
Scenarios involving two or more sources/states (e.g., faulty coins, binary signal transmission through noisy channels, disease/job tests) where an observed signal/result requires computing the posterior probability of the underlying cause.
Linearity of Expectation and Indicator Variables
Commonmixed1.5 marks2026, 2021, 2017, 2014
Computing the expected number of structures (e.g., triangles/cycles in random graphs, matching bins, word lengths, or game strategies) using the sum of expectations of indicator variables.
Continuous Distribution Normalization and Moment Integrals
CommonNAT1.5 marks2025, 2021, 2016, 2014
Given a PDF with an unknown constant $C$ or boundary $a$ over an interval (polynomial, uniform, or exponential), finding the constant by setting the integral to 1, or computing subinterval probabilities and expected lengths of sub-segments.
Sequential Urn / Coin / Dice Trials with Conditional Stopping
Commonmixed2 marks2026, 2024, 2021, 2018
Multi-stage drawing with/without replacement, Pólya urn processes, coin toss sequences with overlapping condition events, or repeating until non-tie scenarios.
Distribution Identification and Parameter Properties
Occasionalmixed1 marks2026, 2021, 2017, 2013
Recognizing standard distributions (Poisson, Binomial, Normal, Exponential) from their PMF/PDF or PGF, and computing mean, variance, or standard deviation directly from algebraic parameters.
Set-Theoretic Probability and Inclusion-Exclusion
OccasionalMSQ1.5 marks2024, 2023, 2014
Determining union, intersection, and complement probabilities or divisibility counts over finite sets using the Principle of Inclusion-Exclusion and event independence definitions.
Conditional Probability with Reduced Sample Space
OccasionalMCQ1 marks2011
A random experiment (e.g., multiple fair coin tosses) is conducted with partial outcome information provided as a conditioning event (e.g., 'at least one head'). The candidate must evaluate the posterior probability of a specific joint outcome given the conditioned sample space.
Recursive Probabilistic Experiment / First-Step Conditioning
RareNAT2 marks2016
An experiment with distinct stopping and loopback conditions defined across coin tosses or dice rolls, requiring the determination of the eventual probability of reaching a particular outcome state.
Evaluation of Cumulative Distribution Function (CDF)
RareMCQ1 marks2012
Given a discrete random variable with specified outcomes and probabilities, determine the values of its CDF $F(x)$ at specific points.
Sum Distribution of Discrete Uniform Random Variables
RareNAT2 marks2014
Finding the probability or number of favorable outcomes for the sum of independent discrete random variables (e.g., $n$ fair six-sided dice) equaling a specific target value.

Formulas GATE uses

Bayes' Theorem

P(A_i \mid B) = \frac{P(B \mid A_i) P(A_i)}{\sum_{j} P(B \mid A_j) P(A_j)}

Used when computing posterior probabilities given an observed condition across mutually exclusive causes.

Variance and Second Moment Relation

\mathrm{Var}(X) = E[X^2] - (E[X])^2

Used to compute variance or second moments, especially for Poisson, Binomial, and discrete linear transformations.

Linearity of Expectation

E\left[\sum_{i=1}^n c_i X_i\right] = \sum_{i=1}^n c_i E[X_i]

Used for expected counts (triangles in random graphs, word lengths, multi-step exam rewards) regardless of independence.

PDF Normalization and Interval Probability

\int_{-\infty}^{\infty} f(x) \, dx = 1, \quad P(a \le X \le b) = \int_a^b f(x) \, dx

Used to evaluate unknown parameters in continuous probability density functions and calculate slice probabilities.

Binomial Variance & Standard Deviation

\sigma = \sqrt{n p (1 - p)}

Used when computing the spread/standard deviation of independent Bernoulli trial counts.

Cauchy-Schwarz Inequality for Random Variables

(\mathrm{Cov}(X, Y))^2 \le \mathrm{Var}(X) \, \mathrm{Var}(Y)

Used to evaluate the validity of theoretical statements bounding covariance and correlation.

Poisson Distribution PMF and Variance

P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}, \quad E[X] = \mathrm{Var}(X) = \lambda

Used to find probabilities of rare event counts and expectations involving shifted quadratic terms.

Gaussian (Normal) PDF Form

f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)

Used to match standard Gaussian parameters from an exponential-quadratic density equation.

Conditional Probability Definition

P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{|A \cap B|}{|B|}

Used to find the probability of event $A$ occurring given that event $B$ has already occurred, especially when all elementary outcomes in sample space $S$ are equally likely.

Law of Total Probability (First-Step Analysis)

P(E) = P(\text{Stop with } E) + P(\text{Restart}) \cdot P(E)

Used to set up a linear equation for the overall success probability when an experiment can restart identically from the initial state.

Sum of Infinite Geometric Series for Repeated Trials

P(E) = \sum_{k=0}^{\infty} P(\text{Restart})^k \cdot P(\text{Stop with } E) = \frac{P(\text{Stop with } E)}{1 - P(\text{Restart})}

Used to compute total probability over indefinitely repeated independent trials.

Cumulative Distribution Function (CDF)

F(x) = P(X \le x) = \sum_{x_i \le x} P(X = x_i)

Used to compute cumulative probabilities up to a given real value $x$ for a discrete random variable.

Stars and Bars (Non-negative Integer Solutions)

\binom{n + k - 1}{k} = \binom{n + k - 1}{n - 1}

Used to find the number of ways to distribute a sum $k$ across $n$ non-negative integer variables $y_1 + y_2 + \dots + y_n = k$ .

Symmetry of Dice Sums

\text{Ways}(S) = \text{Ways}(n(m + 1) - S)

Used to simplify counting the number of ways to achieve a large sum $S$ with $n$ fair $m$ -sided dice by reflecting it about the mean.

Dice Transformation for Upper Bounds

y_i = 6 - d_i \quad \text{where } 1 \le d_i \le 6 \implies 0 \le y_i \le 5

Used to shift bounded dice variables into non-negative integers summing to a small number.

Examiner traps in Probability and Statistics

Assuming independence implies mutual exclusivity or vice versa (e.g., P(A ∩ B) = 0 vs P(A ∩ B) = P(A)P(B)).
Forgetting that standard deviation is invariant under constant shifts: (aX + b) = |a| (X), omitting +b.
Overlooking overlap/dependence in conditional coin-toss problems (e.g., treating shared coin tosses across two events as independent).
Rounding errors in NAT questions when computing rational probabilities like 5/42 or powers of (0.99)5.
In continuous interval questions (e.g., subinterval containing a point), failing to split the integral at the boundary point where the piecewise definition changes.
Confusing covariance definition Cov(X, Y) = E[(X- X)(Y- Y)] (which can be negative) with the non-negative expectation of absolute differences.
Assuming unconditional independence (e.g., assuming the second coin is independently 1/2 without accounting for the condition 'at least one outcome is a head')
Confusing the conditioned event denominator with the total original sample space (e.g., obtaining 1/4 instead of 1/3)
Failing to account for permutations in compound outcomes (e.g., counting HT and TH as a single event)
Stopping at the first step and answering the single-step probability (e.g., (1)/(4) = 0.25) instead of solving the recurrence for the infinite trial sequence ((1)/(3) 0.33).
Misidentifying the sample space of multiple independent coin flips (e.g., treating (TAILS, HEADS) and (HEADS, TAILS) as indistinguishable single outcomes).
Rounding errors in NAT questions when converting fractions like (1)/(3) to decimal form.
Confusing P(X < x) with P(X ≤ x), which causes an off-by-one probability calculation at discontinuity points.
Assuming F(xmin) = 0 instead of F(xmin) = P(X = xmin) for discrete distributions.
Applying standard stars-and-bars without enforcing the upper bound constraint (di ≤ 6), which is only valid when the transformed sum is less than 6.
Off-by-one errors when shifting bounds (e.g., confusing yi = 6 - di with yi = di - 1).
Mistaking the total sample space size (64 = 1296) when setting up the probability ratio.

How questions are asked

Compute posterior probability via Bayes' rule in channel transmission or urn problems
Calculate expected value or variance of a discrete/continuous random variable
Find normalisation constant or bounds for a given PDF
Evaluate truth values of multiple theoretical statements on independence and covariance
Count favorable outcomes using permutations/combinations and Inclusion-Exclusion
Determine the optimal sequential strategy to maximize expected score/reward
Word problem computing conditional probability of multi-stage Bernoulli trials
Step-by-step algorithmic flow with conditional branches and loopback
Direct evaluation of CDF values at discrete support points
Find the numerator X in the probability expression X / 6n for a specified sum.

Paper trends

Shift from simple standard MCQ formulas towards NAT and multi-select MSQs testing precise definitions of independence, mutual exclusivity, and covariance bounds.
2024, 2023, 2021, 2015, 2014
Introduction of geometric/continuous piecewise random variables (such as stick-breaking, subinterval containment, and median of rectified Gaussian variables).
2025, 2017, 2014
Growing use of CS-specific applications including bit transmission over noisy channels, parity of random binary vectors, and random graph substructures.
2025, 2021, 2020, 2019, 2015, 2013
Direct, single-step sample space reduction and conditional probability testing was posed as a foundational 1-mark MCQ in earlier GATE exams.
2011
Appeared as a 2-mark NAT question testing recursive probability and law of total probability modeled via an algorithmic sequence of coin flips.
2016
Appeared as a direct, foundational 1-mark multiple choice question evaluating definition-level knowledge of CDFs.
2012
In 2014, discrete probability was tested via a 2-mark NAT problem requiring combinatorial variable transformation rather than straightforward brute-force enumeration.
2014

Difficulty notes

Easy questions (1 mark) typically involve direct plug-and-chug into distribution formulas (Binomial variance, Poisson mass function, Gaussian PDF parameter identification) or basic 2-event set formulas. Medium questions (2 marks) require conditioning on intermediate random variables (e.g., common toss outcome in coin sequences, multi-stage Pólya urn analysis), piecewise definite integrals for continuous expectations, or applying Bayes' theorem to transition diagrams.