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., 14 = 0.25) instead of solving the recurrence for the infinite trial sequence ( 13 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 13 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.

Probability and Statistics

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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' theorem, discrete a… Guide

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GATE CSE 2026 Set 1Q11MCQ1 markMediumAn 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 put back in the urn. The probability that there are equal number of red and blue balls after two steps is
GATE CSE 2026 Set 1Q58NAT2 marksEasyLet $X$ 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 $\Pr(X = 3) = \Pr(X = 4) = \Pr(X = 6) = \Pr(X = 8) = \frac{1}{12}$ . The expected value of $X$ , denoted by $E[X]$ , is equal to ________.…
GATE CSE 2026 Set 2Q14MCQ1 markEasyThe probability density function $f(x)$ of a random variable $X$ which takes real values is $f(x) = \frac{1}{3\sqrt{2\pi}} \exp\left(-\frac{x^2}{18}\right), \quad x \in (-\infty, +\infty)$ Which one of the following statements is correct about the random variable $X$ ?
GATE CSE 2026 Set 2Q63NAT2 marksMediumSuppose 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 tosses, there are at least two heads. Let $E_2$ be the event that among the first, second, third, and fifth coin tosses, there are equal number…
GATE CSE 2025 Set 1Q32NAT1 markEasyA 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 pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is ________. (rounded off to two…
GATE CSE 2025 Set 1Q56NAT2 marksEasySuppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is $0.01$ . Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is…
GATE CSE 2025 Set 1Q58NAT2 marksEasyConsider a probability distribution given by the density function $P(x)$ . $P(x) = \begin{cases} Cx^2, & \text{for } 1 \le x \le 4 \\ 0, & \text{for } x < 1 \text{ or } x > 4 \end{cases}$ The probability that $x$ lies between 2 and 3, i.e., $P(2 \le x \le 3)$ is __________. (rounded off to three decimal places)
GATE CSE 2025 Set 2Q64NAT2 marksMediumA quadratic polynomial $(x - \alpha)(x - \beta)$ over complex numbers is said to be square invariant if $(x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2)$ . Suppose from the set of all square invariant quadratic polynomials we choose one at random. The probability that the roots of the chosen polynomial are…
GATE CSE 2025 Set 2Q65NAT2 marksMediumThe unit interval $(0,1)$ is divided at a point chosen uniformly distributed over $(0,1)$ in $\mathbb{R}$ into two disjoint subintervals. The expected length of the subinterval that contains $0.4$ is ________. (rounded off to two decimal places)
GATE CSE 2024 Set 1Q14MCQ1 markEasyConsider a permutation sampled uniformly at random from the set of all permutations of $\{1, 2, 3, \cdots, n\}$ for some $n \ge 4$ . Let $X$ be the event that 1 occurs before 2 in the permutation, and $Y$ the event that 3 occurs before 4. Which one of the following statements is TRUE?
GATE CSE 2024 Set 1Q27MSQ1 markEasyLet $A$ 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?
GATE CSE 2024 Set 1Q63NAT2 marksEasyA 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 decimal places) that both balls drawn are red is ________
GATE CSE 2024 Set 2Q18MCQ1 markEasyWhen 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 2Q44MCQ2 marksMediumLet $x$ and $y$ be random variables, not necessarily independent, that take real values in the interval $[0,1]$ . Let $z = xy$ and let the mean values of $x,y,z$ be $\bar{x},\bar{y},\bar{z}$ , respectively. Which one of the following statements is TRUE?
GATE CSE 2023Q53MSQ2 marksEasyConsider 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, and $C$ be the event that denotes HEAD on the second throw. Which of the following statements is/are TRUE?
GATE CSE 2021 Set 1Q18NAT1 markEasyThe lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter 2. For a randomly picked component of this type, the probability that its lifetime exceeds the expected lifetime (rounded to 2 decimal places) is _________.
GATE CSE 2021 Set 1Q35MCQ2 marksMediumConsider the two statements. $\begin{aligned} S_1 :& \quad \text{There exist random variables } X \text{ and } Y \text{ such that} \\ & \quad \left(\mathbb{E}\left[(X - \mathbb{E}(X)) (Y - \mathbb{E}(Y))\right]\right)^2 > \mathrm{Var}[X] \, \mathrm{Var}[Y] \\ S_2 :& \quad \text{For all random variables } X \text{ and } Y, \\ & \quad \mathrm{Cov}[X,Y] = \mathbb{E}\left[|X - \mathbb{E}[X]| \, |Y - \mathbb{E}[Y]|\right] \end{aligned}$ …
GATE CSE 2021 Set 1Q54NAT2 marksMediumDiagramA 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 edge $(u, v)$ is the probability of receiving $v$ when $u$ is transmitted, where $u, v \in \{H, L\}$ . For example, the…
GATE CSE 2021 Set 2Q22NAT1 markEasyFor 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 number of times that head appeared in these 1,000 tosses. The standard deviation of X (rounded to 2 decimal places) is ________.
GATE CSE 2021 Set 2Q29MCQ2 marksMediumDiagramIn an examination, a student can choose the order in which two questions ( $\text{QuesA}$ and $\text{QuesB}$ ) must be attempted. - If the first question is answered wrong, the student gets zero marks. - If the first question is answered correctly and the second question is not answered correctly, the student gets…
GATE CSE 2021 Set 2Q33MCQ2 marksMediumA 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 ball is placed back into the bag along with another ball of the same colour. Note that the number of balls in the bag will increase by one,…
GATE CSE 2020Q45NAT2 marksEasyFor $n > 2$ , 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 $\sum_{i=1}^n a_i x_i$ is an odd number is _________.
GATE CSE 2019Q22NAT1 markMediumTwo 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) binary representations have the same most significant bit is _______.
GATE CSE 2018Q15NAT1 markMediumTwo 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 dice repeatedly until there is no tie. Define a trial as a throw of the dice by P and Q. Assume that all 6 numbers on each dice are…
GATE CSE 2017 Set 1Q19NAT1 markMediumLet $X$ 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 $a$ and $b$ . The median of $Y$ is ________.
GATE CSE 2017 Set 2Q26MCQ2 marksMediumP 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 job is $\frac{1}{2}$ , and the probability that Q applies for the job given that P applies for the job is $\frac{1}{3}$ . Then the…
GATE CSE 2017 Set 2Q31MCQ2 marksMediumFor any discrete random variable $X$ , with probability mass function $P(X = j) = p_j$ , $p_j \ge 0$ , $j \in \{0, \dots, N\}$ , and $\sum_{j=0}^N p_j = 1$ , define the polynomial function $g_X(z) = \sum_{j=0}^N p_j z^j$ . For a certain discrete random variable $Y$ , there exists a scalar $\beta \in [0, 1]$ …
GATE CSE 2017 Set 2Q48NAT2 marksEasyIf a random variable $X$ has a Poisson distribution with mean 5, then the expectation $E[(X + 2)^2]$ equals ________.
GATE CSE 2016 Set 1Q4NAT1 markEasyA 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 ________ .
GATE CSE 2016 Set 1Q29NAT2 marksEasyConsider 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, HEADS) or (HEADS, TAILS), then output $N$ and stop. Step 4. If the outcomes are (TAILS, TAILS), then go to Step 1. The probability…
GATE CSE 2016 Set 2Q5NAT1 markEasySuppose 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 given that it is of Type 2 is 0.4. The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is __________.
GATE CSE 2015 Set 3Q37NAT2 marksMediumSuppose $X_i$ 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 $i = 1,2,3$ . Define another random variable $Y = X_1 X_2 \oplus X_3$ , where $\oplus$ denotes XOR. Then $\Pr[Y = 0 | X_3 = 0] =$ …
GATE CSE 2014 Set 1Q2NAT1 markMediumSuppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .
GATE CSE 2014 Set 1Q48NAT2 marksMediumFour 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 2Q1NAT1 markMediumThe 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 computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are…
GATE CSE 2014 Set 2Q2NAT1 markEasyEach 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 of the pieces is drawn at random from the box. The *expected length* of the word drawn is _________. (The answer should be rounded to one…
GATE CSE 2014 Set 2Q48NAT2 marksMediumThe 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 3Q48NAT2 marksEasyLet $S$ 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 $P(A)P(B)$ is _________.
GATE CSE 2013Q2MCQ1 markEasySuppose $p$ 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 probability of observing fewer than 3 cars during any given minute in this interval?
GATE CSE 2013Q24MCQ1 markMediumConsider 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 of length three?
GATE CSE 2012Q21MCQ1 markEasyConsider 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
GATE CSE 2012Q33MCQ2 marksMediumSuppose 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 turn up is at least 6?
GATE CSE 2011Q3MCQ1 markEasyIf 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 2011Q18MCQ1 markEasyIf the difference between the expectation of the square of a random variable $\left(E\left[X^2\right]\right)$ and the square of the expectation of the random variable $\left(E[X]\right)^2$ is denoted by $R$ , then
GATE CSE 2011Q33MCQ2 marksEasyConsider a finite sequence of random values $X = [x_1, x_2, \dots, x_n]$ . Let $\mu_x$ be the mean and $\sigma_x$ be the standard deviation of $X$ . Let another finite sequence $Y$ of equal length be derived from this as $y_i = a * x_i + b$ , where $a$ and $b$ are positive constants. Let $\mu_y$ be the…
GATE CSE 2011Q34MCQ2 marksEasyA 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 two cards are selected with the number on the first card being one higher than the number on the second card?

Topic guide

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.

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
common · NAT · 2 marks · 2025, 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
common · mixed · 1.5 marks · 2026, 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
common · NAT · 1.5 marks · 2025, 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
common · mixed · 2 marks · 2026, 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
occasional · mixed · 1 marks · 2026, 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
occasional · MSQ · 1.5 marks · 2024, 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
occasional · MCQ · 1 marks · 2011
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
rare · NAT · 2 marks · 2016
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)
rare · MCQ · 1 marks · 2012
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
rare · NAT · 2 marks · 2014
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., 14 = 0.25) instead of solving the recurrence for the infinite trial sequence ( 13 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 13 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.