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GATE DA Probability & Statistics Previous Year Questions

40 solved GATE DA questions on Probability & Statistics, drawn from 3 exam years and grouped by year. Every question shows the official answer and a step-by-step solution.

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Probability & Statistics: define the experiment before calculating

Define outcomes, events, and a sampling rule before choosing a formula. Conditional probabilities depend on observations; statistical conclusions depend on data collection. These selected foundations connect counting, distributions, and inference without replacing the full syllabus.

Our study notes and original examples support the PYQs below; they are not official exam questions or a replacement for the current syllabus.

Before you start

  • Set operations, fractions, factorials, and sums of finite sequences.
  • Basic differentiation and integration for continuous probability densities.

Concepts to revise before solving

Counting and the sample space

Decide whether order matters and whether sampling permits replacement. Count favorable and total outcomes using the same representation. Their ratio gives a probability only when elementary outcomes are equally likely. For overlapping events, subtract the intersection once when applying inclusion-exclusion.

Check yourself: Are ordered sequences being compared with unordered selections?

Conditioning and Bayes' rule

P(A given B) = P(A ∩ B)/P(B), provided P(B) is positive. For a partition of possible causes, total probability combines each cause's prior with its likelihood. Bayes' rule normalizes those joint probabilities after observing evidence; it does not discard the priors.

Check yourself: Does your denominator include every way the observation can occur?

Random variables and distributions

A PMF assigns probability to discrete values; a PDF must be integrated over intervals. Both normalize to one over their support. A CDF gives P(X ≤ x). A binomial count requires independent Bernoulli trials with the same success probability, not merely a fixed trial count.

Check yourself: Does the stated experiment satisfy the distribution's assumptions?

Moments and dependence

Compute variance as E(X²) − E(X)². Expectation is linear even for dependent variables, but Var(X + Y) includes twice their covariance. Independence implies zero covariance when these moments exist; zero covariance alone does not establish independence. Scaling X by a scales its variance by a².

Check yourself: Have you justified dropping the covariance term?

Sampling and inference

For independent, identically distributed observations with finite variance σ², the sample mean has variance σ²/n. The central limit theorem motivates a large-sample normal approximation. Confidence levels describe repeated interval construction; a p-value measures tail extremeness under the null model, not the null hypothesis's probability.

Check yourself: Are you distinguishing data variability from the sample mean's uncertainty?

Mistakes to avoid

Reversing the direction of a conditional probability.
Write the conditioned event in the denominator and reconstruct the joint probability before simplifying.
Treating mutually exclusive positive-probability events as independent.
Their intersection is empty, whereas independence would require the positive product of their probabilities.
Using a density height as a point probability.
Integrate over an interval; a variable with a PDF has zero probability at any single point.

Original teaching example · not a PYQ

Work through the reasoning

Choose bag A with probability 1/3 and bag B with probability 2/3. A contains three red balls and one blue; B contains one red and three blue. Draw one ball uniformly from the chosen bag. Given a red draw, what is the probability that A was chosen?

  1. Let R mean a red draw. The likelihoods are P(R given A) = 3/4 and P(R given B) = 1/4.
  2. Weight by bag selection: P(A ∩ R) = (1/3)(3/4) = 1/4; P(B ∩ R) = (2/3)(1/4) = 1/6.
  3. The bag events partition the experiment, so P(R) = 1/4 + 1/6 = 5/12.
  4. Normalize the A branch among red draws: P(A given R) = (1/4)/(5/12) = 3/5.

P(A given red) = 3/5 = 0.6.

Try it before reading the answer

Let X count successes in four independent trials, each with success probability 1/4. Find E(X), Var(X), and P(X = 0).

Show answer and reasoning

E(X) = 1; Var(X) = 3/4; P(X = 0) = 81/256.

X is binomial with n = 4 and p = 1/4. Its mean is np and variance np(1 − p). Zero successes requires four failures, whose independent probabilities multiply to (3/4)^4 = 81/256.

Go deeper with free learning resources

Supplemental reading, not an official GATE reading list or an endorsement of these notes.

Apply this to the previous-year questions

Previous-year questions by year

This page shows 40 recent questions from the released archive, newest first. For older questions and complete papers, browse all GATE DA papers. Questions can carry more than one subject tag; counts are not marks weightage.

GATE DA 202614 questions

  1. Set 1 Q7Five integers are picked from 0 to 20, with possible repetitions, such that their mean is 12, median is 18, and they have a single mode of 20. Ignoring…MCQ · +2 marks · Medium
  2. Set 1 Q19Let MM be a randomly chosen non-empty subset of S={1,2,3,…,2026}S = \{1, 2, 3, \dots, 2026\}. Which of the following is the probability that the product of all the elements…MCQ · +1 marks · Medium
  3. Set 1 Q20Suppose that a computer program provides a non-negative and integer-valued random solution to the equation n1+n2+n3+n4=20n_1 + n_2 + n_3 + n_4 = 20. Which of the following…MCQ · +1 marks · Medium
  4. Set 1 Q28Suppose a random variable ZZ follows Normal(μ=0,σ2=1)Normal(\mu = 0, \sigma^2 = 1) distribution with probability density function g(z)g(z) and cumulative distribution…MSQ · +1 marks · Hard
  5. Set 1 Q33The number of bijections f(⋅)f(\cdot) from the set S={1,2,3,4}S = \{1, 2, 3, 4\} to itself such that f(f(n))=nf(f(n)) = n, for all n∈Sn \in S, is __________ . (Answer in integer)NAT · +1 marks · Medium
  6. Set 1 Q34Let XX be an exponentially distributed random variable with mean λ(>0)\lambda (> 0). If P(X>5)=0.35P(X > 5) = 0.35, then the conditional probability…NAT · +1 marks · Easy
  7. Set 1 Q44Let XX and YY be two independent random variables. XX follows Bernoulli(p=0.3)Bernoulli(p = 0.3) distribution and YY follows Normal(μ=0,σ2=100)Normal(\mu = 0, \sigma^2 = 100)…MCQ · +2 marks · Medium
  8. Set 1 Q45Let L=lim⁡n→∞∑k=0ne−nnkk!L = \lim_{n \to \infty} \sum_{k=0}^{n} \frac{e^{-n} n^k}{k!} Which of the following is the value of LL?MCQ · +2 marks · Hard
  9. Set 1 Q53Let X1,X2,…,XnX_1, X_2, \dots, X_n be nn independent random variables. Each of the random variables follows Normal(μ=0,σ2=1)Normal(\mu=0, \sigma^2=1) distribution. Define…MSQ · +2 marks · Medium
  10. Set 1 Q54Let XX be a discrete valued random variable with cumulative distribution function F(x)F(x). Which of the following statements is/are correct?MSQ · +2 marks · Medium
  11. Set 1 Q57A clinic specializes in testing for a disease D. The result of the test can be either positive or negative. A study revealed that if a person suffers from the…NAT · +2 marks · Medium
  12. Set 1 Q62For a given data set {x1,x2,…,xn}\{x_1, x_2, \dots, x_n\}, where n=100n = 100, it is known that 12000∑i=1n∑j=1n(xi−xj)2=99\frac{1}{2000} \sum_{i=1}^{n} \sum_{j=1}^{n} (x_i - x_j)^2 = 99 Let us…NAT · +2 marks · Medium
  13. Set 1 Q63Let XX be a random variable that follows Uniform(−1,1)Uniform(-1,1) distribution. The conditional distribution of the random variable YY given X=xX = x is the…NAT · +2 marks · Medium
  14. Set 1 Q64Let A5×5A_{5 \times 5} be a matrix such that each of its elements follows Bernoulli(p=0.50)Bernoulli(p = 0.50) distribution independently. The probability that the row-sum of…NAT · +2 marks · Medium

GATE DA 202514 questions

  1. Set 1 Q10The number of patients per shift (XX) consulting Dr. Gita in her past 100 shifts is shown in the figure. If the amount she earns is ₹ 1000(X−0.2)1000(X - 0.2), what is…MCQ · +2 marks · Medium
  2. Set 1 Q11Suppose XX and YY are random variables. The conditional expectation of XX given YY is denoted by E[X∣Y]E[X|Y]. Then E[E[X∣Y]]E[E[X|Y]] equalsMCQ · +1 marks · Easy
  3. Set 1 Q19Let XX be a continuous random variable whose cumulative distribution function (CDF) FX(x)F_X(x), for some tt, is given as follows:…MCQ · +1 marks · Easy
  4. Set 1 Q20Let X=aZ+bX = aZ + b, where ZZ is a standard normal random variable, and a,ba, b are two unknown constants. It is given that…MCQ · +1 marks · Medium
  5. Set 1 Q21It is given that P(X≥2)=0.25P(X \geq 2) = 0.25 for an exponentially distributed random variable XX with E[X]=1λE[X] = \frac{1}{\lambda}, where E[X]E[X] denotes the…MCQ · +1 marks · Easy
  6. Set 1 Q31There are three boxes containing white balls and black balls. Box-1 contains 2 black and 1 white balls. Box-2 contains 1 black and 2 white balls. Box-3…NAT · +1 marks · Medium
  7. Set 1 Q36Let Y=Z2,Z=X−μσY = Z^2, Z = \frac{X - \mu}{\sigma}, where XX is a normal random variable with mean μ\mu and variance σ2\sigma^2. The variance of YY isMCQ · +2 marks · Medium
  8. Set 1 Q39Consider the cumulative distribution function (CDF) of a random variable XX:…MCQ · +2 marks · Medium
  9. Set 1 Q40A random variable XX is said to be distributed as Bernoulli(θ)Bernoulli(\theta), denoted by X∼Bernoulli(θ)X \sim Bernoulli(\theta), if…MCQ · +2 marks · Medium
  10. Set 1 Q41For x∈Rx \in \mathbb{R}, the floor function is denoted by f(x)=⌊x⌋f(x) = \lfloor x \rfloor and defined as follows ⌊x⌋=k,k≤x<k+1,\lfloor x \rfloor = k, \quad k \le x < k + 1,…MCQ · +2 marks · Medium
  11. Set 1 Q45A random experiment consists of throwing 100 fair dice, each die having six faces numbered 1 to 6. An event AA represents the set of all outcomes where at…MCQ · +2 marks · Easy
  12. Set 1 Q54Consider a coin-toss experiment where the probability of head showing up is pp. In the ithi^{th} coin toss, let Xi=1X_i = 1 if head appears, and Xi=0X_i = 0 if…MSQ · +2 marks · Easy
  13. Set 1 Q60Let D={x(1),…,x(n)}D = \{x^{(1)}, \dots, x^{(n)}\} be a dataset of nn observations where each x(i)∈R100x^{(i)} \in \mathbb{R}^{100}. It is given that…NAT · +2 marks · Medium
  14. Set 1 Q61A bag contains 5 white balls and 10 black balls. In a random experiment, nn balls are drawn from the bag one at a time with replacement. Let SnS_n denote the…NAT · +2 marks · Easy

GATE DA 202412 questions

  1. Set 1 Q3How many 4-digit positive integers divisible by 3 can be formed using only the digits {1,3,4,6,7}\{1, 3, 4, 6, 7\}, such that no digit appears more than once in a…MCQ · +1 marks · Medium
  2. Set 1 Q7The probability of a boy or a girl being born is 1/21/2. For a family having only three children, what is the probability of having two girls and one boy?MCQ · +2 marks · Easy
  3. Set 1 Q11Consider the following statements: (i) The mean and variance of a Poisson random variable are equal. (ii) For a standard normal random variable, the mean is…MCQ · +1 marks · Easy
  4. Set 1 Q12Three fair coins are tossed independently. TT is the event that two or more tosses result in heads. SS is the event that two or more tosses result in tails.…MCQ · +1 marks · Easy
  5. Set 1 Q27Let the minimum, maximum, mean and standard deviation values for the attribute income of data scientists be ₹46000, ₹170000, ₹96000, and ₹21000,…MCQ · +1 marks · Easy
  6. Set 1 Q34The sample average of 50 data points is 40. The updated sample average after including a new data point taking the value of 142 is ______.NAT · +1 marks · Easy
  7. Set 1 Q36A fair six-sided die (with faces numbered 1, 2, 3, 4, 5, 6) is repeatedly thrown independently. What is the expected number of times the die is thrown until…MCQ · +2 marks · Medium
  8. Set 1 Q56Let XX be a random variable uniformly distributed in the interval [1,3][1, 3] and YY be a random variable uniformly distributed in the interval [2,4][2, 4]. If XX…NAT · +2 marks · Medium
  9. Set 1 Q57Let XX be a random variable exponentially distributed with parameter λ>0\lambda > 0. The probability density function of XX is given by:…NAT · +2 marks · Easy
  10. Set 1 Q58Consider two events TT and SS. Let Tˉ\bar{T} denote the complement of the event TT. The probability associated with different events are given as follows:…NAT · +2 marks · Easy
  11. Set 1 Q59Consider a joint probability density function of two random variables XX and YY…NAT · +2 marks · Medium
  12. Set 1 Q65Two fair coins are tossed independently. XX is a random variable that takes a value of 1 if both tosses are heads and 0 otherwise. YY is a random variable…NAT · +2 marks · Medium

Other GATE DA topics

Programming, Data Structures & Algorithms(32)General Aptitude(29)Machine Learning(28)Database Management & Warehousing(23)Linear Algebra(21)Artificial Intelligence(18)Calculus & Optimization(15)

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