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GATE probability and statistics: 5 practice questions
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Probability and statistics is in the engineering-mathematics block of CS, EC, EE, ME and CE, and RA includes it too (with hypothesis testing). The questions are usually short: apply Bayes' theorem, use the mean and variance of a standard distribution, or count cases under a condition. The five original questions below cover common patterns.
What GATE tests in probability and statistics
The 2027 CS syllabus lists random variables; the uniform, normal, exponential, Poisson and binomial distributions; mean, median, mode and standard deviation; conditional probability and Bayes' theorem. The other engineering papers list probability and statistics as one item in their maths block, and the DA paper treats it as a full section reaching to the central limit theorem, confidence intervals and hypothesis tests. See probability for data science for that deeper version, and check your own paper's list on the official syllabus page.
Typical shapes are Bayes' theorem with two or three causes, expected value and variance of sums, a Poisson or exponential probability, conditional probability with dice, cards or coins, and quick statistics questions on mean, median and mode. The real paper also has MSQ (more than one correct option) and NAT; all five below are MCQ.
Practice questions
Question 1
Machine M1 makes 60% of the items in a factory and M2 makes the remaining 40%. 2% of M1's items are defective and 5% of M2's items are defective. An item picked at random is found to be defective. The probability that it came from M2 is:
- A. 3/8
- B. 1/2
- C. 5/8
- D. 3/4
Question 2
X is a Poisson random variable with P(X = 1) = P(X = 2). The value of P(X = 0) is:
- A. e^(−1)
- B. 2e^(−2)
- C. e^(−2)/2
- D. e^(−2)
Question 3
X and Y are independent random variables with Var(X) = 4 and Var(Y) = 9. The variance of 2X − 3Y + 5 is:
- A. 97
- B. 13
- C. 35
- D. 122
Question 4
The lifetime of a component is exponentially distributed with mean 100 hours. Given that it has already worked for 50 hours, the probability that it works for at least another 100 hours is:
- A. e^(−0.5)
- B. e^(−1)
- C. e^(−1.5)
- D. e^(−2)
Question 5
A fair coin is tossed four times. Given that at least one head appears, the probability that exactly two heads appear is:
- A. 3/8
- B. 7/15
- C. 1/2
- D. 2/5
Answers and explanations
- C. By Bayes' theorem, P(M2 | defective) = (0.4 × 0.05) / (0.6 × 0.02 + 0.4 × 0.05) = 0.020 / 0.032 = 5/8. M2 makes fewer items but has a higher defect rate, so it contributes more of the defective ones.
- D. P(X = 1) = λe^(−λ) and P(X = 2) = λ²e^(−λ)/2. Setting them equal gives λ = 2, because the parameter of a Poisson variable is positive and λ = 0 is excluded. Then P(X = 0) = e^(−2) ≈ 0.135.
- A. For independent variables, Var(aX + bY + c) = a²Var(X) + b²Var(Y). That gives 4 × 4 + 9 × 9 = 16 + 81 = 97. The constant 5 does not change the variance, and the signs of the coefficients do not matter because they are squared.
- B. The exponential distribution is memoryless: P(T > 150 | T > 50) = P(T > 100) = e^(−100/100) = e^(−1). Option C is the unconditional chance of surviving 150 hours, and A is the unconditional chance of surviving 50 hours.
- D. The sample space has 16 outcomes. Exactly two heads occur in C(4, 2) = 6 outcomes, and "at least one head" has 16 − 1 = 15 outcomes. The conditional probability is 6/15 = 2/5. The value 3/8 = 6/16 forgets to condition.
How to prepare probability and statistics
Memorise the mean and variance of the standard distributions, because you will use them in almost every question.
| Distribution | Mean | Variance |
|---|---|---|
| Binomial (n, p) | np | np(1 − p) |
| Poisson (λ) | λ | λ |
| Uniform (a, b) | (a + b)/2 | (b − a)²/12 |
| Exponential (rate λ) | 1/λ | 1/λ² |
| Normal (μ, σ²) | μ | σ² |
Then build the habits that save marks:
- Draw a tree for Bayes' theorem. Write the prior probabilities on the first branches and the likelihoods on the second, multiply along each path, and divide the path you want by the sum of all paths that meet the condition.
- Know when to use which rule: E[aX + b] = aE[X] + b always holds, but Var(X + Y) = Var X + Var Y only when X and Y are independent (or uncorrelated).
- Practise the memoryless property of the exponential (and geometric) distribution, which hides behind "given that it has survived" wording.
- For "at least one" questions, compute the complement first: 1 − P(none).
- Revise median and mode definitions with small data sets. They are quick marks if you are careful.
The GATE engineering mathematics guide lists which probability topics each branch includes. Pair this page with the calculus and linear algebra practice pages so the whole maths block is covered.
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