Probability and Statistics GATE Practice Set
Mixed GATE DA style questions on counting, conditional probability and Bayes, random variables, distributions, the central limit theorem, confidence intervals and hypothesis tests, with answers and explanations.
These questions mix every topic in this track, in the GATE formats MCQ, MSQ and NAT. Each numerical answer is recomputed by a program when the site is built.
Probability
[Counting] How many 4-digit numbers (no leading zero) have all digits distinct?
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Answer: 4536The first digit has 9 choices (1 to 9). The second has 9 (any digit except the first, now including 0), the third 8 and the fourth 7: 9 × 9 × 8 × 7 = 4536.
[Conditional probability] Two fair dice are rolled. Given that the sum is 8, what is the probability that both dice show even numbers? Give the answer to one decimal place.
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Answer: 0.6Sum 8 has 5 outcomes: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2). Three of them have both dice even, so the probability is 3/5 = 0.6.
[Bayes] Box A has 2 white and 3 black balls; box B has 4 white and 1 black. A box is chosen with probability 1/2 each and a ball is drawn; it is white. What is the probability that it came from box B? Give the answer to three decimal places.
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Answer: 0.667P(white) = ½ × 2/5 + ½ × 4/5 = 0.2 + 0.4 = 0.6. P(B | white) = 0.4 / 0.6 = 2/3 ≈ 0.667.
[Independence] P(A) = 0.4 and P(B) = 0.5, and A and B are independent. What is P(A ∪ B)?
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Answer: 0.7P(A ∩ B) = 0.4 × 0.5 = 0.2, so P(A ∪ B) = 0.4 + 0.5 − 0.2 = 0.7.
Random variables and distributions
[Expectation] X has pdf f(x) = 3x² on [0, 1]. What is E[X]?
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Answer: 0.75E[X] = ∫₀¹ x · 3x² dx = 3/4 = 0.75.
[Binomial] A multiple-choice test has 5 questions with 4 options each. A student guesses every answer at random. What is the probability of getting at least 4 right? Give the answer to four decimal places.
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Answer: 0.0156X ~ Binomial(5, 0.25). P(4) = 5 × 0.25⁴ × 0.75 = 0.014648 and P(5) = 0.25⁵ = 0.000977. The total is about 0.0156.
[Poisson] A page of a book has on average 0.5 typos (Poisson). What is the probability that a page has at least one typo? Give the answer to four decimal places.
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Answer: 0.3935P(X ≥ 1) = 1 − P(X = 0) = 1 − e^(−0.5) ≈ 1 − 0.6065 = 0.3935.
[Normal] X ~ N(50, 25). What is P(45 < X < 60)? Give the answer to four decimal places.
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Answer: 0.8186σ = 5. Standardising: P(−1 < Z < 2) = Φ(2) − Φ(−1) = 0.9772 − 0.1587 = 0.8186 (to four decimal places, using exact values).
[Covariance] Var(X) = 4, Var(Y) = 1 and ρ(X, Y) = 0.5. What is Var(X + 2Y)?
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Answer: 12Cov(X, Y) = ρσXσY = 0.5 × 2 × 1 = 1. Var(X + 2Y) = Var(X) + 4 Var(Y) + 2 × 2 × Cov(X, Y) = 4 + 4 + 4 = 12.
Inference
[CLT] The time to serve a customer has mean 3 minutes and standard deviation 2 minutes. What is the approximate probability that 100 customers take more than 330 minutes in total? Give the answer to four decimal places.
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Answer: 0.0668The total has mean 100 × 3 = 300 and standard deviation √100 × 2 = 20. Z = (330 − 300)/20 = 1.5, and P(Z > 1.5) = 1 − 0.9332 = 0.0668.
[Confidence interval] A 95% z interval for μ, based on n = 25 and known σ = 10, is (46.08, 53.92). What was the sample mean?
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Answer: 50The interval is centred at X̄: (46.08 + 53.92)/2 = 50. (The half-width 3.92 = 1.96 × 10/5, as expected.)
[Hypothesis test] In a right-tailed z-test at α = 0.05, the test statistic is 1.5. Which statements are true? (Select all that apply.)
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Answer: A, B, DThe critical value at 5% is 1.645 > 1.5, so H₀ is not rejected. p = 1 − Φ(1.5) ≈ 0.0668, which is below 0.10, so H₀ would be rejected at the 10% level (critical value 1.282).
[Chi-squared] For a 3 × 3 contingency table with 90 observations, row totals 30, 30, 30 and column totals 20, 30, 40, what is the expected count in row 1, column 3?
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Answer: 13.3333E = row total × column total / grand total = 30 × 40 / 90 = 13.3333.