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JEE Advanced Statistics and Probability — practice questions — page 2

Explore 125 free Statistics and Probability questions in JEE Advanced Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Statistics and Probability questions and exercises with answers

Showing questions 31–60 of 125.

A coin tossed 10 times. P(exactly 5 heads):Read the answer and worked solution →If P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2: A and B independent?Read the answer and worked solution →For random variable X uniform on [0, 1]: E(X²) =Read the answer and worked solution →For uniform U[0,1]: variance =Read the answer and worked solution →E[X² + Y²] when X, Y are independent identically distributed with E(X) = 0, Var(X) = σ²:Read the answer and worked solution →Cov(X, X) =Read the answer and worked solution →Correlation coefficient r:Read the answer and worked solution →5 cards from deck of 52. P(all spades):Read the answer and worked solution →Coefficient of variation (CV):Read the answer and worked solution →Mean of frequency distribution Σf_i x_i / Σf_i. Find mean of: 2(×3), 4(×5), 6(×2): values 2,4,6 with frequencies 3, 5, 2:Read the answer and worked solution →Standard deviation property: σ(aX + b) =Read the answer and worked solution →For Poisson distribution with rate λ: P(X = k) =Read the answer and worked solution →Three-card hand probability of a flush (any same suit):Read the answer and worked solution →At least one '6' in 4 throws of a fair die:Read the answer and worked solution →Variance of fair die roll (1-6):Read the answer and worked solution →P(A) = 1/3, P(B) = 1/2, P(A ∩ B) = 1/6. P(A ∪ B) =Read the answer and worked solution →From 7 men and 5 women, choose committee of 4 with at least 1 woman:Read the answer and worked solution →A box has 6 balls, 4 are red. Two balls drawn (without replacement). P(both red):Read the answer and worked solution →Mean Square Error (MSE) of estimator decomposes as:Read the answer and worked solution →Properties of normal distribution: ___ % of data within 1 SD of mean.Read the answer and worked solution →X ~ Binomial(50, 0.4). Find E[X] and Var[X]:Read the answer and worked solution →For two events A and B, P(A | B) = 0.6, P(B) = 0.5, P(A) = 0.4. P(B | A) =Read the answer and worked solution →Mean Absolute Deviation (MAD):Read the answer and worked solution →Chebyshev's inequality: P(|X - μ| ≥ kσ) ≤Read the answer and worked solution →Two independent X, Y with Var(X) = 4, Var(Y) = 9. Var(X + Y) =Read the answer and worked solution →Three fair coins are tossed. Let E = "at least two heads" and F = "first coin shows tail". Then P(E|F) isRead the answer and worked solution →The conditional probability P(E|F) is defined byRead the answer and worked solution →The multiplication rule for probability states P(E ∩ F) equalsRead the answer and worked solution →Two events E and F are called independent ifRead the answer and worked solution →A die is rolled and a coin is tossed simultaneously. If E = "die shows an even number" and F = "coin shows head", then P(E ∩ F) isRead the answer and worked solution →

Worked practice questions

A coin tossed 10 times. P(exactly 5 heads):

  1. 5/10
  2. 1/2
  3. C(10, 5) × (1/2)¹⁰ = 252/1024 ≈ 0.246
  4. 1/1024
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Correct answer: C. C(10, 5) × (1/2)¹⁰ = 252/1024 ≈ 0.246

Binomial(10, 0.5): P(5) = C(10,5)(1/2)¹⁰ = 252/1024 ≈ 0.246. Most likely single outcome but only ~25% chance.

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If P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2: A and B independent?

  1. Always dependent
  2. No
  3. Yes (P(A)P(B) = 0.2 = P(A ∩ B))
  4. Cannot tell
Show answer and solution

Correct answer: C. Yes (P(A)P(B) = 0.2 = P(A ∩ B))

We have P(A) × P(B) = 0.4 × 0.5. That is P(A) × P(B) = 0.2. So P(A) × P(B) = P(A ∩ B). **Hence independent.**

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For random variable X uniform on [0, 1]: E(X²) =

  1. 0.5
  2. 1/4
  3. 1/2
  4. 1/3 (∫₀¹ x² dx = 1/3)
Show answer and solution

Correct answer: D. 1/3 (∫₀¹ x² dx = 1/3)

E(X²) = ∫₀¹ x² × 1 dx = 1/3. Var(X) = E(X²) - (E(X))² = 1/3 - 1/4 = 1/12.

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