Probability & Statistics
Practice JEE Main Probability & Statistics PYQs online. Master Bayes’ Theorem, variance shift formula, binomial distribution & step-by-step solutions.
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🎯 High-Yield Syllabus Breakdown
These high-frequency concepts account for over 90% of all questions asked in Probability & Statistics:
- Conditional Probability & Multiplication Rule
- Bayes’ Theorem & Law of Total Probability
- Random Variables & Probability Distributions
- Mean, Variance & Standard Deviation (Var(X) = E(X²) - (E(X))²)
- Variance Shift Invariance: Var(aX + b) = a² Var(X)
⚡ Essential Formulas & Speed Shortcuts
Formulas and time-saving shortcuts to solve Probability & Stats questions in under 45 seconds:
Shift Invariance: Adding or subtracting constant c does not change Variance: Var(X + c) = Var(X).
Scaling: Var(k·X) = k² · Var(X) and SD(k·X) = |k| · SD(X).
Independent Events: P(A ∩ B) = P(A) · P(B).
📖 Sample Solved Exam Questions
Preview authentic questions from this chapter with full step-by-step solutions:
💡 View Step-by-Step Solution & Answer
Correct Answer: 0.04
- Step 1: By definition of conditional probability: P(B|A) = P(A ∩ B) / P(A) ⇒ P(A ∩ B) = P(A) · P(B|A).
- Step 2: Calculate P(A ∩ B) = 0.4 × 0.6 = 0.24.
- Step 3: By the union formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.4 + 0.8 - 0.24 = 0.96.
- Step 4: By De Morgan's Law: P(A' ∩ B') = P((A ∪ B)') = 1 - P(A ∪ B) = 1 - 0.96 = 0.04.
⚡ Speed Tip: De Morgan's Shortcut: P(A' ∩ B') = 1 - P(A ∪ B) = 1 - (0.4 + 0.8 - 0.24) = 0.04.
💡 View Step-by-Step Solution & Answer
Correct Answer: 3/10
- Step 1: Total balls = 4 (red) + 6 (black) = 10 balls.
- Step 2: Total number of ways to draw 3 balls from 10 is ¹⁰C₃ = (10 × 9 × 8) / (3 × 2 × 1) = 120.
- Step 3: Favorable ways: Choose 2 red balls from 4 and 1 black ball from 6: ⁴C₂ · ⁶C₁ = 6 × 6 = 36.
- Step 4: Probability = 36 / 120 = 3 / 10.
⚡ Speed Tip: Hypergeometric Formula: P = (⁴C₂ · ⁶C₁) / ¹⁰C₃ = (6 × 6) / 120 = 36 / 120 = 3/10.
💡 View Step-by-Step Solution & Answer
Correct Answer: 2
- Step 1: Let the remaining two observations be x and y. Total observations N = 7.
- Step 2: Mean = 8 ⇒ (2 + 4 + 10 + 12 + 14 + x + y)/7 = 8 ⇒ 42 + x + y = 56 ⇒ x + y = 14.
- Step 3: Variance = 16 ⇒ (∑ xᵢ²)/7 - (Mean)² = 16 ⇒ (∑ xᵢ²)/7 - 64 = 16 ⇒ (∑ xᵢ²)/7 = 80 ⇒ ∑ xᵢ² = 560.
- Step 4: 2² + 4² + 10² + 12² + 14² + x² + y² = 4 + 16 + 100 + 144 + 196 + x² + y² = 460 + x² + y² = 560 ⇒ x² + y² = 100. With x + y = 14 and x² + y² = 100, the numbers are 8 and 6 (8² + 6² = 100). Absolute difference = |8 - 6| = 2.
⚡ Speed Tip: Pythagorean pair: x + y = 14 and x² + y² = 100 immediately yields {6, 8}, so |x - y| = 2.
❓ Frequently Asked Questions (FAQs)
Does change of origin affect variance in JEE Main questions?
No! Adding or subtracting any constant from all data points leaves variance and standard deviation completely unchanged.