Complete Solutions and Summary of Probability – NCERT Class 10, Mathematics, Chapter 14 – Summary, Questions, Answers, Extra Questions
Comprehensive summary and explanation of Chapter 14 'Probability', covering theoretical and empirical probability concepts, equally likely outcomes, calculation of probabilities for basic experiments such as coin toss, dice throw, card selection, complementary events, compound events, and a variety of examples and exercises based on NCERT Class X Mathematics.
Updated: 8 months ago
Probability
Chapter 14: Mathematics - Ultimate Study Guide | NCERT Class 10 Notes, Questions, Examples & Quiz 2025
Full Chapter Summary & Detailed Notes - Probability Class 10 NCERT
Overview & Key Concepts
- Chapter Goal: Understand probability as measure of uncertainty. Exam Focus: Theoretical vs empirical, equally likely outcomes, calculations. 2025 Updates: Real-life applications. Fun Fact: Probability from gambling. Core Idea: Predict outcomes. Real-World: Coin toss, dice, cards.
- Wider Scope: Statistics foundation, decision-making.
Introduction
- Quote: Probability theory's mathematical interest, practical importance (R.S. Woodward).
- Probability: Measure uncertainty in experiments like coin toss, die roll.
- History: 16th century J. Cardan 'Book on Games of Chance'. Bernoulli, de Moivre, Laplace contributions. Laplace's 1812 book greatest. Used in biology, economics, etc.
14.1 Probability — A Theoretical Approach
- Equally Likely Outcomes: Fair coin: Head/tail equal chance. Fair die: 1-6 equal. Assume unbiased, random.
- Not Always Equal: Bag 4 red, 1 blue: Red more likely. But color draw equal if specified.
- Assumption: Chapter assumes equally likely outcomes.
- Empirical Probability: From Class IX: P(E) = trials happened / total trials. Limitations: Expensive, unfeasible (e.g., satellite launch, earthquake).
- Theoretical Probability: P(E) = favorable outcomes / total possible. Assumes equal likelihood. Defined by Laplace 1795.
Example 1: Coin Toss Probability
- Outcomes: H, T. P(head) = 1/2, P(tail) = 1/2.
Example 2: Bag with Balls
- Red, blue, yellow. P(yellow) = 1/3, P(red) = 1/3, P(blue) = 1/3.
Remarks
- Elementary event: One outcome. Sum probabilities all elementary = 1.
Example 3: Die Throw
- P(>4) = 2/6 = 1/3 (5,6). P(≤4) = 4/6 = 2/3 (1,2,3,4).
Complementary Events
- P(E) + P(not E) = 1. Not E denoted Ē.
Impossible and Sure Events
- P(impossible) = 0 (e.g., die show 8). P(sure) = 1 (e.g., <7 on die).
- 0 ≤ P(E) ≤ 1.
Example 4: Cards Deck
- 52 cards, 4 suits, 13 each. P(ace) = 4/52 = 1/13. P(not ace) = 48/52 = 12/13.
Example 5: Tennis Match
- P(Sangeeta win) = 0.62, P(Reshma win) = 0.38.
Example 6: Birthdays
- 365 days (non-leap). P(different) = 364/365, P(same) = 1/365.
Example 7: Class Representative
- 40 students, 25 girls. P(girl) = 25/40 = 5/8, P(boy) = 15/40 = 3/8.
Example 8: Marbles Box
- 3 blue, 2 white, 4 red. P(white) = 2/9, P(blue) = 3/9 = 1/3, P(red) = 4/9.
Example 9: Two Coins
- Outcomes: HH, HT, TH, TT. P(at least one head) = 3/4.
Example 10*: Musical Chair (Continuous)
- Stop within 2 min. P(within 0.5 min) = 0.5/2 = 1/4.
Example 11*: Helicopter Crash
- Region 4.5x9=40.5 km², lake 2.5x3=7.5 km². P(lake) = 7.5/40.5 = 5/27.
Example 12: Shirts Carton
- 100 shirts, 88 good, 8 minor, 4 major. P(Jimmy accept good) = 88/100 = 0.88. P(Sujatha accept non-major) = 96/100 = 0.96.
Example 13: Two Dice
- 36 outcomes. P(sum 8) = 5/36, P(13) = 0, P(≤12) = 36/36 = 1.
Exercise 14.1
- 25 questions: Complete statements, equally likely, coin toss fair, invalid probabilities, etc.
14.2 Summary
- Theoretical P(E) = favorable / total possible (equally likely).
- Sure event P=1, impossible P=0.
- 0 ≤ P(E) ≤ 1.
- Elementary events sum P=1.
- P(E) + P(Ē) = 1.
Why This Guide Stands Out
Complete chapter coverage: Notes, examples, Q&A (all NCERT + extras), quiz. Student-centric, exam-ready for 2025. Free & ad-free.
Key Themes & Tips
- Probability: Theoretical, empirical.
- Outcomes: Equally likely.
- Events: Impossible, sure, complementary.
- Tip: Assume equal unless stated; sum P=1.
Exam Case Studies
Cards, dice, bags; continuous probability.
Project & Group Ideas
- Simulate coin tosses; compare empirical/theoretical.
This section provides over 3 pages of detailed notes, covering all subtopics from introduction to summary, with examples explained in depth for comprehensive understanding.
Further details: In theoretical approach, emphasis on assumptions like fair coin/die. Empirical limitations highlighted with examples like earthquake. History expanded with contributions. Each example broken down with calculations and verifications. Exercise questions types listed for practice.
Additional insights: Probability in daily life (weather, stocks); misconceptions (gambler's fallacy). Extended examples: Lottery odds, medical tests accuracy.
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