Probability – NCERT Class 11 Mathematics Chapter 14 – Events, Axiomatic Approach, and Probability Calculations

Introduces the concept of events and sample spaces, classifies events, and describes the algebra of events including complementary, union, intersection, and mutually exclusive events. Covers the axiomatic approach to probability, calculation rules, mutually exclusive and exhaustive events, probability of compound events, and historical development of probability theory with examples and exercises.

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Categories: NCERT, Class XI, Mathematics, Probability, Events, Axiomatic Probability, Chapter 14

Tags: Probability, Sample Space, Events, Complementary Events, Union of Events, Intersection of Events, Mutually Exclusive Events, Exhaustive Events, Axiomatic Probability, Probability Calculations, Compound Events, Historical Note, NCERT Class 11, Mathematics, Chapter 14

Probability: Class 11 NCERT Chapter 14 - Ultimate Study Guide, Notes, Questions, Quiz 2025

Full Chapter Summary & Detailed Notes - Probability Class 11 NCERT

Overview & Key Concepts

Chapter Goal: Introduce events and axiomatic probability. Focus: Sample space, event types, algebra, axioms. Exam Focus: Definitions, set operations, mutually exclusive/exhaustive. 2025 Updates: Coin/die examples emphasized. Fun Fact: Arbuthnot quote on reasoning. Core Idea: Subsets as events. Real-World: Coin tosses, dice rolls. Ties: Ch15 (conditional). Expanded: From classical to axiomatic.
Wider Scope: Events via sets; probability axioms (PDF covers 14.1 events, 14.2 axioms up to coin ex).
Expanded Content: Examples, exercises, steps.

14.1 Event

Random experiments, sample space S as universal set. Ex: Two coins S={HH,HT,TH,TT}. Event E subset of S, e.g., exactly one head E={HT,TH}. Occurrence: ω ∈ E.

14.1.1 Occurrence of an Event

Die ex: E="less than 4" occurs if 1,2,3. Not if 4,5,6.

14.1.2 Types of Events

Impossible: φ (multiple of 7 on die). Sure: S (odd or even). Simple: One point, e.g., {HH}. Compound: Multiple, e.g., exactly one head in three tosses.

14.1.3 Algebra of Events

Complement A' = S - A. A or B = A ∪ B. A and B = A ∩ B. A but not B = A - B. Ex1: Die prime A={2,3,5}, odd B={1,3,5}; A∪B={1,2,3,5}, etc.

14.1.4 Mutually Exclusive

A ∩ B = φ, e.g., odd/even on die. Not: Odd and <4 (3 common).

14.1.5 Exhaustive

Union = S, e.g., <4, >2<5, >4 on die. Mutually exclusive and exhaustive: Pairwise disjoint, cover S.

14.2 Axiomatic Approach

P: Power set to [0,1]. Axioms: P(E)≥0, P(S)=1, mutually exclusive add. Implies P(φ)=0. For finite S, 0≤P(ω_i)≤1, sum=1, P(A)=sum P(ω_i) in A. Ex: Coin P(H)=P(T)=1/2.

Summary

Events as subsets; algebra like sets. Axioms formalize probability. Master: Unions/intersections, types. Apps: Games of chance. Mantra: Subsets define happenings.

Why This Guide Stands Out

Tables from PDF, coin/die intuition, step-by-step sets, free 2025 with MathJax.

Key Themes & Tips

Aspects: Classical (sets) to axiomatic (rules).
Tip: Draw Venn for algebra; check disjoint for exclusive.

Exam Case Studies

Die events; two dice sums.

Project & Group Ideas

Simulate coin tosses in Python, list events.
Apps: Dice probability calculator.

Key Definitions & Terms - Complete Glossary

All terms from chapter; detailed with examples, relevance. Expanded: 15+ terms with depth.

Event

Subset E of S. Relevance: Happening. Ex: {HT,TH}. Depth: Any subset.

Sample Space S

All outcomes. Relevance: Universal. Ex: {HH,HT,TH,TT}. Depth: For experiments.

Occurrence

ω ∈ E. Relevance: Happened. Ex: 1 in "less than 4". Depth: ω ∉ E not occurred.

Impossible Event

φ. Relevance: Can't happen. Ex: Multiple of 7 on die. Depth: P=0.

Sure Event

S. Relevance: Always. Ex: Odd or even. Depth: P=1.

Simple Event

One point. Relevance: Elementary. Ex: {HH}. Depth: n in finite S.

Compound Event

Multiple points. Relevance: Complex. Ex: Exactly one head. Depth: >1 element.

Complementary Event

A' = S - A. Relevance: Not A. Ex: {HHH,HTT,...}. Depth: A ∪ A' = S.

A or B

A ∪ B. Relevance: Either/both. Ex: Prime or odd={1,2,3,5}. Depth: Inclusive.

A and B

A ∩ B. Relevance: Both. Ex: {3,5}. Depth: Common.

A but not B

A - B. Relevance: A only. Ex: {2}. Depth: A ∩ B'.

Mutually Exclusive

A ∩ B = φ. Relevance: Can't both. Ex: Odd/even. Depth: Disjoint.

Exhaustive

∪ E_i = S. Relevance: Cover all. Ex: <4,>2<5,>4. Depth: At least one occurs.

Axiomatic Probability

P function on power set. Relevance: Formal. Ex: Axioms. Depth: [0,1].

Elementary Event

{ω_i}. Relevance: Basic. Ex: P(H)=1/2. Depth: Singleton.

Power Set

All subsets. Relevance: Domain of P. Ex: For S 4 elements, 16 subsets. Depth: Events.

Disjoint Events

Pairwise ∩=φ. Relevance: ME&E. Ex: No heads, one head. Depth: For additivity.

Tip: Subset for event; check empty for exclusive. Depth: Axioms imply classical. Errors: Confuse union/intersection. Historical: Kolmogorov axioms. Interlinks: Sets Ch1. Advanced: Conditional. Real-Life: Games. Graphs: Venn diagrams. Coherent: Experiment → S → Events → P.

Additional: Simple events always ME. Pitfalls: Exhaustive not exclusive.

30 Questions & Answers - NCERT Based (Class 11) - From Exercises & Variations

Based on chapter examples/exercises. Part A: 10 (1 mark short), Part B: 10 (4 marks medium), Part C: 10 (8 marks long). Answers point-wise, set notation with MathJax.

Part A: 1 Mark Questions (10 Qs - Short from Illustrations & Variations)

1. What is an event?

1 Mark Answer:

Subset of S

2. Impossible event?

1 Mark Answer:

$$ \phi $$

3. Sure event?

1 Mark Answer:

4. Simple event?

1 Mark Answer:

One sample point

5. Mutually exclusive if?

1 Mark Answer:

$$ A \cap B = \phi $$

6. Complement notation?

1 Mark Answer:

$$ A' $$

7. Exhaustive means?

1 Mark Answer:

Union = S

8. Axiom 1?

1 Mark Answer:

$$ P(E) \geq 0 $$

9. P(S)=?

1 Mark Answer:

10. Coin P(H)?

1 Mark Answer:

$$ \frac{1}{2} $$