NCERT Solutions for Class 11 Maths Chapter 16 – Probability
Subject: Mathematics | Class: 11 | Chapter: 16 | Board: CBSE | Curriculum: New NCERT
What Are NCERT Solutions for Class 11 Maths Chapter 16 – Probability?
Every decision made under uncertainty involves probability. Whether a doctor interprets a test result, a meteorologist forecasts rain, or a financial analyst assesses investment risk — all of them use the language and logic of probability. Chapter 16 of Class 11 Mathematics lays the formal groundwork for this language.
You have seen basic probability in Classes 9 and 10 — the simple formula of favourable outcomes divided by total outcomes. Chapter 16 rebuilds that foundation more rigorously. It introduces sample spaces and events using set notation (connecting beautifully back to Chapter 1), distinguishes between different types of events, and develops the axiomatic approach to probability. The three axioms of probability — non-negativity, normalization, and additivity — are presented and used to derive all the familiar probability results from first principles. For all Chapters must, read NCERT Solutions for Class 11 Maths and subject-wise NCERT Solutions for Class 11.
The NCERT solutions for chapter is demanding not because the calculations are hard (they usually are not) but because the logical setup of each problem is critical. Identifying the correct sample space, listing the right event, and recognising whether events are mutually exclusive or exhaustive — these conceptual steps determine whether your answer is correct far more than arithmetic does. The solutions here give particular attention to writing out sample spaces completely, because incomplete sample spaces are the single most common source of wrong answers in this chapter.
Download PDF – All Exercises of NCERT Solutions for Class 11 Maths Chapter 16 Probability
| Exercise | Topic Covered | Number of Questions |
|---|---|---|
| Exercise 16.1 | Sample Space and Events for Various Experiments | 16 Questions |
| Exercise 16.2 | Types of Events; Algebra of Events | 7 Questions |
| Exercise 16.3 | Axiomatic Probability; Addition Theorem; Conditional Ideas | 21 Questions |
| Miscellaneous | Mixed Probability Problems | 10 Questions |
Chapter 16 – Probability: Concepts, Explanation and Key Tables
Sample Space and Events
A random experiment is one whose outcome cannot be predicted with certainty in advance but whose set of all possible outcomes is known.
The sample space S is the complete set of all possible outcomes of the experiment.
An event is any subset of the sample space.
| Experiment | Sample Space S |
|---|---|
| Tossing one coin | {H, T} |
| Tossing two coins | {HH, HT, TH, TT} |
| Tossing three coins | {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} — 8 outcomes |
| Rolling one die | {1, 2, 3, 4, 5, 6} |
| Rolling two dice | 36 ordered pairs: {(1,1), (1,2), ..., (6,6)} |
| Drawing one card from 52 | 52 individual cards |
| Drawing one card; noting suit only | {Hearts, Diamonds, Clubs, Spades} |
Types of Events
| Type of Event | Definition | Example |
|---|---|---|
| Impossible Event | Event that can never occur; P = 0 | Getting 7 on a single die toss |
| Sure Event | Event that always occurs; P = 1 | Getting a number ≤ 6 on a die |
| Simple Event | Event with exactly one outcome in the sample space | Getting Head in one coin toss |
| Compound Event | Event with more than one outcome | Getting an even number on a die: {2,4,6} |
| Complementary Event | Event A′ = all outcomes not in A | A = {1,2,3} → A′ = {4,5,6} |
| Mutually Exclusive Events | A ∩ B = ∅; cannot occur simultaneously | Getting Head AND Tail on one toss |
| Exhaustive Events | A₁ ∪ A₂ ∪ ... ∪ Aₙ = S; collectively cover all outcomes | {1,2,3} and {4,5,6} from a die roll |
| Equally Likely Events | Each outcome has the same probability | Any two faces on a fair die |
Probability Formulas
| Formula | Expression | Condition |
|---|---|---|
| Classical Probability | P(A) = n(A) / n(S) | All outcomes equally likely |
| Complement Rule | P(A′) = 1 – P(A) | Always valid |
| Addition Theorem (General) | P(A ∪ B) = P(A) + P(B) – P(A ∩ B) | For any two events |
| Addition Theorem (Mutually Exclusive) | P(A ∪ B) = P(A) + P(B) | Only when A ∩ B = ∅ |
| Three Events Addition | P(A∪B∪C) = P(A)+P(B)+P(C)–P(A∩B)–P(B∩C)–P(A∩C)+P(A∩B∩C) | Inclusion-Exclusion |
| Probability range | 0 ≤ P(A) ≤ 1 | Always |
| Sum of all simple event probabilities | Σ P(eᵢ) = 1 | Axiom |
Axiomatic Approach — Three Axioms of Probability
| Axiom | Statement | What It Means |
|---|---|---|
| Non-negativity | P(A) ≥ 0 for every event A | Probabilities cannot be negative |
| Normalization | P(S) = 1 | The sample space is certain to occur |
| Additivity | If A ∩ B = ∅, then P(A ∪ B) = P(A) + P(B) | Probability of union = sum if mutually exclusive |
All probability rules and theorems — including the addition theorem and complement rule — are derived from these three axioms alone.
Odds For and Against an Event
| Concept | Formula | Example |
|---|---|---|
| Odds in favour of A | n(A) : n(A′) | If P(A) = 3/8, odds in favour = 3:5 |
| Odds against A | n(A′) : n(A) | Odds against = 5:3 |
| P(A) from odds m:n in favour | P(A) = m/(m+n) | Odds 2:3 → P = 2/5 |
Common Probability Calculations from Playing Cards
| Event | Number of Favourable Outcomes | Probability |
|---|---|---|
| Drawing any card | 52 | 52/52 = 1 |
| Drawing a red card | 26 (13 Hearts + 13 Diamonds) | 26/52 = 1/2 |
| Drawing a face card | 12 (3 per suit × 4 suits) | 12/52 = 3/13 |
| Drawing an Ace | 4 | 4/52 = 1/13 |
| Drawing a Heart | 13 | 13/52 = 1/4 |
| Drawing the Ace of Spades | 1 | 1/52 |
| Drawing a King or Queen | 8 | 8/52 = 2/13 |
Algebra of Events (Set Notation Applied to Events)
| Set Operation | Probability Context | Meaning |
|---|---|---|
| A ∪ B | A or B | At least one of A, B occurs |
| A ∩ B | A and B | Both A and B occur |
| A′ | Not A | A does not occur |
| A – B = A ∩ B′ | A but not B | A occurs; B does not |
| A ∩ B = ∅ | A and B mutually exclusive | A and B cannot both occur |
| A ∪ B = S | A and B are exhaustive | Together they cover all outcomes |
Study Tips for Chapter 16
- Write the sample space fully for every problem — especially for two-dice experiments (36 outcomes) and two-coin/three-coin experiments. Incomplete sample spaces lead to wrong denominators.
- The addition theorem P(A∪B) = P(A) + P(B) – P(A∩B) is always valid. The simplified version P(A∪B) = P(A) + P(B) is only valid when you have confirmed A and B are mutually exclusive — state this explicitly.
- For card and die problems, always identify the type of deck (standard 52 cards) or die (standard 6-faced) at the start — and note whether draws are with or without replacement when multiple events are involved.
- The miscellaneous exercise often combines probability with counting (permutations/combinations from Chapter 7) — practise expressing favourable and total outcomes as ⁿCᵣ expressions.
FAQs on NCERT Solutions Class 11 Maths Chapter 16 Probability
Probability is a mathematical measure of the likelihood that a particular event will occur. It helps students analyze situations involving uncertainty and predict possible outcomes. In Chapter 16, students learn about random experiments, sample spaces, events, and probability calculations. Probability is widely used in statistics, economics, insurance, science, and decision-making processes. NCERT Solutions explain these concepts through detailed examples and solved exercises. Understanding probability helps students develop logical reasoning and analytical skills. It also forms the foundation for advanced probability and statistics topics that are studied in higher classes and competitive examinations.
A sample space is the complete set of all possible outcomes of a random experiment. For example, when a coin is tossed, the sample space consists of heads and tails. Understanding sample spaces is essential because probability calculations are based on identifying possible outcomes and favorable outcomes. In Chapter 16, students learn how to represent and analyze sample spaces for different experiments. NCERT Solutions provide detailed explanations and examples that make these concepts easy to understand. A strong understanding of sample spaces helps students solve probability questions accurately and build confidence in mathematical reasoning.
Probability is used extensively in everyday life to make predictions and informed decisions under uncertain conditions. It is applied in weather forecasting, medical research, sports analysis, finance, insurance, and risk management. Businesses and governments also use probability to plan and evaluate outcomes. Learning probability helps students understand how mathematical methods can be used to estimate chances and analyze uncertain events. NCERT Solutions explain practical applications along with textbook exercises, helping students connect theoretical concepts with real-world situations. This makes the chapter both interesting and highly relevant for future studies and careers.
NCERT Solutions provide accurate and detailed answers to all textbook questions, helping students understand every concept clearly. They explain sample spaces, events, probability formulas, and problem-solving techniques through step-by-step methods. Students can use these solutions for homework, revision, and examination preparation. Regular practice improves conceptual understanding, logical reasoning, and accuracy in calculations. Since probability questions are commonly asked in school exams and competitive tests, mastering NCERT Solutions can significantly improve performance. These solutions also help students develop a strong foundation for advanced topics in probability and statistics studied in higher classes.