Last Updated: June 1, 2026

📋 Table of Contents
What is Probability Revision Notes?
Introduction
1. Random Experiments and Sample Space
2. Events and Their Types
3. Relationships Between Events
4. Axiomatic Approach to Probability
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Probability Revision Notes?
Introduction
1. Random Experiments and Sample Space
- Sample Space (S):
2. Events and Their Types
- Types of Events:
3. Relationships Between Events
4. Axiomatic Approach to Probability
- Fundamental Formulas:
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Probability Class 11 Mathematics Revision — JEE 2026 Grandmaster Guide

What is Probability Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Sample Space (S): The set of all possible outcomes of a random experiment.

Event (E): A subset of the sample space.

Mutually Exclusive: $A \cap B = \phi$ (cannot happen together).

Exhaustive Events: $A \cup B = S$ (at least one must happen).

Axiomatic Probability: $0 \leq P(E) \leq 1$ n $P(S) = 1$ .

Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ . 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

Probability is the mathematical measurement of uncertainty, providing the framework for analyzing random experiments and events. Master the Axiomatic Approach, Sample Spaces, n the Addition Rule of sets to excel and advanced statistical modeling and Bayes' Theorem. This class 11 Math Chapter 16 guide ensures you have all the essential foundations for JEE and CBSE exams. Probability is the mathematical way of measuring uncertainty.

1. Random Experiments and Sample Space

A Random Experiment is one where the outome cannot be predicted with certainty, even if the possible outcomes are known.

Sample Space (S):

The set of all possible outcomes.

Tossing a coin: $S = \{H, T\}$ .
Rolling a die: $S = \{1, 2, 3, 4, 5, 6\}$ .
Tossing two coins: $S = \{HH, HT, TH, TT\}$ .

Sample Point: Each element of the sample space is called a sample point.

2. Events and Their Types

An Event is simply a subset of the sample space.

Types of Events:

Impossible Event: The empty set $\phi$ . (e.g., getting a $7 on a standard die).$ 2. Sure Event: The entire sample space $S$ .
Simple Event: An event containing only one sample point.
Compound Event: An event containing more than one sample point.
Complementary Event ( $E'$ ): The event "not E", calculated as $S - E$ .

3. Relationships Between Events

This is where set theory from Chapter 1 meets Probability.

Mutually Exclusive Events: Events $A$ n $B$ are mutually exclusive if they cannot occur at the same time. Mathematically, $A \cap B = \phi$ .
Exhaustive Events: Events $E_1, E_2, \dots, E_n$ are exhaustive if their union equals the sample space. Mathematically, $E_1 \cup E_2 \cup \dots \cup E_n = S$ .
Mutually Exclusive and Exhaustive: If both conditions are met, the probabilities of these events $\sum to exactly 1.$

4. Axiomatic Approach to Probability

Instead of just counting outcomes, we assign a number $P(E)$ to an event $E$ that satisfies:

$P(E) \geq 0$ (Probabilities are never negative).
$P(S) = 1$ (The sure event has 100% probability).
If $A$ n $B$ are mutually exclusive, $P(A \cup B) = P(A) + P(B)$ .

Fundamental Formulas:

** $P(A \text{$ or } B) = P(A \cup B) = P(A) + P(B) - P(A \cap B) $**$
$P(\text{not } A) = P(A') = 1 - P(A)$
$P(A - B) = P(A) - P(A \cap B)$

Comprehensive Exam Strategy (Q&A)

Q1: Two dice are thrown. What is the probability that the $\sum is exactly 7$ ? Answer:

Total outcomes ( $n(S)$ ) = $6 \times 6 = 36$ .
Event $E$ ( $\sum$ is 7) = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}.
$n(E) = 6$ .
$P(E) = 6/36 = \mathbf{1/6}$ .

Q2: Are 'getting an odd number' n 'getting a number > 3' mutually exclusive on a single die roll? Answer:

$A$ (odd) = {1, 3, 5}.
$B$ (>3) = {4, 5, 6}.
$A \cap B = \{5\}$ .
Since the intersection is not empty, they are NOT mutually exclusive.

Q3: If $P(A) = 0.5$ n $P(B) = 0.3$ , what is $P(A \cup B)$ if $A$ n $B$ are mutually exclusive? Answer:

For mutually exclusive events, $P(A \cap B) = 0$ .
$P(A \cup B) = P(A) + P(B) = 0.5 + 0.3 = \mathbf{0.8}$ .

Related Revision Notes

Chapter 15: Statistics
Chapter 7: permutations n Combinations
[External Reference: NCERT Class 11 Math Chapter 16 (Authoritative Source)]

Conclusion

Probability teaches us to look at the world through the lens of logic rather than luck. By mastering the relationships between events and the addition rule, you lay the foundation for advanced statistical modeling and decision-making. Whether you're calculating the odds and a game or analyzing scientific data, these axioms remain your best guide!

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

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Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

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Continue your revision with these related guides:

🪤 The 5 Mistakes That Cost Marks

Independent Events Confusion: Many students get confused between independent and dependent events. For independent events, the occurrence or non-occurrence of one does not affect the probability of the occurrence of the other, whereas in dependent events, the probability of one event is affected by the occurrence of the other.
Neglecting the Concept of Equally Likely Outcomes: A common mistake is not considering the concept of equally likely outcomes when calculating probabilities. If all outcomes are not equally likely, the probability of an event cannot be determined by simply dividing the number of favorable outcomes by the total number of outcomes.
Incorrect Usage of Conditional Probability Formula: Students often incorrectly apply the conditional probability formula P(A|B) = P(A ∩ B) / P(B), forgetting that it's only applicable when P(B) is not equal to 0.
Not Considering the Sample Space: A significant mistake is not carefully defining the sample space for the experiment, leading to incorrect calculations of probabilities. The sample space should include all possible outcomes of the experiment.
Misinterpretation of the Concept of Mutual Exclusivity: Some students mistakenly assume that if two events are mutually exclusive, they cannot occur at the same time, which is correct, but they also incorrectly conclude that if two events cannot occur at the same time, they must be mutually exclusive, ignoring the possibility of dependent events.

🔁 Last 5 Minutes Box

Probability Formulas:

* P(E) = (Number of times event occurs) / (Total number of trials)
* P(E) = 1 - P(E') 
* P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
* P(A ∩ B) = P(A) * P(B) for independent events
* Bayes' Theorem: P(A|B) = P(B|A) * P(A) / P(B)

Last Updated: June 1, 2026

📋 Table of Contents
What is Probability Revision Notes?
Introduction
1. Random Experiments and Sample Space
2. Events and Their Types
3. Relationships Between Events
4. Axiomatic Approach to Probability
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Probability Revision Notes?
Introduction
1. Random Experiments and Sample Space
- Sample Space (S):
2. Events and Their Types
- Types of Events:
3. Relationships Between Events
4. Axiomatic Approach to Probability
- Fundamental Formulas:
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Probability Class 11 Mathematics Revision — JEE 2026 Grandmaster Guide

What is Probability Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Sample Space (S): The set of all possible outcomes of a random experiment.

Event (E): A subset of the sample space.

Mutually Exclusive: $A \cap B = \phi$ (cannot happen together).

Exhaustive Events: $A \cup B = S$ (at least one must happen).

Axiomatic Probability: $0 \leq P(E) \leq 1$ n $P(S) = 1$ .

Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ . 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

1. Random Experiments and Sample Space

A Random Experiment is one where the outome cannot be predicted with certainty, even if the possible outcomes are known.

Sample Space (S):

The set of all possible outcomes.

Tossing a coin: $S = \{H, T\}$ .
Rolling a die: $S = \{1, 2, 3, 4, 5, 6\}$ .
Tossing two coins: $S = \{HH, HT, TH, TT\}$ .

Sample Point: Each element of the sample space is called a sample point.

2. Events and Their Types

An Event is simply a subset of the sample space.

Types of Events:

Impossible Event: The empty set $\phi$ . (e.g., getting a $7 on a standard die).$ 2. Sure Event: The entire sample space $S$ .
Simple Event: An event containing only one sample point.
Compound Event: An event containing more than one sample point.
Complementary Event ( $E'$ ): The event "not E", calculated as $S - E$ .

3. Relationships Between Events

This is where set theory from Chapter 1 meets Probability.

Mutually Exclusive Events: Events $A$ n $B$ are mutually exclusive if they cannot occur at the same time. Mathematically, $A \cap B = \phi$ .
Exhaustive Events: Events $E_1, E_2, \dots, E_n$ are exhaustive if their union equals the sample space. Mathematically, $E_1 \cup E_2 \cup \dots \cup E_n = S$ .
Mutually Exclusive and Exhaustive: If both conditions are met, the probabilities of these events $\sum to exactly 1.$

4. Axiomatic Approach to Probability

Instead of just counting outcomes, we assign a number $P(E)$ to an event $E$ that satisfies:

$P(E) \geq 0$ (Probabilities are never negative).
$P(S) = 1$ (The sure event has 100% probability).
If $A$ n $B$ are mutually exclusive, $P(A \cup B) = P(A) + P(B)$ .

Fundamental Formulas:

** $P(A \text{$ or } B) = P(A \cup B) = P(A) + P(B) - P(A \cap B) $**$
$P(\text{not } A) = P(A') = 1 - P(A)$
$P(A - B) = P(A) - P(A \cap B)$

Comprehensive Exam Strategy (Q&A)

Q1: Two dice are thrown. What is the probability that the $\sum is exactly 7$ ? Answer:

Total outcomes ( $n(S)$ ) = $6 \times 6 = 36$ .
Event $E$ ( $\sum$ is 7) = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}.
$n(E) = 6$ .
$P(E) = 6/36 = \mathbf{1/6}$ .

Q2: Are 'getting an odd number' n 'getting a number > 3' mutually exclusive on a single die roll? Answer:

$A$ (odd) = {1, 3, 5}.
$B$ (>3) = {4, 5, 6}.
$A \cap B = \{5\}$ .
Since the intersection is not empty, they are NOT mutually exclusive.

Q3: If $P(A) = 0.5$ n $P(B) = 0.3$ , what is $P(A \cup B)$ if $A$ n $B$ are mutually exclusive? Answer:

For mutually exclusive events, $P(A \cap B) = 0$ .
$P(A \cup B) = P(A) + P(B) = 0.5 + 0.3 = \mathbf{0.8}$ .

Related Revision Notes

Chapter 15: Statistics
Chapter 7: permutations n Combinations
[External Reference: NCERT Class 11 Math Chapter 16 (Authoritative Source)]

Conclusion

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

📚 Related Topics

Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

📚 Related Topics

Continue your revision with these related guides:

🪤 The 5 Mistakes That Cost Marks

Independent Events Confusion: Many students get confused between independent and dependent events. For independent events, the occurrence or non-occurrence of one does not affect the probability of the occurrence of the other, whereas in dependent events, the probability of one event is affected by the occurrence of the other.
Neglecting the Concept of Equally Likely Outcomes: A common mistake is not considering the concept of equally likely outcomes when calculating probabilities. If all outcomes are not equally likely, the probability of an event cannot be determined by simply dividing the number of favorable outcomes by the total number of outcomes.
Incorrect Usage of Conditional Probability Formula: Students often incorrectly apply the conditional probability formula P(A|B) = P(A ∩ B) / P(B), forgetting that it's only applicable when P(B) is not equal to 0.
Not Considering the Sample Space: A significant mistake is not carefully defining the sample space for the experiment, leading to incorrect calculations of probabilities. The sample space should include all possible outcomes of the experiment.
Misinterpretation of the Concept of Mutual Exclusivity: Some students mistakenly assume that if two events are mutually exclusive, they cannot occur at the same time, which is correct, but they also incorrectly conclude that if two events cannot occur at the same time, they must be mutually exclusive, ignoring the possibility of dependent events.

🔁 Last 5 Minutes Box

Probability Formulas:

* P(E) = (Number of times event occurs) / (Total number of trials)
* P(E) = 1 - P(E') 
* P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
* P(A ∩ B) = P(A) * P(B) for independent events
* Bayes' Theorem: P(A|B) = P(B|A) * P(A) / P(B)