Probability Class 11 Math Quick Recall / Short Notes (2026-27)

Probability Concepts and Sample Spaces Diagram

[!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: $0 \leq P(E) \leq 1$ and $P(S) = 1$ .

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, and the Addition Rule of sets to excel in 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).
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 $A$ and $B$ 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 $A$ and $B$ 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' and '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 $P(A) = 0.5$ and $P(B) = 0.3$ , what is $P(A \cup B)$ if $A$ and $B$ 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 and 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 in a game or analyzing scientific data, these axioms remain your best guide!