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CBSE • Class XI • Mathematics • Ch 14
Estimated Time: 45 Mins
Study Progress: In Progress

Probability

In Class 11 Mathematics, "Probability" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

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🎲 Have You Ever Wondered?

How do casinos calculate house edge down to four decimal places, or how do AI speech recognition models predict which word you are going to say next? ...

How do casinos calculate house edge down to four decimal places, or how do AI speech recognition models predict which word you are going to say next? Axiomatic probability bridges empirical outcomes with set theory.

Why This Chapter Matters

In Class 11 Mathematics, "Probability" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Before You Begin (Prerequisites)

  • Theoretical probability from Class 10.
  • Set operations (Union, Intersection, Complement).
  • Permutations and combinations.

What You Will Learn (Core Objectives)

  • Define Random Experiment, Sample Space ($S$), and Sample Points.
  • Classify Events: Impossible, Sure, Simple (Elementary), Compound, Mutually Exclusive, and Exhaustive events.
  • State the three Kolmogorov Axioms of Probability.
  • Calculate probabilities using set algebra: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
  • Apply complementary probability: $P(A') = 1 - P(A)$ and mutually exclusive additions.

Chapter Roadmap & Progression

1 1. Sample Space & Types of Events
2 2. Axiomatic Definition of Probabil...
3 3. Probability Addition Theorems

Complete Concept Guide (100% Curriculum Coverage)

1. Sample Space & Types of Events

The set of all possible outcomes of a random experiment is the Sample Space ($S$):
• Tossing 2 coins: $S = \{HH, HT, TH, TT\}$.
• Mutually Exclusive Events: Two events $A$ and $B$ that cannot occur simultaneously: $\mathbf{A \cap B = \emptyset \implies P(A \cap B) = 0}$.
• Exhaustive Events: Events whose union covers the entire sample space: $\mathbf{A \cup B = S}$.

2. Axiomatic Definition of Probability

For any event $E$ of sample space $S = \{\omega_1, \omega_2, \dots, \omega_n\}$:
(1) $0 \le P(\omega_i) \le 1$ for all outcomes.
(2) $\sum_{i=1}^{n} P(\omega_i) = 1$.
(3) For any event $E$, $P(E) = \sum_{\omega \in E} P(\omega)$.

3. Probability Addition Theorems

For any two events $A$ and $B$: $$\mathbf{P(A \cup B) = P(A) + P(B) - P(A \cap B)}$$ If $A$ and $B$ are mutually exclusive: $$\mathbf{P(A \cup B) = P(A) + P(B)}$$ Complement: $\mathbf{P(A') = 1 - P(A)}$.

Visual Learning & Conceptual Map

Probability Master Matrix

Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture

1. Sample Space & Types of Events • 2. Axiomatic Definition of Probability

Chapter Summary & 10 Key Takeaways

Takeaway 1
Sample Space ($S$): Universal set containing all possible random experiment outcomes.
Takeaway 2
Mutually Exclusive: Disjoint events unable to occur concurrently ($A \cap B = \emptyset$).
Takeaway 3
Exhaustive Events: Complete outcome set whose cumulative union forms $S$.
Takeaway 4
Kolmogorov Axioms: Foundation establishing non-negativity and unit total probability.
Takeaway 5
Addition Theorem: Set-theoretic formula adjusting for double-counted intersection outcomes.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
A coin is tossed three times. Write the sample space and find the probability of getting at least two heads.
Reveal Answer & Explanation
Answer: $S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$ ($n(S) = 8$). Event of at least 2 heads $= \{HHH, HHT, HTH, THH\}$ (4 outcomes). Probability $= \frac{4}{8} = \frac{1}{2}$.
Sample space has 8 outcomes; P = 1/2.
2
If $P(A) = \frac{3}{5}$ and $P(B) = \frac{1}{5}$, find $P(A \text{ or } B)$ if $A$ and $B$ are mutually exclusive events.
Reveal Answer & Explanation
Answer: For mutually exclusive events, $P(A \cap B) = 0$. $P(A \cup B) = P(A) + P(B) = \frac{3}{5} + \frac{1}{5} = \frac{4}{5}$.
4/5.
3
If $P(A) = 0.5$, $P(B) = 0.3$, and $P(A \cap B) = 0.2$, find: (i) $P(A \cup B)$, (ii) $P(A')$.
Reveal Answer & Explanation
Answer: (i) $P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.5 + 0.3 - 0.2 = 0.6$. (ii) $P(A') = 1 - P(A) = 1 - 0.5 = 0.5$.
(i) 0.6; (ii) 0.5.
4
Two dice are thrown simultaneously. What is the probability of getting a sum greater than 9?
Reveal Answer & Explanation
Answer: Total outcomes $= 36$. Favorable outcomes with sum $> 9$ (sums 10, 11, 12): $(4,6), (5,5), (6,4), (5,6), (6,5), (6,6)$ (6 pairs). $P = \frac{6}{36} = \frac{1}{6}$.
1/6.
5
Explain why $P(A \cup B) = P(A) + P(B)$ is true only for mutually exclusive events.
Reveal Answer & Explanation
Answer: Because for mutually exclusive events, the intersection $A \cap B = \emptyset$, so $P(A \cap B) = 0$. When events overlap, the intersection probability must be subtracted to prevent double-counting shared elements.
Shared intersection is empty; otherwise double-counting occurs.
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