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Probability

Class 11 Maths • Chapter 14 • Comprehensive Interactive Notes

1. Random Experiments & Sample Space

An experiment is called random if it has more than one possible outcome and the outcome cannot be predicted in advance. The set of all possible outcomes is called the Sample Space (S).

Sample Space Generator

Select an experiment to see its outcomes.

Solve NCERT Ex 14.1 Solve Exemplar Ex 14.1

2. Events

Any subset of a sample space is called an event.

Type Description
Impossible Event Empty set \( \phi \). (e.g., Rolling 7 on a die).
Sure Event Whole sample space S.
Mutually Exclusive Events A and B cannot happen together. \( A \cap B = \phi \).
Exhaustive Events Their union forms the sample space. \( A \cup B = S \).
Notation Meaning
\( A \cup B \) A or B
\( A \cap B \) A and B
\( A' \) (or \( \bar{A} \)) Not A
\( A - B \) A but not B

📌 Important Types of Events (CBSE)

  • Equally Likely Events: All outcomes have equal chance. (e.g., tossing a fair coin)
  • Independent Events: Occurrence of one does not affect the other. (e.g., tossing two coins)
  • Dependent Events: Occurrence of one affects the other. (e.g., drawing cards without replacement)

CBSE Tip: Independent ≠ Mutually Exclusive (students often confuse these).

Solve NCERT Ex 14.2 Solve Exemplar Ex 14.2

3. Axiomatic Probability

For finite sample space, \( P(E) = \frac{\text{Number of outcomes in E}}{\text{Total outcomes in S}} \).

Experimental vs Theoretical

Flip a coin many times. Does Heads % approach 50%?

Heads: 0 Tails: 0 Total: 0
Exp. P(H) = 0%

🧮 How to Solve Probability Questions

  1. Write the sample space S
  2. Count total outcomes n(S)
  3. Define event E clearly
  4. Count favourable outcomes n(E)
  5. Apply formula: \( P(E) = \frac{n(E)}{n(S)} \)

CBSE Tip: Writing sample space gives full method marks.

4. Addition Theorems

\( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
If \( A \cap B = \phi \), then \( P(A \cup B) = P(A) + P(B) \).
\( P(A') = 1 - P(A) \)

Conditional Probability \( P(A|B) \): Probability of event A given that event B has already occurred.

\( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where \( P(B) \neq 0 \).

Multiplication Rule:

\( P(A \cap B) = P(B) \cdot P(A|B) \)

If A and B are independent events, then \( P(A \cap B) = P(A) \cdot P(B) \).

Probability Law Solver

Calculate \( P(A \cup B) \).

Solve Exemplar Ex 14.3

🧠 One-Page Probability Revision

  • ✔ Meaning of random experiment
  • ✔ Sample space & notation S
  • ✔ Types of events revised
  • ✔ Algebra of events clear
  • ✔ Complement rule \( P(A') = 1 - P(A) \)
  • ✔ Addition theorem formula
  • ✔ Difference between mutually exclusive & independent

Self-Test:

  • ❓ What is equally likely?
  • ❓ Can two events be both independent & mutually exclusive?
  • ❓ Why probability is always between 0 and 1?

Concept Mastery Quiz

1. Probability of an impossible event is:


2. If P(A) = 0.3, then P(not A) is:


3. Two events are mutually exclusive if:


4. Total outcomes in tossing 3 coins is:


5. Max value of probability is:

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