Currently Empty: $0.00
2.Independent Events
Class 12 • Mathematics • NCERT
Independent Events
[ Embed Your Video Lecture Here ]
This lesson explains independent events in probability, where the occurrence of one event does not affect the occurrence of another, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand the concept of independent events.
- Learn the condition for independence.
- Differentiate independent and dependent events.
- Solve NCERT board-level questions.
1. Meaning of Independent Events
Two events A and B are said to be independent if the occurrence of one event does not affect the probability of occurrence of the other event.
Independence means no influence.
[ Image Placeholder: Two Unrelated Events ]
2. Definition (NCERT)
Two events A and B are independent if:
P(A ∩ B) = P(A) × P(B)
Equivalently, P(A | B) = P(A) and P(B | A) = P(B).
[ Image Placeholder: Independent Events Representation ]
3. Independent vs Dependent Events
It is important to distinguish between independent and dependent events.
• Independent: P(A | B) = P(A)
• Dependent: P(A | B) ≠ P(A)
• Dependent: P(A | B) ≠ P(A)
[ Image Placeholder: Comparison Diagram ]
4. Important Properties (NCERT)
Independent events satisfy the following properties:
• P(A ∩ B) = P(A)P(B)
• P(A’ ∩ B) = P(A’)P(B)
• P(A ∩ B’) = P(A)P(B’)
• P(A’ ∩ B’) = P(A’)P(B’)
• P(A’ ∩ B) = P(A’)P(B)
• P(A ∩ B’) = P(A)P(B’)
• P(A’ ∩ B’) = P(A’)P(B’)
[ Image Placeholder: Complementary Events ]
5. NCERT Example
A coin is tossed and a die is thrown.
Let A = getting a head, B = getting a 3.
P(A) = 1/2, P(B) = 1/6
P(A ∩ B) = 1/2 × 1/6 = 1/12
Hence, A and B are independent.
[ Image Placeholder: Coin and Die Experiment ]
6. Important NCERT Notes
• Independence is a strong condition
• Mutually exclusive events are NOT independent (except trivial cases)
• Used in multiplication theorem
• Frequently tested in boards
• Mutually exclusive events are NOT independent (except trivial cases)
• Used in multiplication theorem
• Frequently tested in boards
Practice Questions (NCERT)
- Define independent events.
- State the condition for independence of two events.
- Write the relation between independent events and conditional probability.
- Are mutually exclusive events independent?
- Find P(A ∩ B) if P(A)=0.4 and P(B)=0.5.
- If P(A | B)=P(A), what can you conclude?
- Are complements of independent events also independent?
- Is this topic part of NCERT Class 12 syllabus?
- Is this concept important for board exams?
- Is independence used in Bayes’ theorem?
✅ Show Answer Key
- Events that do not affect each other
- P(A ∩ B)=P(A)P(B)
- P(A | B)=P(A)
- No
- 0.2
- A and B are independent
- Yes
- Yes
- Yes
- Yes
© Aviate Learning – Independent Events (NCERT Class 12)