Chapter 14: Probability | SJMaths Class 11

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.

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).

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

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

CBSE Tip: Writing sample space gives full method marks.

4. Addition Theorems

General Addition Rule

\( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

Mutually Exclusive Events

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

Probability of 'Not A'

\( P(A') = 1 - P(A) \)

Probability Law Solver

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

🧠 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: