🔹 1. Introduction

Probability measures the likelihood of an event occurring.

👉 Value lies between:

• 0 (impossible event)
• 1 (certain event)

This chapter builds the foundation for advanced probability in Class 12 and is important for CBSE + JEE exams.

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🔹 2. Random Experiment

An experiment whose outcome cannot be predicted with certainty.

✔️ Examples:

• Tossing a coin
• Rolling a dice
• Drawing a card

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🔹 3. Sample Space

The set of all possible outcomes.

Notation: S

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✔️ Example

Coin toss:

S = {H, T}

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🔹 4. Event

A subset of sample space.

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✔️ Types of Events

1. Simple Event

Contains one outcome

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2. Compound Event

Contains more than one outcome

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3. Sure Event

Occurs always → P = 1

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4. Impossible Event

Never occurs → P = 0

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🔹 5. Classical Probability

✔️ Definition

$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$P(E)=Total number of outcomesNumber of favourable outcomes​

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✔️ Conditions

• Outcomes must be equally likely
• Total outcomes ≠ 0

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🔹 6. Important Results

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✔️ Range of Probability

$0 \le P(E) \le 1$0≤P(E)≤1

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✔️ Complementary Event

If E is an event:

$P(E’) = 1 – P(E)$P(E′)=1−P(E)

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🔹 7. Algebra of Events

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✔️ Union of Events

$P(A \cup B) = P(A) + P(B) – P(A \cap B)$P(A∪B)=P(A)+P(B)−P(A∩B)

$P(A)$P(A)

$P(B)$P(B)

$P(A\cap B)$P(A∩B)

$P(A\cup B)=P(A)+P(B)-P(A\cap B)\approx 0.80$P(A∪B)=P(A)+P(B)−P(A∩B)≈0.80AB0.20

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✔️ Mutually Exclusive Events

If A and B cannot occur together:

$P(A \cap B) = 0$P(A∩B)=0

$P(A)$P(A)

$P(B\mid A)$P(B∣A)

$P(A\cap B)=P(A)\cdot P(B\mid A)\approx 0.21$P(A∩B)=P(A)⋅P(B∣A)≈0.21P(A) = 0.60P(B|A) = 0.350.21

Then:

$P(A \cup B) = P(A) + P(B)$P(A∪B)=P(A)+P(B)

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🔹 8. Equally Likely Events

All outcomes have equal probability.

Example:
Fair dice → each outcome = 1/6

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🔹 9. Exhaustive Events

Set of all possible outcomes.

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🔹 10. Favourable Outcomes

Outcomes that satisfy the given condition.

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🔹 11. Important Examples

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✔️ Coin Toss

• P(Head) = 1/2
• P(Tail) = 1/2

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✔️ Dice

• Total outcomes = 6
• P(any number) = 1/6

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✔️ Cards

• Total cards = 52
• Hearts = 13

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🔹 12. Types of Events

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✔️ Mutually Exclusive

Cannot occur together

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✔️ Independent Events

Occurrence of one does not affect the other

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✔️ Dependent Events

One event affects another

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🔹 13. Conditional Probability (Basic Idea)

Probability of event A given B:

$P(A|B) = \frac{P(A \cap B)}{P(B)}$P(A∣B)=P(B)P(A∩B)​

$P(B)$P(B)

$P(A\cap B)$P(A∩B)

$P(A\mid B)=\frac{P(A\cap B)}{P(B)}\approx 0.46$P(A∣B)=P(B)P(A∩B)​≈0.46P(B)=0.65P(A∩B)=0.30P(A|B) ≈ 0.46A∩B is the part of B where A also happens

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🔹 14. Important Applications

✔️ Games of chance
✔️ Statistics
✔️ Decision making
✔️ Risk analysis

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🔹 15. JEE & CBSE Important Points

✔️ Understand sample space clearly
✔️ Use correct counting method
✔️ Complement method saves time
✔️ Practice card & dice problems
✔️ Conditional probability basics important

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🔹 16. Common Mistakes

❌ Wrong sample space
❌ Counting errors
❌ Ignoring mutually exclusive condition
❌ Misusing formulas

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🔹 17. Practice Questions

1. Find probability of getting a head
2. Probability of even number on dice
3. Draw a card → find probability
4. Solve union of events
5. Use complementary probability

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🔹 18. Quick Revision Sheet

• P(E) = favourable / total
• 0 ≤ P ≤ 1
• P(E’) = 1 − P(E)
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• Independent events → no effect

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🔹 19. Conclusion

Probability is a scoring and concept-based chapter. Strong understanding of:

• Sample space
• Event types
• Basic formulas

👉 ensures high marks in CBSE & JEE exams.