Class 12 Maths Probability Notes

13.1 Introduction

Probability measures the likelihood of occurrence of an event.

• Values lie between 0 and 1
  • 0 → Impossible event
  • 1 → Certain event
• Widely used in statistics, finance, genetics, and risk analysis

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13.2 Conditional Probability

The probability of an event A given that event B has occurred is called conditional probability:$$P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) \neq 0$$P(A∣B)=P(B)P(A∩B)​,P(B)=0

• Helps in updating probability when new information is known
• Used in real-life scenarios, such as medical tests or weather predictions

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13.3 Multiplication Theorem on Probability

For any two events A and B:$$P(A \cap B) = P(A|B) \cdot P(B) = P(B|A) \cdot P(A)$$P(A∩B)=P(A∣B)⋅P(B)=P(B∣A)⋅P(A)

• Can be extended to three or more events

$$P(A \cap B \cap C) = P(A) \cdot P(B|A) \cdot P(C|A \cap B)$$P(A∩B∩C)=P(A)⋅P(B∣A)⋅P(C∣A∩B)

• Useful for joint probabilities of multiple events

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13.4 Independent Events

Two events A and B are independent if occurrence of one does not affect the probability of the other:$$P(A|B) = P(A) \quad \text{or equivalently} \quad P(A \cap B) = P(A) \cdot P(B)$$P(A∣B)=P(A)or equivalentlyP(A∩B)=P(A)⋅P(B)

• Example: Tossing a coin and rolling a die are independent events
• Independence simplifies calculations of complex probabilities

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13.5 Bayes’ Theorem

Bayes’ theorem allows us to reverse conditional probabilities:$$P(A_i | B) = \frac{P(A_i) P(B|A_i)}{\sum_{j} P(A_j) P(B|A_j)}$$P(Ai​∣B)=∑j​P(Aj​)P(B∣Aj​)P(Ai​)P(B∣Ai​)​

• Useful when we know effects (B) and want to find causes (A)
• Widely applied in diagnosis, quality control, and decision-making

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✅ Key Points to Remember

• Probability measures chance of an event (0 ≤ P ≤ 1)
• Conditional probability updates likelihood given new information
• Multiplication theorem helps find joint probabilities
• Independent events have no influence on each other
• Bayes’ theorem is used to find reverse probabilities