4.6 Independent Events and Unions of Events

Part of Probability, Random Variables, and Probability Distributions in AP® Statistics. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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Independent Events

Independent events

Mathematical definition:

A \text{ and } B \text{ are independent} \iff P(A \mid B) = P(A) \iff P(B \mid A) = P(B)

Equivalently: ${P(A \cap B) = P(A) \cdot P(B)}$

Multiplication rule for independent events:

\boxed{P(A \cap B) = P(A) \cdot P(B)} \quad \text{(if independent)}

How to verify independence: Check if ${P(A \mid B) = P(A)}$ , or equivalently if ${P(A \cap B) = P(A) \cdot P(B)}$ .

Examples:

Flipping two fair coins: getting heads on flip 1 and flip 2 are independent
Drawing cards WITH replacement: each draw is independent
Drawing cards WITHOUT replacement: draws are NOT independent (dependent)

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