Independence (Probability)

Updated 01 Oct 2026

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Definition
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  two events $A, B$ are independent if knowing one happened tells you nothing about the other: $\Pr[A \cap B] = \Pr[A] \cdot \Pr[B]$
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  two random variables $X, Y$ are independent if $\Pr[X{=}a \text{ and } Y{=}b] = \Pr[X{=}a] \cdot \Pr[Y{=}b]$ for all $a, b$
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Example
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  sampling a uniform 2-bit string $R$ is equivalent to flipping two independent coins; the least-significant bit and most-significant bit of $R$ are independent random variables
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Operational meaning
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  for positive-probability conditioning events, independence gives $P(A\mid B)=P(A)$ .
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  multiply unconditional probabilities only when independence is justified.
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Sampling
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  independent, identivally performed draws with replacement preserve the category probabilities
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  without replacement, probabilities generally change after each draw
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Source clarification
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  Combining a $0.4$ chance of one win with a $0.7$ chance of another person’s win does not determine a joint probability by itself.
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  For example, $0.4(1-0.7)=0.12$ for “first wins, second loses” requires independence.
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Related
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  Random Variable
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  XOR Uniformity Lemma
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Linked references 5