Independence (Probability)

Updated 12 Aug 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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Related
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  Random Variable
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  XOR Uniformity Lemma
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Linked references 3