Two independent events

$$ \newcommand{\bP}{\mathbb{P}} $$

We say that two events, \(A\) and \(B\), are in­de­pen­dent when learn­ing that \(A\) has oc­curred does not change your prob­a­bil­ity that \(B\) oc­curs. That is, \(\bP(B \mid A) = \bP(B)\). Equiv­a­lently, \(A\) and \(B\) are in­de­pen­dent if \(\bP(A)\) doesn’t change if you con­di­tion on \(B\): \(\bP(A \mid B) = \bP(A)\).

Another way to state in­de­pen­dence is that \(\bP(A,B) = \bP(A) \bP(B)\).

All these defi­ni­tions are equiv­a­lent:

$$\bP(A,B) = \bP(A)\; \bP(B \mid A)$$

by the chain rule, so

$$\bP(A,B) = \bP(A)\; \bP(B)\;\; \Leftrightarrow \;\; \bP(A)\; \bP(B \mid A) = \bP(A)\; \bP(B) \ ,$$

and similarly for \(\bP(B)\; \bP(A \mid B)\).

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  • Probability theory

    The logic of sci­ence; co­her­ence re­la­tions on quan­ti­ta­tive de­grees of be­lief.