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 \(\bP(B)\; \bP(A \mid B)\).



  • Probability theory

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