Proposition: Characterization of Independent Events

Let \(B\) be an event with the probability \(0 < p(B) < 1\). Another event \(A\) is independent from \(B\), if any only if the probability of \(A\) equals the conditional probability of \(A\) given \(B\):

\[p(A)=p(A|B).\]

Loosely speaking, the frequency of occurrence of \(A\) does not change, if \(B\) also happened.