Two independent events
We say that two \(A\) 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:
by the, so
and similarly for \(\bP(B)\; \bP(A \mid B)\).