\[A\]
and \[B\]
, independence holds every which that, that is \[A'\]
and \[B'\]
(not \[A\]
and not \[B\]
respectively, or \[A\]
and \[B'\]
etc).If
\[A, \: B\]
are independent, then \[(P(A \cap B)=P(A) \times P(B)\]
.But
\[P(A)=1-P(A'), \: P(B)=1-P(B')\]
and\[1=P(A \cap B)-P(A' \cap B')+P(A')+P(AB')\]
so\[P(A \cap B) =1+P(A' \cap B')-P(A')-P(B')\]
.Then
\[1+P(A' \cap B')-P(A')-P(B')=(1-P(A')) \times (1-P(B'))\]
\[1+P(A' \cap B')-P(A')-P(B')=1-P(A')-P(B')+P(A')P(B')\]
\[P(A' \cap B')=P(A')P(B')\]