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Independence Every Which Way

For independent events  
\[A\]
  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
\[1+P(A' \cap B')-P(A')-P(B')=(1-P(A')) \times (1-P(B'))\]

\[1+P(A' \cap B')-P(A')-P(B')=1-P(A')-P(B')+P(A')P(B')\]

\[P(A' \cap B')=P(A')P(B')\]

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