SOLUTION: If A and B are two independent events, then show that A' and B' are also independent, where A' amd B' are complementary events.

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Question 935080: If A and B are two independent events, then show that A' and B' are also independent, where A' amd B' are complementary events.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Given: P(A⋂B) = P(A)P(B)
To prove: P(A'⋂B') = P(A')P(B')

     First we use DeMorgan's theorem: A'⋂B' = (A⋃B)'

P(A'⋂B') = P[(A⋃B)'] = 1-P(A⋃B) = 1-[P(A)+P(B)-P(A⋂B)] = 

= 1-P(A)-P(B)+P(A⋂B) = 1-P(A)-P(B)+P(A)P(B) = 

     Group the first two terms and factor -P(B) out 
     of the last two terms

= [1-P(A)] - P(B)[1-P(A)] =

     [1-P(A)] is a common factor.  Factor it out:

= [1-P(A)][1-P(B)] 

= P(A')P(B') 

Therefore:  P(A'⋂B')=P(A')P(B') 

Edwin