If event a and b are independent then prove that a and bc is independent
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A') = 1-P(A)
P(B') = 1- P(B)
Now clearly P(A′)P(B′)=1−[P(A)+P(B)]+P(A∩B)
From set algebra we know that P(A)+P(B)=P(A∪B)+P(A∩B) Substituting, we have P(A′)P(B′)=P([A∪B]′)
Now from De morgans law we know that:
[A∪B]′=[A′∩B′]
Substituting, we have P(A′)P(B′)=P(A′∩B′) , as required.
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