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Question

For two events if $$A$$ and $$B$$ are independent then prove that $$A',B'$$ are also independent.


Solution

Given $$A,B$$ are independent then $$P(A\cap B)=P(A)P(B)$$.......(1).
Now 
$$P(A'\cap B')=P\{(A\cup B)'\}$$ [ Using De Morgan's Law]
or, $$P(A'\cap B')=1-P(A\cup B)$$
or, $$P(A'\cap B')=1-\{P(A)+P(B)-P(A\cap B)\}$$
or, $$P(A'\cap B')=1-\{P(A)+P(B)-P(A)P(B)\}$$ [ Using (1)]
or, $$P(A'\cap B')=1-P(A)-P(B)+P(A)P(B)$$
or, $$P(A'\cap B')=(1-P(A))(1-P(B))$$
or, $$P(A'\cap B')=P(A')P(B')$$
This given $$A',B'$$ are independent.

Mathematics

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