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Suppose $A$ and $B$ are independent. Show that $A^{c}$ and $B$ are independent.

So $$P(A^{c} \cap B) = P(B)-P(B \cap A) $$ $$= P(B)-P(B)P(A)$$ $$= P(B)[1-P(A)]$$ $$= P(B)P(A^{c})$$

Is this right? I couldn't write it as $P(A^{c} \cap B) = P(A^{c})-P(A^{c} \cap B^{c})$ because we do not know that $A^{c}$ and $B^{c}$ are independent yet.

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    I see no issues with your proof.2012-01-23
  • 3
    Yes, it is right. It is possible that someone might expect justification of the line $P(A^c\cap B)=P(B)-P(B\cap A)$. I wouldn't.2012-01-23

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