If A and B are independent events, show that $\displaystyle \bar{A} \ \ \text{and} \ \ \bar{B}$ are independent.

I haven't a clue what to do for this one.

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- Feb 3rd 2011, 04:51 PMdwsmithComplement Proof of independent events
If A and B are independent events, show that $\displaystyle \bar{A} \ \ \text{and} \ \ \bar{B}$ are independent.

I haven't a clue what to do for this one. - Feb 3rd 2011, 05:10 PMharish21
I usually write $\displaystyle A^c$ for complements. You can show $\displaystyle P(A^c \cap B^c) = P(A^c) P(B^c)$.

$\displaystyle P(A^c \cap B^c)$

$\displaystyle = P[(A \cup B)^c] $

$\displaystyle = 1-P(A \cup B) $

$\displaystyle = 1-P(A)-P(B)+P(A\cap B)$

now use $\displaystyle P(A\cap B) = P(A) P(B)\; and\; P(A^c)=1-P(A)\;and\;P(B^c)=1-P(B) $ and simplify