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.
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