If we know that $A,B$ are independent events, how can we proof that $\bar{A},\bar{B}$ are independent events to?
We should using the definition: $\Pr(A\cap B)=\Pr(A)\cdot \Pr(B)$
\begin{align}\operatorname{Pr}(\bar A\cap\bar B)&=1-\operatorname{Pr}(A)-\operatorname{Pr}( B)+\operatorname{Pr}(A\cap B)\\ &=1-\operatorname{Pr}(A)-\operatorname{Pr}( B)+\operatorname{Pr}(A)\operatorname{Pr}(A)\\ &=(1-\operatorname{Pr}(A))(1-\operatorname{Pr}(B))\\ &=\operatorname{Pr}(\bar A)\operatorname{Pr}(\bar B).\end{align}
• Since the OP might be a novice, it is worth adding that DeMorgan's law has been used to write $$\Pr(\bar{A}\cap\bar{B}) = 1 - \Pr(A \cup B) = 1 - [\Pr(A) + \Pr(B) - \Pr(A \cap B)]$$ in the first line. Also, there is a minor typo ($\Pr(A)\Pr(A)$ instead of $\Pr(A)\Pr(B)$) in the second line. – Dilip Sarwate Jul 12 '13 at 15:26