\problemdata                                                             
        {Independence} % latex-friendly label for the prob.
        {Independence} % The topic(s) of the problem content.  
        {?} % Source (if known)                                          
        {S01 Tutorial 11, 1} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}           
Is it true that ``$A$ and $B$ independent" $\Longleftrightarrow$ ``$A$ and $\bar{B}$ independent"?

\solution{Yes
\proof

($\Longrightarrow$)Assume that $\Pr[A\intersect B]~=~\Pr[A]\cdot\Pr[B]$. Then:
\[
\Pr[A\intersect\bar{B}] = \Pr[A] - \Pr[A\intersect B]
 = \Pr[A](1-\Pr[B]) = \Pr[A]\Pr[\bar{B}]
\]

($\Longleftarrow$)Assume that $\Pr[A\intersect \bar{B}]~=~\Pr[A]\Pr[\bar{B}]~=~\Pr[A]\cdot(1-\Pr[B])$. Then:
\[
\Pr[A\intersect B]~=~\Pr[A] - \Pr[A\intersect \bar{B}]=\Pr[A] - \Pr[A](1-\Pr[B])=\Pr[A]\cdot \Pr[B]
\]


}
\end{problem}
                                 