\problemdata      
        {probability-intro-8} % latex-friendly label for the prob.
        {total probability theorem} % The topic(s) of the problem content.
        {?} % Source (if known)
        {S02 cp9W} % Usage (list ones you are aware of).
        {Albert Meyer, S02} % Last revision info (author, date).

\begin{problem}
Prove that if $A$ and $B$ are independent events, then so are $\bar{A}$
and $B$.

\solution{From the Law of Total Probability we have
\[
\pr{B} = \pr{B \intersect \bar{A}} + \pr{B \intersect A},
\]
so
\begin{align}
\pr{B \intersect \bar{A}} & =  \pr{B} - \pr{B \intersect A}\\
 & = \pr{B} - \pr{B}\pr{A} & \text{because $A$ and $B$ are independent}\\
 & = \pr{B}(1 - \pr{A}) = \pr{B}\pr{\bar{A}}.
\end{align}
}

\end{problem}
