TITLE: Are two independent events $A$ and $B$ also conditionally independent given the event $C$?
QUESTION [5 upvotes]: If we know that two events $A$ and $B$ are independent, can we say that $A$ and $B$ are also conditionally independent given an arbitrary event $C$?
$$P(A\cap B) = P(A)P(B) \overset{?}{\Rightarrow} P(A\cap B|C) = P(A|C)P(B|C)$$

REPLY [2 votes]: Hint: Consider the example of
$${{\text{Event }A}\atop \begin{array}{|c|c|c|}
\hline\strut& & \\\hline
\strut& & \\\hline
\strut\Large\color{red}{\bullet} & \Large\color{red}{\bullet} & \Large\color{red}{\bullet}\\\hline
\end{array}}\qquad {{\text{Event }B}\atop \begin{array}{|c|c|c|}
\hline\strut& & \Large\color{blue}{\bullet}\\\hline
\strut\;\;&\;\; & \Large\color{blue}{\bullet}\\\hline
\strut &  & \Large\color{blue}{\bullet}\\\hline
\end{array}}\qquad {{\text{Event }C}\atop \begin{array}{|c|c|c|}
\hline\strut& & \\\hline
\strut\;\;&\Large\color{green}{\bullet} & \\\hline
\strut & \Large\color{green}{\bullet} & \Large\color{green}{\bullet}\\\hline
\end{array}}$$