Are two independent events $A$ and $B$ also conditionally independent given the event $C$?

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 $C$?

$$P(A\cap B) = P(A)P(B) \overset{?}{\Rightarrow} P(A\cap B|C) = P(A|C)P(B|C)$$

Solution 1:

No, Toss an unbiased coin twice. Let $A$ be the event that the first toss is a head, let $B$ be the event that the second toss is a head and let $C$ be the event that the results of both tosses are the same (that is $C=\{HH,TT\}$).

Then $A$ and $B$ are independent but $\mathbb P(A\cap B|C) = \mathbb P(A|C) = \mathbb P(B|C) = \frac 12$.

Solution 2:

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