I'm trying to solve an exercise:
Consider three random variables, $A,B,C$. Give an example of a distribution in which $A \perp\kern-5pt\perp B ,B \perp\kern-5pt\perp C, C \perp\kern-5pt\perp A $ but $A$ and $B$ are not conditionally independent when given $C$.
My attempt:
Let $A = \{\text{flipping coin A}\}$ and $B\{\text{flipping coin B}\}$ and $C = \begin{cases} 1 & \text{both heads or both tails} \\ 0 & \text{otherwise}\end{cases}$
Now, $A \perp\kern-5pt\perp B$ and $A\perp\kern-5pt\perp B | C$ is not true, but I can't say that $C \perp\kern-5pt\perp A,B$ since $P(A,B,C) = P(C|A,B)P(A)P(B)$ and $P(A,C)= \sum_B P(C|A,B)P(B)P(A) = P(C|A)P(A) \neq P(A)P(C)$
So how could I find such an example?