# $A \perp\kern-5pt\perp B ,B \perp\kern-5pt\perp C, C \perp\kern-5pt\perp A$ but $A$ and $B$ are not conditionally independent when given $C$

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?

• Your example is correct, have you tried working out the probabilities directly from the four possible outcomes?
– Tim
Jul 25, 2022 at 13:50
• @Tim didn't realize that. That's indeed true
– user
Jul 25, 2022 at 13:59