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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?

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

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