Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 221 / 3 / b

Yes. Every path from $D$ to $G$ is blocked by a collider, such as $D\to A\leftarrow G$ or $D\to C\to M\leftarrow A\leftarrow G$. Thus D-separation gives $D\perp G$.

No. Conditioning on the collider $A$ opens $D\to A\leftarrow Z$, so $D$ and $Z$ are not conditionally independent given $A$.

Yes. Conditioning on $A$ blocks the directed paths from $Z$ to $Y$. Although conditioning on $A$ opens $Z\to A\leftarrow G\to Y$ and $Z\to A\leftarrow D\to C\to M\to Y$, conditioning additionally on $G$ and $D$ blocks those paths. Hence $Z\perp Y\mid (A,G,D)$.

Yes. Conditioning on $D$ blocks $C\leftarrow D\to A$, while $C\to M\leftarrow A$ is blocked at the collider $M$. Therefore $C\perp A\mid D$.