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

The independence fails after conditioning on $S=1$. Since $S$ is a descendant of collider $M$, selection opens $D\to C\to M\leftarrow A\leftarrow G$.

The independence already failed, and it still fails because conditioning on $A$ opens $D\to A\leftarrow Z$.

The independence still holds. Every path from $Z$ starts through $A$; paths on which $A$ is a noncollider are blocked by conditioning on $A$, while paths opened at collider $A$ are blocked by the conditioned variables $G$ or $D$. Conditioning on $S$ creates no route avoiding these blocks.

The independence fails. Conditioning on $S$, a descendant of $M$, opens the collider path $C\to M\leftarrow A$ even after conditioning on $D$.