causal graph - counting the number of backdoor paths in a DAG

Question

I am following "A Crash Course in Causality: Inferring Causal Effects from Observational Data" on Coursera.

I am struggling at correctly identifying backdoor paths in causal graphs (or DAG for Directed Acyclic Graph).

Example #1 :

The following DAG is given in example in week $2$'s video on the "backdoor path criterion".

The course states that there are $3$ backdoor paths from $A$ to $Y,$ but I see $4$ of them:

\begin{align*} &A \leftarrow W \leftarrow Z \leftarrow V \to Y\\ &A \leftarrow W \to M \to Y\\ &A \leftarrow Z \leftarrow V \to Y\\ &A \leftarrow Z \to W \to M \to Y \quad (\text{not pointed out}) \end{align*}

Example #2 :

In the same week quiz, we are asked to count the number of backdoor paths on this DAG:

Apparently, there is only $1$ but once again I see more of them (I count $2$ of them):

\begin{align*} &G \to A \to B\\ &G \leftarrow E \leftarrow D \to A \to B \end{align*}

Where am I wrong?

Answer 1

For Example 1, you are correct. $A\leftarrow Z\to W\to M\to Y$ is a valid backdoor path with no colliders in it (which would stop the backdoor path from being a problem).

In Example 2, you are incorrect. The definition of a backdoor path implies that the first arrow has to go into $G$ (in this case), or it's not a backdoor path. Only $G\leftarrow E\leftarrow D\to A\to B$ satisfies that criterion.