Normal Half-Plane by Symmetry

Since $X$ and $Y$ are independent centred normals, the linear combination $Z = Y - 4X$ is normal with

so $Z \sim \mathcal{N}(0,17)$. A centred normal is symmetric about $0$ and puts no mass at $0$. Hence

Alternative. The pair $(X,Y)$ is rotationally symmetric about the origin. The event $\{Y > 4X\}$ is an open half-plane whose boundary is a line through the origin, so it has probability $\tfrac12$.

Generalisation. The same argument gives $\mathbb{P}[aX + bY > 0] = \tfrac12$ for any $(a,b) \neq (0,0)$.