Square Billy

Billy is at $(1, 2)$ and begins a 2D symmetric random walk (each turn, Billy has a $\frac{1}{4}$ chance each of going up, down, right, or left). There is a square with vertices $(0, 0), (0, 4), (4, 4), (4, 0)$. Billy ends his random walk when he hits the square. What is the probability that Billy ends up on a vertical side of the square?