Problem: We have a bowling ball and a lane that is 50 meters long and 2.5 meters wide and is surrounded by 50 meter long gutters on both sides. For every meter forward that the ball travels, given that it was previously in the center of the lane, there is a 1/4 chance of it going 1 meter to the left, 1/4 chance of it going to the 1 meter to the right, and 1/2 chance of it remaining in the center. Given that the bowling ball was previously not in the center of the lane, for every meter forward it travels, there is a 3/8 chance of it sliding into the gutter, a 1/8 chance of it moving 1 meter back to the center, and a 1/2 chance of it remaining in its horizontal place. If the ball goes into the gutter, it will permanently stay there. If the ball is not in the gutter by the end of the 50 meters, we will score some points by knocking over some pins. Given that we start by launching the ball in the center of the lane, what is the probability of scoring some nonzero number of points?
I'm 99% sure we have to solve this with states/markov chains just based on my intuition, so I went ahead to write the following equations: $$C = \dfrac{1}{2}C + \dfrac{1}{8}S$$ $$S = \dfrac{1}{2}C + \dfrac{1}{2}S$$ $$G = \dfrac{3}{8}S + G$$
where $C$ is the probability of the ball being in the center after traveling $1$ meter, and similarly $S$ is the probability of the ball being in one of the sides of the lane after traveling $1$ meter, and $G$ is the probability of being in the gutter. We want to find $C+S$.
However, I set something up wrong because I get that $C$ and $S$ are both equal to $0$, which is wrong. Could someone help me find my mistake?