The probability that it is dead, i.e. is surrounded by white stones, is the sum, over all finite connected sets $C$ containing $(0,0)$, of $3^{-(|C|+|C'|)}$. I don't expect this to have a closed form.

The probability that it is dead, i.e. is surrounded by white stones, is the sum, over all finite connected sets $C$ containing $(0,0)$, of $3^{-(|C|+|C'|)}$. I don't expect this to have a closed form. Simulation seems to indicate it is about $0.0152$. The contribution of $C = \{(0,0)\}$ to this is $3^{-4} \approx 0.0123$.