Suppose on an infinite two dimensional Go board the tengen is occupied by a black stone, and every other grid point is occupied by a black stone, or a white stone, or nothing, with probability 1/3 respectively. What is the probability for the group belonging to the central black stone to live on the board?

More formally, suppose $f:\mathbb Z^2\to\{1,0,-1\}$ is a random function such that $f(0,0)=1$ and for all $(x,y)\neq(0,0)$ and all $a\in\{1,0,-1\}$, $\mathbb P(f(x,y)=a)=1/3$. Let $C$ be the connected component (adjacency is defined as 4 immediate neighbors, a la Von Neumann) of $f^{-1}(1)$ containing the point $(0,0)$. What is $\mathbb P(\forall (x,y)\in C', f(x,y)=-1)$, where $C'=\{(x,y)\in\mathbb Z^2\backslash C: (x,y)$ is adjacent to some point in $C\}$?

Related, but not identical: Probability that a randomly filled Go board has a set of white stones connected through their von Neumann neighborhoods