We can solve this with a general approach by doing some linear algebra. The nice thing about a general approach like this is, that even irregular, 42-dimensional (finite) shapes can still be solved.
In this case, it will involve a matrix with 81 rows and 117 columns and compute for all starting fields the exact probability distribution of rest states (at the border). And while a matrix with 9477 entries may look a bit daunting at first, it can be easily handled by a computer program.
For illustration purposes, first consider the region of tuples $x, y \in 0,1,2,3$. Respectively only the inner square of four points $\{1,2\}×\{1,2\}$ is not filled with honey. It looks like this:
FOOD ab
F..D or cαβd , to give some variable names.
F..D eγδf
FOOD gh
For each of the inner fields $α, β, γ, δ$, the probability distribution of final states called $D$ is given by a linear equation, e.g.:
$$Dα = \frac{a+c+β+γ}{4}\text{.}$$
The resulting matrix is (after multiplication by $-4$):
α β γ δ a b c d e f g h
Dα -4,1,1,0, 1,0,1,0,0,0,0,0;
Dβ 1,-4,0,1, 0,1,0,1,0,0,0,0;
Dγ 1,0,-4,1, 0,0,0,0,1,0,1,0;
Dδ 0,1,1,-4, 0,0,0,0,0,1,0,1;
And using a handy online tool by Kardi Teknomo we can compute the reduced row echelon form of this matrix: (Multiplied by $-24$)
α β γ δ a b c d e f g h
Dα -24,0,0,0, 7,2,7,2,2,1,2,1;
Dβ 0,-24,0,0, 2,7,2,7,1,2,1,2;
Dγ 0,0,-24,0, 2,1,2,1,7,2,7,2;
Dδ 0,0,0,-24, 1,2,1,2,2,7,2,7;
We can now extract the information that
$$Dγ = \frac{2a+b+2c+d+7e+2f+7g+2h}{24}$$
And since we would like to know what the chance of turning up at $x=0$ or $y=0$ is, we just add up the chances to end up at $c, e, g$ or $h$ which is $\frac{2+7+2+7}{24} = \frac{9}{12}$.
What does this tell us about the actual problem at hand? Well we can write a little program which generates the redistribution matrix $D$ for us. Next we tell it to calculate the reduced row echelon form and finally we extract our information. For a lack of motivation though, writing that program is left as an exercise to the reader :P.
