An ant walking on a ...(guess what) a plane

I stole this question from math.stackexchange and puzzling.stackexchange shamelessly as this is my question (and it was ignored on math.stackexchange and got tumbleweed), but let's go on with the question...

An ant is on a cartesian plane at point (7,7). Every time, he randomly moves up, down, left or right each turn with equal chances for each. Find the probability of his x or y coordinate to reach 10.before either reach zero.

For instance, there are lines of honey at x=10, y=10, x=0 and y=0. What are the chances of the ant to reach the honey at x=10 or y=10 first?

The answer is very strange and in case you are wondering, the answer is not 7/10, which was shown to be true in 1-D.

For this question, just find the numerator of the fraction give that the numerator and denominator are coprime.

Note: if the ant reaches the intersection of 2 lines straight away... Oh wait, it shouldn't happen!