A random walk on the 2-dimensional integer lattice begins at the origin. At each step, the walker moves one unit either left, right, or up, each with probability \(\frac13\). (No downward steps ever.) A walk is a success if it reaches the point \((1,1)\). What is the probability of success?
Note: One can vary the problem by varying the target point. Eg., use \((1, 0)\) or \((0,1)\) instead. Perhaps there is a good method to resolve the general case of target \((a, b)\).
Source: Bruce Torrence, Randolph-Macon College