If the random walk hits points A or B it stops. Consider the symmetric RW on a graph constructed from the two complete graphs $K_{m}$ and $K_{n}$ in such a way that the two complete graphs have exactly 2 common vertices. S is one of the common vertices. Assume that A and B are end points in the graph, and let m = $10^{6}$ and n = 2 * $10^{6}$ (i.e., million and two-million). I am interested in the probability that the symmetric random walk, starting from S, dies in point A. Give an approximation of the value of this probability.
Consider this figure for a visual reference, where m=4,n=5.
I am interested in solving the case m = $10^{6}$ and n = 2 * $10^{6}$ and calculating the probability the random walk ends at Point A. Is the answer simply 1/3 as $K_{n}$ is twice as big $K_{m}$?
I improved the wording on my previous question.