Title: A biased variant of the ladybug clock problem.

Abstract: Instead of the common assumption of uniform probabilities for steps in a random walk on a graph, we generalize the steps to be biased and compute several statistics related to biased random walks on a cycle graph, more fancifully known as the “ladybug clock problem,” courtesy of Richard Stanley. We obtain new explicit formulas for the distribution of probabilities for a walk to end on a specific node, the expected cover time for the cycle and other similar statistics. We carry out these computations solely with generating functions related to the classic gambler’s ruin problem (and Dyck paths).