I and J Polynomials in a Potpourri of Probability Problems

Abstract

Some new methodology is developed for Network Reliability problems and for random paths on finite lattices. In terms of stopping sets which define different (random) ways of reaching a goal in a geometrical setting, certain I and J polynomials are developed which give rise to the probability distribution (and its moments) of the waiting time (WT) needed to reach the preassigned goal. These new techniques have many different applications from Network Reliability to Recreational problems of tic-tac-toe and attacking all the squares on a chess board with randomly placed rooks or knights or queens, etc. Failure probabilities need not be equal and random sampling can be carried out with replacement, without replacement or by Pólya sampling schemes.