We apply the concept of stopping times of martingales to problems in classical probability theory regarding the occurrence of sequence patterns in repeated experiments. For every finite collection of sequences of possible outcomes, we compute the expected waiting time till one of them is observed in a run of experiments. Also we compute the probability for each sequence to be the first to appear. The main result, with a transparent proof, is a generalization of some well-known facts on Bernoulli process including formulas of Feller and the "leading number" algorithm of Conway.