This paper is concerned with elementary methods for evaluating the distribution of the time to system failure, following a particular sequence of events from a Markov chain. After discussing a simple example in which a specific sequence from a two-state Markov chain leads to failure, the method is generalized to a sequence from a (k>2)-state chain. The expectation and variance of the time T to failure can be obtained from the probability generating function (p.g.f.) of T. The method can be extended to the case of continuous time.