A method of obtaining a fairly good lower approximation to the expected duration time of a project whose individual job times are discrete random variables is described. It is assumed that jobs that immediately precede any given job may have a joint distribution of times, with independence between immediate predecessors of different jobs. The method provides an estimate that is usually much better, and never worse, than the one obtained by replacing each random job time by its expected value. An application to expected minimal path lengths in networks whose arc lengths are random variables is discussed.