The general effect of redundancy on system reliability is analyzed and discussed. Three types of redundancy are described—active redundancy, standby redundancy, and activestandby redundancy—and equations are developed for each type of redundancy. The most significant contribution is the development of a Poisson-binomial probability distribution function (PDF) which applies to active-standby redundancy and is the general case of the Poisson and binomial PDF's. Special attention is given to the most general case (activestandby redundancy with start and switching; i.e., a modified Poisson-binomial PDF). The modified Poisson-binomial is applicable to systems where 1) the operating units have a constant failure rate characteristic of the operating mode, 2) the standby units have a different failure rate (also constant) characteristic of the standby mode, 3) all units are subject to failure when started, and 4) the standby units are also subject to failure when switched into the system. A rapid and accurate approximation technique for analyzing active-standby redundancy is also presented.