This paper applies, to ballistic missile systems, replacement policies for equipment subject to stochastic failures in which the actual state of the system--good or failed--is not known with certainty. Although it applies these policies only to the missile itself, optimal maintenance policies for the system as a whole are conceptually similar. The criterion employed for comparison of alternative maintenance policies is to minimize the total cost per expected piece of equipment good per period of time. Maximization of system reliability is a special case of this criterion which ignores the financial costs of maintenance actions. The range of maintenance policies examined may be subdivided in two ways. First, it may be assumed that the distribution of times to failure for parts not subject to inspection is known. Secondly, it may be assumed that the distribution is known but that one or more parameters characterizing the distribution are unknown. The central feature of these latter policies is that the interval between replacements is adapted to new information as it accumulates. A second subdivision encompasses those maintenance policies in which the time to replacement for the non-monitored part depends on the state of the remaining parts of the system, and those in which only the failure rate of the uninspected part and the cost of maintenance actions are taken into account in determining the replacement interval. The first class of policies is called "opportunistic," since optimal policies are designed to take advantage of opportunities arising through failures of the remaining parts. Opportunistic policies have a structure which may be described by a critical number N and a set of such critical numbers n1, n2 ⋅⋅⋅, each corresponding to one of the parts under inspection. The optimal policy may be described as follows: For 0 ? x ? ni, the non-monitored part is not replaced if the i-th monitored part fails; for ni ? x ? N, the non-monitored part and the i-th monitored part are replaced whenever the i-th monitored part fails. For illustrative purposes, optimal policies of each type are computed for a hypothetical ballistic missile system. They are then compared with each other and with a rule of thumb frequently proposed for replacement policy. Replace at the mean time to failure. [ABSTRACT FROM AUTHOR]