Nicolas Caritat de Condorcet (1743–94) was an early proponent of the use of majority systems in voting procedures. In his Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité de voix (Condorcet, 1785), he demonstrated what is now known as the ‘Condorcet Jury Theorem'. This theorem states that if n=2m+1 jurists act independently, each with probability p>12 of making the correct decision, then the probability h2m+1(p) that the jury (deciding by majority rule) makes the correct decision increases monotonically to 1 as m increases to infinity. Condorcet argued therefore that there are situations in which it is advisable to entrust a decision to a group of individuals of lesser competence than to a single individual of greater competence. Of course, Condorcet's theorem makes the assumption of independence and homogeneity within the group—assumptions which are seldom realistic. Some generalisations of this theorem will be presented in which (a) voter competencies vary, and (b) there is a dependence between voter decisions. Furthermore, the concept of an indirect majority system will be discussed and compared with a simple majority system.

If τ is the lifetime of a coherent system, then the signature of the system is the vector of probabilities that the lifetime coincides with the ith order statistic of the component lifetimes. The signature can be useful in comparing different systems. In this treatment we give a characterization of the signature of a system with independent identically distributed components in terms of the number of path sets in the system as well as in terms of the number of what we call ordered cut sets. We consider, in particular, the signatures of indirect majority systems and compare them with the signatures of simple majority systems of the same size. We note that the signature of an indirect majority system of size r × s = n is symmetric around , and use this to show that the expected lifetime of an r × s = n indirect majority system exceeds that of a simple (direct) majority system of size n when the components are exponentially distributed with the same parameter.

The problem of where to allocate a redundant component in a system in order to optimize the lifetime of a system is an important problem in reliability theory which also poses many interesting questions in mathematical statistics. We consider both active redundancy and standby redundancy, and investigate the problem of where to allocate a spare in a system in order to stochastically optimize the lifetime of the resulting system. Extensive results are obtained in particular for series and parallel systems.

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The problem of where to allocate a redundant component in a system in order to optimize the lifetime of a system is an important problem in reliability theory which also poses many interesting questions in mathematical statistics. We consider both active redundancy and standby redundancy, and investigate the problem of where to allocate a spare in a system in order to stochastically optimize the lifetime of the resulting system. Extensive results are obtained in particular for series and parallel systems.