Theory concerned with determining the probability that a system (with n components) is working. Let x_j=1 if the jth component is working and let x_j=0 if it has failed. The vector x=(x₁ x₂…x_n) is called the state vector. The function ϕ(x), which takes the value 1 when the system is working and 0 when it has failed, is called the structure function. For n components in series,

For n components in parallel,

If ϕ(x)=1 then x is a path vector: it traces a set of connected working components. If failure of any of its working components results in system failure, the vector is a minimal path vector. Correspondingly, if ϕ(x)=0 then x is a cut vector and, if it is the case that repair of any of the failed components in x leads to the system working, then the vector is a minimal cut vector.

If p_j denotes the probability that the jth component continues to work during the next unit of time, then the probability that a structure consisting of n components in series continues to work is p₁×p₂×⋯×p_n with the corresponding probability for n components in parallel being 1−(1−p₁)(1−p₂)⋯(1−p_n).

Reliability theory. Complicated systems can always be subdivided into combinations of components arranged either in series or in parallel.