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Reliability theory uses probability theory and statistical inference to describe whether a component or system continues to perform its specified function. For a nonnegative failure time $T$, its reliability at time $t$ is the survival function $R(t)=\mathbb{P}(T>t)$. If $T$ has probability density function $f$, the hazard function $h(t)=f(t)/R(t)$ describes the instantaneous failure rate conditional on survival, where $R(t)>0$.

System structure matters. For independent components with reliabilities $R_i(t)$, a series reliability system fails at the first component failure, so $R_{\mathrm{series}}(t)={\prod }_i{R}_i(t)$. A parallel reliability system works while at least one component works, so $R_{\mathrm{parallel}}(t)=1-{\prod }_i(1-R_i(t))$. Two independent components each having reliability $0.9$ therefore give $0.81$ in series and $0.99$ in parallel. These product formulas depend on independence; a common cause of failure can invalidate them.^[ref]

A series reliability system works only when every required component works. With component lifetimes $T_i$, its lifetime is $T={\min }_i{T}_i$. For independent component survival events,Two independent components each reliable with probability $0.9$ therefore give system reliability $0.81$. A pump that requires both a motor and a controller is a functional series system even if its wiring is not literally in series. For independent exponential distributions of lifetimes with rates ${\lambda }_i$, $R(t)=e^{-t{\sum }_i{\lambda }_i}$: adding required components adds failure rates. Shared causes of failure require a joint probability distribution rather than this product.^[ref]