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A parallel reliability system works whenever at least one of its components works. With component lifetimes $T_i$, its lifetime is $T={\max }_i{T}_i$. If component survival events at time $t$ are independent, their survival functions giveThe formula follows because system failure requires every component to have failed. For two independent units each reliable with probability $0.9$, the system reliability is $1-0.1^2=0.99$. Two backup power supplies illustrate the intended redundancy, provided either can carry the required load. A common power surge can make failures dependent and invalidate the product formula; parallel structure alone does not establish independence.^[ref]

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]