How much does a common cause failure hurt redundancy?

Substantially. A beta-factor model splits failure rate into independent and common-cause portions; with a beta of just 5%, achievable system unreliability is floored near that 5% no matter how many redundant units you add. In practice common cause, not component count, sets the limit of redundancy.

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k-out-of-n Redundancy Calculator

A k-out-of-n system works as long as at least k of its n components work. It generalises the two extremes: n-out-of-n is a pure series system, 1-out-of-n is fully parallel. Voting logic in safety instrumented systems, redundant pumps and multi-string power supplies are all k-out-of-n problems.

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Formula

System reliability (identical components)Rs = Σ (i = k to n) C(n, i) · R^i · (1 − R)^(n − i)

Binomial coefficientC(n, i) = n! / [ i! · (n − i)! ]

Series special case (k = n)Rs = R^n

Parallel special case (k = 1)Rs = 1 − (1 − R)^n

Worked example

Five pumps each with R = 0.90 over the mission. At least three must run to meet demand — a 3-out-of-5 configuration.

i = 3: C(5,3) · 0.90³ · 0.10² = 10 × 0.729 × 0.01 = 0.07290
i = 4: C(5,4) · 0.90⁴ · 0.10¹ = 5 × 0.6561 × 0.10 = 0.32805
i = 5: C(5,5) · 0.90⁵ = 1 × 0.59049 = 0.59049
Rs = 0.07290 + 0.32805 + 0.59049 = 0.99144

System reliability is 99.14%, against 90% for a single pump. For comparison, the same five pumps in pure series would give 0.90⁵ = 59.0%, and fully parallel (1-out-of-5) would give 99.999%. The voting requirement is what sets the cost-benefit point.

Run this with your own numbers

Common mistakes

Assuming independence when a common cause exists. Five identical pumps on one power bus, one control system or one maintenance crew share failure modes the binomial model cannot see — this is where redundancy calculations most often overstate reality.
Forgetting that redundancy raises the probability of at least one component failing, which increases maintenance load even as availability improves.
Using the identical-component formula for components with different reliabilities — that requires enumerating states or a Markov model.

Frequently asked questions

How much does a common cause failure hurt redundancy?: Substantially. A beta-factor model splits failure rate into independent and common-cause portions; with a beta of just 5%, achievable system unreliability is floored near that 5% no matter how many redundant units you add. In practice common cause, not component count, sets the limit of redundancy.

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