Advanced Analysis
System Reliability Calculator
Reliability of series, parallel, and k-out-of-n configurations from the reliability of each component.
R = 1 − ∏(1 − Rᵢ)System configurations
Series — all components must work
R = R₁ × R₂ × … × RₙSystem reliability is always lower than the weakest component. Like a chain, it breaks at its weakest link.
Parallel — only one needs to work
R = 1 − ∏(1 − Rᵢ)System reliability is always higher than the best component. Redundancy dramatically improves reliability.
k-out-of-n — at least k of n must work
R = Σ C(n, i) × pⁱ × (1 − p)ⁿ⁻ⁱ, for i = k … nThat formula is for identical components (each with reliability p). With different reliabilities the calculator sums the probability of every combination in which at least k work.
Common with voting systems, engine configurations (2-of-3), and RAID arrays.
Configuration comparison example
Three identical components, each with R = 0.90:
| Configuration | System reliability | Working |
|---|---|---|
| Series (all 3 must work) | 72.9% | 0.9 × 0.9 × 0.9 |
| 2-out-of-3 | 97.2% | 3(0.9)² − 2(0.9)³ — one can fail |
| Parallel (any 1 works) | 99.9% | 1 − (0.1)³ |
These assume independent failures; common-cause failures reduce the real benefit of redundancy.
Frequently asked questions
When should I use redundancy?
When the cost of failure exceeds the cost of adding redundant components. Common in safety systems (dual brakes), critical infrastructure (backup generators), and high-availability IT (RAID, load balancers).
What is the difference between active and standby redundancy?
Active redundancy: all units run simultaneously (hot standby). Standby redundancy: backup units activate only when the primary fails (cold/warm standby). Active is simpler but ages all units; standby preserves backup life but requires switching logic.