A First-principles Derivation of the Beta Factor Common Cause Failure (CCF) Model: Boolean Expansion, Worked Example, and the Irreducible CCF Floor for a Two-Pump Redundant System
The Beta Factor model is the simplest and most widely deployed mechanism for representing common cause failure (CCF) in fault tree analysis. Its single parameter, the beta factor, partitions the unavailability of every component in a common cause group into an independent share and a shared CCF share. This note develops the model analytically from first principles using the canonical two-pump example, demonstrates that the Beta Factor implementation in fault tree software is shorthand for an explicit three-event topology that can be built by hand, derives the closed-form expression for the top event unavailability in both its rare-event-approximated form and the exact form retaining the cross term, computes the irreducible CCF floor that constrains every redundant system, and reports a sensitivity table that quantifies how rapidly the CCF term overwhelms the independent term as beta increases. A closing argument explains why the rare-event approximation, although introducing an eighth-significant-figure error against the exact form, is the appropriate level of detail for routine engineering work given the parametric uncertainty inherent in the inputs.