Birnbaum importance
Birnbaum importance measures the marginal sensitivity of a system's top-event probability to one basic event. Formally, it's the partial derivative of P(top) with respect to P(event) — equivalently, how much P(top) changes if you nudge that one event's probability by one unit. Designers use it to identify which basic events most repay engineering effort.
Formula
I_B(i) = ∂ P(top) / ∂ P(i)
= P(top | event i fails) − P(top | event i works)
The two forms are mathematically equivalent in a coherent fault tree because P(top) is linear in each P(i) when the others are held fixed.
Worked example
Tree: TOP = OR(AND(a, b), c). Let P(a) = 0.01, P(b) = 0.02, P(c) = 0.005.
P(top | c works) = P(AND(a,b)) = 0.01·0.02 = 2e-4 P(top | c fails) = 1 = 1 I_B(c) = 1 − 2e-4 ≈ 0.9998 P(top | a works) = P(c) = 0.005 P(top | a fails) = P(OR(b, c)) = 1 − (1−0.02)(1−0.005) ≈ 0.0249 I_B(a) ≈ 0.0249 − 0.005 = 0.0199
Even though a is the more probable failure, c dominates Birnbaum importance because it sits alone at the top OR — it's a single point of failure.
How to use it
Birnbaum importance ranks events by marginal influence. It's most useful for design questions: "if I could halve any one component's failure rate, which one moves the needle most?" The answer is the event with the highest Birnbaum importance.
The trade-off is that Birnbaum doesn't account for how likely the event is to happen — only the structural significance. For "where is risk actually accumulating today?", Fussell-Vesely importance is usually a better lens.