What is a minimal cut set?

The smallest combination of basic events that together cause the top event, such that removing any one of them stops it from causing the top event. Order-1 cut sets are single points of failure and are the first thing to look for in any completed tree.

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Fault Tree Analysis Calculator

Fault tree analysis works backwards from an undesired top event to the combinations of basic events that can cause it. It is deductive where FMEA is inductive: FMEA asks what happens if this part fails, FTA asks what must fail for this outcome to occur.

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Formula

AND gate (all inputs required)P = P₁ × P₂ × … × Pₙ

Assumes independence.

OR gate (any input sufficient)P = 1 − (1 − P₁)(1 − P₂)…(1 − Pₙ)

OR gate rare-event approximationP ≈ P₁ + P₂ + … + Pₙ

Acceptable when all probabilities are small; always overestimates.

Top event from minimal cut setsP(top) = 1 − Π (1 − P(cut setᵢ))

Worked example

Loss of cooling occurs if the pump fails AND the backup pump fails, OR if the controller fails. P(pump) = 0.02, P(backup) = 0.05, P(controller) = 0.01.

AND branch: 0.02 × 0.05 = 0.001
OR of {0.001, 0.01}: 1 − (1 − 0.001)(1 − 0.01)
= 1 − (0.999 × 0.99) = 1 − 0.98901
P(top) = 0.01099

Top event probability is 1.10%. The controller — a single component with no redundancy — contributes ten times more risk than the entire redundant pump pair. Redundancy applied to the wrong branch buys nothing; the minimal cut set of order 1 is where the risk lives.

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Common mistakes

Building the tree without a precisely defined top event, including its operating state and time window.
Multiplying AND-gate probabilities for events that share a common cause. Independence must be argued, not assumed.
Stopping at a component level where failure data does not exist, leaving branches unquantifiable.
Overlooking order-1 cut sets. Any single basic event that alone causes the top event dominates everything else in the tree.

Frequently asked questions

What is a minimal cut set?: The smallest combination of basic events that together cause the top event, such that removing any one of them stops it from causing the top event. Order-1 cut sets are single points of failure and are the first thing to look for in any completed tree.

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