Fault Tree AND / OR Gate Calculator
Combine up to five independent basic event probabilities through an AND gate (all must happen) or an OR gate (any one happens), as used in fault tree analysis and LOPA checks.
- OR: P = 1 − (1 − P₁)(1 − P₂)… = 0.03465
- Rare event approximation for OR: P ≈ ΣPi = 0.035
Gate Equations
OR: P = 1 − (1 − P₁)(1 − P₂)…(1 − Pn)
Rare event approximation: POR ≈ ΣPi
P are the probabilities of independent basic events over the same time period. For an AND gate combining a frequency with probabilities (such as an initiating event and protection layer failures), multiply the frequency by the probabilities, as in LOPA.
Worked example
Three independent events of 0.01, 0.005 and 0.02 through an OR gate give 0.0347 (sum approximation 0.035). Through an AND gate they give 1 × 10⁻⁶.
Frequently Asked Questions
What is a fault tree?
A top down diagram that shows how combinations of basic failures, through AND and OR gates, lead to a top event.
When can I use the rare event approximation?
When all probabilities are small, below about 0.1. It slightly overestimates the OR result.
How do common cause failures affect an AND gate?
They make events dependent, so the true probability can be much higher than the product. Model them as a separate OR branch.
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