Calculator Inputs
Example Data Table
This example has three minimal path sets and five components.
| Item | Input | Meaning |
|---|---|---|
| Component A | 0.98 | A works with 98% probability. |
| Component B | 0.95 | B works with 95% probability. |
| Component C | 0.97 | C works with 97% probability. |
| Component D | 0.93 | D is shared by two paths. |
| Component E | 0.96 | E appears in the third path. |
| Path P1 | A,B | Both A and B must work. |
| Path P2 | A,C,D | A, C, and D must work. |
| Path P3 | D,E | D and E must work. |
Formula Used
Path event: If path set Pj contains components i, then Pr(Pj works) = ∏ p_i.
System success: The system works when at least one minimal path set works.
R = Pr(P1 ∪ P2 ∪ ... ∪ Pm)
R = Σ Pr(Pj) - Σ Pr(Pj ∩ Pk) + Σ Pr(Pj ∩ Pk ∩ Pl) - ...
Intersection term: For selected paths, merge their components. Multiply each unique component probability once.
Failure probability: Q = 1 - R.
Component importance: I_i = R(p_i = 1) - R(p_i = 0).
How to Use This Calculator
- Enter component success probabilities, one component per line.
- Enter each minimal path set on its own line.
- Use the same component labels in both boxes.
- Set a default probability for missing components if needed.
- Choose decimal precision for the displayed output.
- Press the calculate button.
- Review exact reliability, failure probability, bounds, charts, and criticality.
- Export the report as CSV or PDF for records.
Understanding Minimal Path Set Probability
Why minimal path sets matter
Minimal path sets describe the smallest working routes in a system. A route works only when every component inside that route works. The whole system works when at least one minimal path set works. This idea is common in reliability studies, network statistics, safety models, and fault tolerant design.
Why overlap must be handled
A direct sum of path probabilities can be misleading. Two paths may share the same component. When they share parts, their success events overlap. Inclusion-exclusion corrects that overlap. It adds single path probabilities. It subtracts pair overlaps. It adds triple overlaps, and continues until every selected group is tested.
Input meaning
This calculator uses independent component probabilities. Each component should represent the chance that the part works during the selected mission, test, or time period. You can enter path sets as lines. Each line lists the components that must all work together. You can also enter component probabilities in a separate box.
Result meaning
The exact result is the probability that at least one path set succeeds. The failure probability is one minus that value. The page also shows the sum bound, the strongest single path bound, and a disjoint path approximation. The approximation is only a guide. The exact inclusion-exclusion value is the main statistic.
Criticality insight
Criticality values help explain the result. A component is important when changing it from failed to perfect strongly changes system reliability. This is useful for maintenance choices. It can show which shared component controls many routes. It can also reveal a weak part with a small individual probability.
Good modeling practice
Use clean labels for components. Letters, numbers, dashes, and underscores are easy to read. Avoid duplicate components inside the same path. The calculator removes duplicates automatically. Keep the number of path sets reasonable, because exact inclusion-exclusion grows quickly. More paths create more intersection terms.
Practical workflow
For best results, start with an example. Change one probability at a time. Watch how the chart moves. Then add more paths. Compare exact reliability with the simple sum. If the gap is large, path overlap is important. That tells you the structure has shared risk. Such insight is often more useful than the final probability alone.
It supports audit notes, teaching checks, and quick design reviews.
FAQs
What is a minimal path set?
It is the smallest group of components that can make a system work. If any component is removed from that group, the path no longer works.
What probability should I enter for a component?
Enter the chance that the component works during the mission, test, or time period being studied. Use values from 0 to 1.
Why not just add all path probabilities?
Adding paths double counts shared components and overlapping success events. Inclusion-exclusion removes that double counting and gives the exact union probability.
Does this calculator assume independent components?
Yes. It assumes component states are independent. Correlated failures, common cause failures, or shared environments need a different model.
What does failure probability mean here?
It is the chance that no minimal path set works. It equals one minus the calculated system reliability.
What does component importance show?
It shows how much system reliability changes when a component moves from failed to perfect. Higher values indicate stronger structural influence.
Why is there a path set limit?
Exact inclusion-exclusion grows exponentially. Sixteen paths already require many combinations, so the limit keeps the page responsive.
Can I export the results?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a compact report with key statistics.