Two checks. Same scores. Different protection.
Imagine a process that receives 100 faulty cases. Check A misses 10. Check B also misses 10. The process catches a fault if either check catches it.
How many faults slip past both? That depends on whether they miss the same cases.
Constructed example
Both checks evaluate the same 100 cases. Their individual miss counts stay at 10. Move only the overlap.
Use arrow keys or the buttons. All values are constructed; these are not measurements of a real system.| Outcome | Count |
|---|---|
| Both miss | 10 |
| A misses; B catches | 0 |
| A catches; B misses | 0 |
| Both catch | 90 |
| Total faulty cases | 100 |
Together they miss 10 of 100. Adding B to A catches 0 additional faulty cases.
Identical misses: adding B catches none of the faults A missed.
A check can be small. A process can be large.
A check might be a form validator, a human review, a test, a sensor, a classifier, an audit, or an entire subsystem that produces a pass/fail decision. The arithmetic is about the defined events and how they combine, rather than the technology or size of the components.
For example, compare two invoice checks on the same known incorrect invoices, two tests on the same known defective builds, or two safeguards on the same known harmful inputs. These are suggested applications, not empirical findings about those systems.
For a larger composition, first state the outcome and the wiring: does a failure require both components to miss, either component to fail, a routing decision, or a sequence of changing states? This page uses an either-check-catches rule, so process failure means both miss. It does not establish end-to-end reliability for every integration.
If A changes the case, blocks it, routes it, or changes B’s later behavior, B may face a different workload. Measure those conditional or sequential outcomes in the actual workflow. Do not multiply standalone rates and call that a measured system result. Pairwise tables also do not generally determine a joint event across three or more components.
Ask what the second check adds
On this fixed faulty-case pool, the additional catches from adding B are the cases A misses and B catches. A useful real evaluation also needs the extra false alarms on valid cases, the operating settings, and the added time or cost. The 100 faulty cases here contain no information about those tradeoffs.
For two miss probabilities p and r on a common population, the both-miss probability lies between max(0, p + r − 1) and min(p, r). For 10% and 10%, that is 0% to 10%. Independence chooses 1%; it is one assumption, not a consequence of the separate scores.
Run the existing constructed example · Inspect the mathematical claim · Inspect the endpoint witnesses
Prior work: Dung and Mai on shared failures across alignment techniques. The probability bound is established mathematics, not a novelty claim.