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Locate the assumption in the multiplication
The chance that all tests avoid errors equals a product of marginal no-error probabilities only under the stated independence assumption. Reusing the same data or applying related specifications may create dependence. The number of tests alone does not settle it.
Compare perfectly linked error events
Consider ten true-null tests whose error events are identical: either all reject incorrectly or none do. If that single event has probability 0.05, the family chance of at least one false rejection is 0.05. The independent model would give about 0.4012631. Both scenarios have expected false-rejection count 0.5.
Use a bound when only marginal sizes are known
The union bound gives family risk at most the sum of individual error probabilities. For ten tests each bounded by 0.05, that bound is 0.5. It is an upper bound, not a claim that the actual probability equals 50%.
Report assumptions with any family-risk number
| Reported quantity | Necessary context |
|---|---|
| Exact independent family probability | True nulls, event sizes and independence |
| Expected false-rejection count | Marginal false-rejection probabilities |
| Union upper bound | Valid marginal bounds and defined family |