Can the sum exceed one?

Yes, but then the upper bound of one is more informative.

Must every test receive the same allowance?

No. Nonnegative, appropriately specified allowances can differ provided their total meets the target.

Is this the same as the Bonferroni test?

The inequality is a probability result; the testing procedure applies it to false-rejection events.

AI Glossary / Statistics and probability / Advanced

Bonferroni inequality

Definition

The Bonferroni inequality, in the sense used here, states that the probability of at least one of several events occurring cannot exceed the sum of their individual probabilities. This upper bound, also called the union bound, applies whether or not the events are independent. It underpins a widely used method of controlling errors across multiple statistical comparisons.

What it means

A team evaluating an AI system may care about several possible failures at once. Knowing the probability of each failure does not generally reveal the probability that any failure occurs, because failures can overlap. The inequality provides a conservative bound without requiring a complete model of that overlap.
For events E1 through Em, P(at least one E) is no greater than P(E1) + ... + P(Em). Since probabilities cannot exceed one, the useful upper bound is the smaller of one and that sum. A bound of 0.08 means at most eight percent under the stated probabilities, not exactly eight percent.

How it works

Think of counting outcomes that belong to any of several event sets. Adding the counts for each set counts overlapping outcomes more than once. Counting each outcome only once therefore cannot produce a larger total. The same argument applies to probabilities.
Taking complements gives an equivalent statement: the probability that all desired events hold is at least one minus the sum of their failure probabilities. A negative lower bound can be replaced by zero.
For multiple tests, let each event mean a false rejection of a true null hypothesis. If the error allowances across the planned family sum to no more than alpha, the probability of any false rejection is bounded by alpha. The individual tests must actually respect those allowances; the inequality cannot repair invalid tests.

Three practical examples

An evaluator considers three distinct alert failures in an AI monitoring workflow. Individual probability bounds can be added to bound the chance of encountering at least one failure on a defined run.
A research team constructs several confidence intervals for model-performance differences. It allocates a small noncoverage probability to each interval so their combined noncoverage bound remains within its chosen allowance.
A product experiment compares several planned outcomes. The analysis accounts for the complete comparison family instead of interpreting every unadjusted threshold as though only one outcome had been examined.

Think of it this way

Imagine counting people carrying umbrellas and people wearing raincoats. Adding the two counts can count the same person twice. It therefore bounds, rather than necessarily equals, the number carrying an umbrella or wearing a raincoat. Statistical failure events can overlap in the same way.

Worked example

Consider an invented workflow with three failure events: A, B, and C. Suppose their probabilities for one run are 0.02, 0.03, and 0.01. The inequality gives P(A or B or C) no greater than 0.06, so the probability of avoiding all three is at least 0.94.
Different overlap structures produce different exact answers. If the events are mutually exclusive, the union probability is exactly 0.06. If A and C are both contained in B, the union is simply B, with probability 0.03. Both possibilities have the same individual probabilities and satisfy the bound.
If, instead, all three events are mutually independent, the probability of avoiding them is 0.98 times 0.97 times 0.99, which equals 0.941094. The probability of at least one failure is then 0.058906. Independence permits this product calculation; it is unnecessary for the upper bound of 0.06.
Now consider four valid statistical tests with an overall false-rejection allowance of 0.05. Equal allocation gives each test 0.0125. The total allowance is 4 times 0.0125 = 0.05, bounding the familywise error probability by five percent. If all four null hypotheses are true and the rejection events are mutually independent with probability exactly 0.0125 each, the exact probability is 1 minus 0.9875 to the fourth power, approximately 0.049070. The inequality remains valid when those independence assumptions do not hold.

Illustrative case study

Fictional company Birch Evaluation reports twenty performance checks for a document assistant. Its initial draft highlights whichever checks pass an ordinary significance threshold, without identifying how many were examined.
The research lead defines the full comparison family and records the intended overall error allowance before the next evaluation. The team uses valid individual tests with allowances that add to the family total. Its report explains the distinction between the mathematical bound, the observed effect sizes, and practical usefulness. No claim is made that adjustment removes measurement bias or proves the assistant is suitable for deployment. This scenario is illustrative, not an actual company outcome.

Common misconceptions and mistakes

The sum is not usually the exact union probability. Overlap determines the difference.
Independence is not required for the inequality. It is required for the simple product-of-complements calculation used in the independent example.
A five-percent familywise error bound is not a five-percent probability that a particular rejected hypothesis is true. The bound concerns the testing procedure across repetitions under the relevant true null hypotheses.
An allowance computed for four comparisons does not automatically cover a larger, selectively reported search. The family and the validity of the individual procedures matter.

Business use

The inequality helps teams explain how several individually small risks or statistical error allowances combine. Document the events, the population or experiment to which their probabilities apply, and whether those probabilities are known, estimated, or themselves bounded. Avoid presenting uncertain operational estimates as certified guarantees.

Check your understanding

What bounds three event probabilities of 0.02, 0.03, and 0.01?

Answer: Their union is at most 0.06.

Could the actual union be 0.03?

Answer: Yes, with the nesting described above.

Does correlation invalidate the bound?

Answer: No. Dependence affects exact probabilities, not this inequality.

Related terms

Familywise error rate
Multiple comparisons
Independence
Union bound
Boole's inequality
Bonferroni correction
simultaneous confidence intervals

Learn next

Compare familywise error control with false discovery rate control. Then study inclusion-exclusion to see how information about event intersections can sharpen probability calculations.

False discovery rate

Sources

David C. Howell, Statistical Methods for Psychology glossary, Bonferroni inequality: https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html . NIST/SEMATECH, Bonferroni's Method: https://www.itl.nist.gov/div898/handbook/prc/section4/prc473.htm . Numerical examples and company scenario are original illustrations.

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