Bonferroni’s inequality is a conservative way to account for multiplicity in testing: the combined significance level (chance of a false positive) of a collection of tests is at most the sum of their significance levels, even when the tests have arbitrary statistical dependence (provided all the tests considered are included in the sum--Bonferroni’s inequality does not protect against false positives if the tests are chosen after looking at the data). The chance of an arbitrary union of events is at most the sum of their separate chances. However, because it gives universal protection, Bonferroni’s inequality can be quite conservative in some circumstances.