 The reasoning behind this is that when formula_43 is small the procedure is more likely to produce an error (i.e., an interval that does not cover the true value) than when formula_43 is large, and the distribution of formula_43 itself is independent of formula_25.That is, not the over-all error probability should be reported but the conditional probability given the knowledge one has on the number of background events in the sample.This conditional probability is

which correspond to the above definition of CLs.