  It is known that the coupling lemma provides a useful tool in the study of
probability theory and its related areas. It describes the relation between the
variational distance of two probability distributions and the probability that
outcomes from the two random experiments associated with each distribution are
not identical. In this paper, the failure probability interpretation problem
that has been presented by Yuen and Hirota is discussed from the viewpoint of
the application of the coupling lemma. First, we introduce the coupling lemma,
and investigate properties of it. Next, it is shown that the claims for this
problem in the literatures are justified by using the coupling lemma.
Consequently, we see that the failure probability interpretation is not
adequate in the security analysis of quantum key distribution.
