The inactivity time, also known as reversed residual life, has been a
topic of increasing interest in the literature. In this investigation, based
on the comparison of inactivity times of two devices, we introduce
and study a new estimate of the probability of the inactivity time of
one device exceeding that of another device. The problem studied in
this paper is important for engineers and system designers. It would
enable them to compare the inactivity times of the products and,
hence to design better products. Several properties of this probability
are established. Connections between the target probability and the
reversed hazard rates of the two devices are established. In addition,
some of the reliability properties of the new concept are investigated
extending the well-known probability ordering. Finally, to illustrate the
introduced concepts, many examples and applications in the context
of reliability theory are included.