  Given a finite sequence of events and a well-defined notion of events being
interesting, the Odds-theorem (Bruss (2000)) gives an online strategy to stop
on the last interesting event. It is optimal for independent events. Here we
study questions in how far optimal win probabilities mirror monotonicity
properties of the underlying sequence of probabilities of events. We make these
questions precise, motivate them, and then give complete answers. This note,
concentrating on the original Odds-theorem, is elementary, and the answers are
hoped to be of interest. We include several applications.
