This paper presents a formal characterisation of safety and liveness
properties \`a la Alpern and Schneider for fully probabilistic systems. As for
the classical setting, it is established that any (probabilistic tree) property
is equivalent to a conjunction of a safety and liveness property. A simple
algorithm is provided to obtain such property decomposition for flat
probabilistic CTL (PCTL). A safe fragment of PCTL is identified that provides a
sound and complete characterisation of safety properties. For liveness
properties, we provide two PCTL fragments, a sound and a complete one. We show
that safety properties only have finite counterexamples, whereas liveness
properties have none. We compare our characterisation for qualitative
properties with the one for branching time properties by Manolios and Trefler,
and present sound and complete PCTL fragments for characterising the notions of
strong safety and absolute liveness coined by Sistla