  Probabilistic powerdomain in domain theory plays an important role in
modeling the semantics of nondeterministic functional programming languages
with probabilistic choice. In this paper, we extend the notion of powerdomain
to directed spaces, which is equivalent to the notion of the T0
monotone-determined space [4]. We construct the probabilistic powerspace of the
directed space, which is defined as a free directed space-cone. In addition,
the relationships between our construction and classical probabilistic
powerdomain are studied.
