  We define Almost Sure Productivity (ASP), a probabilistic generalization of
the productivity condition for coinductively defined structures. Intuitively, a
probabilistic coinductive stream or tree is ASP if it produces infinitely many
outputs with probability 1. Formally, we define almost sure productivity using
a final coalgebra semantics of programs inspired from Kerstan and K\"onig.
Then, we introduce a core language for probabilistic streams and trees, and
provide two approaches to verify ASP: a sufficient syntactic criterion, and a
reduction to model-checking pCTL* formulas on probabilistic pushdown automata.
The reduction shows that ASP is decidable for our core language.
