Stochastic discrete-time systems, i.e., discrete-time dynamic systems subject to stochastic disturbances, are an
essential modelling tool for many engineering systems, and reach-avoid analysis is able to guarantee safety (i.e., via avoiding unsafe sets)
and progress (i.e., via reaching target sets). In this paper we study the reach-avoid problem of stochastic
discrete-time systems over open (i.e., not bounded a priori) time horizons. The stochastic discrete-time system of interest
is modeled by iterative polynomial maps with stochastic disturbances, and the problem addressed is to effectively compute
an inner approximation of its \textit{p}-reach-avoid set. The \textit{p}-reach-avoid set collects those initial states that give
rise to a bundle of trajectories which with probability being larger than \textit{p} eventually hits a designated set of target
states while remaining inside a set of safe states before the first hit. The computation of the \textit{p}-reach-avoid set is
first reduced to the computation of a corresponding strict \textit{p}-super-level set and is then inner-approximated by
solving a semi-definite programming problem obtained from a relaxation of the definition of the super-level set.
Two examples demonstrate the proposed approach.