A class of infinite-state Markov decision processes generated by stateless pushdown automata is considered. This class corresponds to 1 1/2-player games over graphs generated by BPA systems or (equivalently) 1-exit recursive state machines. An extended reachability objective is specified by two sets \(S\) and \(T\) of safe and terminal stack configurations, where the membership to \(S\) and \(T\) depends just on the top-of-the-stack symbol. The question is whether there is a suitable strategy such that the probability of hitting a terminal configuration by a path leading only through safe configurations is equal to (or different from) a given \(x\) in \((0,1)\). It is shown that the qualitative extended reachability problem is decidable in polynomial time, and that the set of all configurations for which there is a winning strategy is effectively regular. More precisely, this set can be represented by a deterministic finite-state automaton with a fixed number of control states. This result is a generalization of a recent theorem by Etessami and Yannakakis which says that the qualitative termination for 1-exit RMDPs (which exactly correspond to our 1 1/2-player BPA games) is decidable in polynomial time. Interestingly, the properties of winning strategies for the extended reachability objectives are quite different from the ones for termination, and new observations are needed to obtain the result. As an application, the EXPTIME-completeness of the model-checking problem is obtained for 1 1/2-player BPA games and qualitative PCTL formulae.