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In this paper, we propose a general framework to design efficient polynomial-time approximation schemes (EPTASs) for fundamental stochastic combinatorial optimization problems. Technically speaking, our approach relies on presenting tailor-made reductions to a newly introduced multidimensional Santa Claus problem. Even though the single-dimensional version of this problem is already known to be APX-Hard, we prove that an EPTAS can be designed for a constant number of machines and dimensions, which hold for each of our applications. To demonstrate the versatility of our framework, we first study selection-stopping settings to derive an EPTAS for the free-order prophets problem and for its cost-driven generalization, Pandora’s box with commitment. These results constitute the first approximation schemes in the nonadaptive setting and improve on known inefficient polynomial-time approximation schemes (PTASs) for their adaptive variants. Next, turning our attention to stochastic probing problems, we obtain an EPTAS for the adaptive ProbeMax problem as well as for its nonadaptive counterpart. In both cases, state-of-the-art approximability results have been inefficient PTASs (Chen et al. [25], Fu et al. [35]). Funding: This work was supported by the National Science Foundation, Division of Computing and Communication Foundations [Grants 2327010 and 2440113].