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We present the induced generalized probabilistic ordered weighted averaging (IGPOWA) operator. It is a new aggregation operator that unifies the probability and the OWA operator in the same formulation considering the degree of importance that each concept has in the formulation. Moreover, it also uses order-inducing variables in order to assess complex reordering processes. Furthermore, it also uses generalized means providing a more general framework that includes a wide range of particular cases such as the probabilistic aggregation, the maximum probabilistic aggregation, the minimum probabilistic aggregation, the arithmetic OWA and the arithmetic probabilistic aggregation. We further generalize this approach by using quasi-arithmetic means obtaining the quasi-arithmetic IPOWA (Quasi-IPOWA) operator. We also present an application of the new approach in a decision making problem.