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We consider an optimal stopping problem with a discrete time stochastic process where a criterion is a threshold probability. We first obtain the fundamental characterization of an optimal value and an optimal stopping time as the result of the classical optimal stopping problem, but the optimal value and the optimal stopping time depend upon a threshold value. We also give the properties of the optimal value with respect to threshold value. These are applied to a secretary problem, a parkingdoi:10.15807/jorsj.46.342 fatcat:edar6arnnjfi7hooz4hudfledi