This paper presents MAPS - a personalized Multi-Attribute Probabilistic Selection framework - to estimate the probability of an item being a user's best choice and rank the items accordingly. The MAPS framework makes three original contributions in this paper. First, we capture the inter-attribute tradeoff by a visual angle model which maps multi-attribute items into points (stars) in a multidimensional space (sky). Second, we model the inter-item competition using the dominating areas of the stars. Third, we capture the user's personal preferences by a density function learned from his/her history. The MAPS framework carefully combines all three factors to estimate the probability of an item being a user's best choice, and produces a personalized ranking accordingly. We evaluate the accuracy of MAPS through extensive simulations. The results show that MAPS significantly outperforms existing multi-attribute ranking algorithms.