The problem of nding the best answers to a query quickly, rather than nding all answers, is of increasing importance as relational databases are applied in multimedia and decision-support domains. An approach to e ciently answering such \Top N" queries is to augment the query with an additional selection that prunes away the unwanted portion of the answer set. The risk is that if the selection returns fewer than the desired number of answers, the execution must be restarted (with a less selective lter). We propose a new, probabilistic approach to query optimization that quanti es this risk and seeks to minimize overall cost including the cost of possible restarts. We also present an extensive experimental study to demonstrate that probabilistic Top N query optimization can signi cantly reduce the average query execution time with relatively modest increases in the optimization time.