This paper presents a new model for online decision making. Motivated by the health care delivery application of dynamically allocating patients to procedure rooms in outpatient procedure centers, the online stochastic extensible bin packing problem is described. The objective is to minimize the combined costs of opening procedure rooms and utilizing overtime to complete a day's procedures. The dynamic patient allocation decisions are made in an uncertain environment where the number of patients scheduled and the procedure durations are not known in advance. The resulting optimization model's tractability focuses the paper's attention on approximation methods and a special case that is amenable to decomposition-based solution methods. Theoretical performance guarantees are presented for list-based approximation methods as well as an approximation that is common in practice where procedure rooms are reserved for patient groups in advance. Numerical results based on a real outpatient procedure center demonstrate the favorable results of the list-based approximations based on their average and worst case performances. Further, the numerical experiments show that the policy of reserving procedure rooms for patient groups in advance can perform very poorly. These results are contrary to common practice and favor alternative, and still easy-to-implement, policies.