Existing distributed opportunistic spectrum management schemes do not consider the inability of today's cognitive transceivers to measure interference at primary receivers. Consequently, optimizing the constrained cognitive radio network performance based only on local interference measurements at cognitive senders does not lead to truly optimal performance due to hidden and exposed primary senders. In this paper, we present the rate-adaptive probabilistic medium access protocol (RAP-MAC) for opportunistic spectrum management. In contrast to existing techniques, our approach addresses the unavoidable inaccuracy in spectrum sensing via a probabilistic spectrum access strategy. Furthermore, RAP-MAC allows multiple cognitive flows to fairly share the available bandwidth without explicit coordination by adopting a non-greedy transmission approach that prevents a single cognitive transmitter from monopolizing an available spectral opportunity. We analytically formulate the cognitive user performance optimization problem as a mixed integer non-linear programming to derive the optimal parameter values. Packet-level simulations show that RAP-MAC achieves between 65% to 119.5% higher goodput with significantly better fairness characteristics compared to existing greedy approaches that overlook the spectrum sensing limitations.