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The design of an efficient medium access control (MAC) for ad hoc networks is challenging. Topology-unaware time-division multiple-access-based schemes, suitable for ad hoc networks, that guarantee a minimum throughput, have already been proposed. These schemes consider a deterministic policy for the utilization of the assigned scheduling time slots that never utilizes nonassigned slots although in such slots collision-free transmissions are possible even under heavy traffic conditions. A simple probabilistic policy, capable of utilizing the nonassigned slots according to an access probability, fixed for all users in the network, is introduced and analyzed here. The conditions under which the system throughput under the probabilistic policy is higher than that under the deterministic policy are derived analytically. Further analysis of the system throughput is shown to be difficult or impossible for the general case and certain approximations have been considered whose accuracy is also investigated. The approximate analysis determines the value for the access probability that maximizes the system throughput, as well as simplified lower and upper bounds that depend only on a topology density metric. Simulation results demonstrate the comparative advantage of the probabilistic policy over the deterministic policy and show that the approximate analysis successfully determines the range of values for the access probability for which the system throughput under the probabilistic policy is not only higher than that under the deterministic policy, but it is also close to the maximum.