Medium Access Control (MAC) policies, in which the scheduling time slots are allocated irrespectively of the underline topology (topology-unaware), are suitable for ad-hoc networks, where nodes can enter, leave or move inside the network at any time. Such topology-unaware MAC policies have been proposed in the past and it has been shown that under certain conditions the system throughput achieved under the Probabilistic Policy is higher than that achieved under the Deterministic Policy. However, as it is shown in this paper, the power consumption under the Probabilistic Policy is higher than that under the Deterministic Policy on a per frame basis. On the other hand, the power consumed per successful transmission is possible to be smaller under the Probabilistic Policy under certain conditions that are established here. For delay constrained packets, the power consumed per successful transmission may be higher when accounting for the wasted power previously consumed by obsolete packets. The conditions under which the power consumed per successful transmission of delay constrained packets under the Probabilistic Policy is smaller than that under the Deterministic Policy, are also established in this work. It is shown analytically that when the system throughput is maximized, the power consumed is close to the minimum. Simulation results support the claims and the expectations of the aforementioned analysis.