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In this paper we consider the following problem for ad hoc networks: assume that n nodes are distributed in a d-dimensional region, with 1≤d≤3, and assume that all the nodes have the same transmitting range r; how large must r be to ensure that the resulting network is strongly connected? We study this problem by means of a probabilistic approach, and we establish lower and upper bounds on the probability of connectedness. For the one-dimensional case, these bounds allow us to determine a suitable magnitude of r for a given number of nodes and displacement region size. In an alternate formulation, the bounds allow us to calculate how many nodes must be distributed should the transmitting range be fixed. Finally, we investigate the required magnitude of r in the two- and three-dimensional cases through simulation. Based on the bounds provided and on the simulation analysis, we conclude that, as compared to the deterministic case, a probabilistic solution to this range assignment problem achieves substantial energy savings. A number of other potential uses for our analyses are discussed as well