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This paper independently derives the probability of any pair of uniformly-distributed nodes to be within transmission range of each other in a square-shaped area. It then explores, via simulation, some new applications of this expression. The applications are relevant for scenarios where node mobility is governed by the popular random walk or random waypoint mobility models (RWkMM and RWPMM). Under the RWPMM with pausing, at any time, some nodes will be mobile and some stationary. The positions of mobile nodes are drawn from a nonuniform distribution, while a uniform distribution applies to the stationary nodes. In various forms of the RWkMM, the node spatial distribution is uniform in its steady state. The studied applications include calculating the expected node degree and the node isolation probability. Simulation results show that the considered model is able to predict these connectivity-related properties near-perfectly under a paused RWPMM and with all mobility scenarios under the RWk with reflection model. With the RWPMM, the accuracy decreases as the fraction of time the nodes spend moving increases. However, it is still generally better than the simple pir2/A disk-covering model, which is often employed for calculating network connectivity-related properties in ad hoc networks. Further application of the considered methods is exemplified by calculation of an accurate upper bound on the per-node transmission capacity for contention-based networks, when the nodes are uniformly distributed.