This article deals with localization probability in a network of randomly distributed communication nodes contained in a bounded domain. A fraction of the nodes denoted as L-nodes are assumed to have localization information while the rest of the nodes denoted as NL nodes do not. The basic model assumes each node has a certain radio coverage within which it can make relative distance measurements. We model both the case radio coverage is fixed and the case radio coverage is determined by signal strength measurements in a Log-Normal Shadowing environment. We apply the probabilistic method to determine the probability of NL-node localization as a function of the coverage area to domain area ratio and the density of L-nodes. We establish analytical expressions for this probability and the transition thresholds with respect to key parameters whereby marked change in the probability behavior is observed. The theoretical results presented in the article are supported by simulations.