Sensor networks hold the promise of many new applications in the area of monitoring and control. Sensor positioning is a fundamental and crucial issue for network operation and management. The motivation for this paper is to explore the probabilistic behavior for the localization problem in sensor networks. Due to the peculiarity of sensor network, its localization behavior needs to be investigated to obtain some general principles by probability theory. The position estimation error and the node connectivity issues are analyzed. First, it is proved that the linear transform of position estimation for node localization is distributed by χ\+2. Secondly, the node connectivity and the number of nodes in a deployed area are distributed by the Poisson, if a sensor network is configured in a uniform fashion over the whole field. Some connectivity requirements should be met to accomplish successfully positioning according to the result of simulation experiments. The model of position errors can be used to evaluate the impact of a location discovery algorithm on subsequent tasks in a multi-hop ad-hoc sensor network. Understanding of behavior characteristics is of importance for implementing localization process and evaluating localization methods.