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There is a growing interest in the use of wireless ad hoc sensor networks to monitor, detect, and track the movement of specified targets in a geographic region. A common concern in the deployment of such networks is whether or not a target can pass or intrude a sensor field without being detected. Recent papers in the literature have defined a measure called exposure to quantify the likelihood of a target passing through a sensor field without being detected. These papers differ in the definition of exposure. Also, the existing definitions of exposure are indicators of the likelihood of intrusion and not a direct measure of it. We directly work with probability of detection instead of its indicators. We also probabilistically account for the presence of noise in the sensor readings. In the presence of noise, there is a trade off between the probability of a target passing through a sensor field without detection and the false alarm probability (i.e., the probability of falsely detecting the presence of target). We analytically characterize this trade off and illustrate the trade off for example deployments. We also introduce a variant of the traversal problem called unauthorized monitoring. This problem is of interest if sensor fields are used to secure an asset. We analytically characterize the probability of detecting unauthorized monitoring. We also show a trade off between this probability and the corresponding false alarm probability in the presence of noise.