The occurrence of false positive detections in presence-absence data, even when they occur infrequently, can lead to severe bias when estimating species occupancy patterns. Building upon previous efforts to account for this source of observational error, we established a general framework to model false positives in occupancy studies and extend existing modeling approaches to encompass a broader range of sampling designs. Specifically, we identified three common sampling designs that are likely to cover most scenarios encountered by researchers. The different designs all included ambiguous detections, as well as some known-truth data, but their modeling differed in the level of the model hierarchy at which the known-truth information was incorporated (site level or observation level). For each model, we provide the likelihood, as well as R and BUGS code needed for implementation. We also establish a clear terminology and provide guidance to help choosing the most appropriate design and modeling approach.