Random encounters in space are central to describing diffusion-limited reactions, animal foraging, search processes, and many other situations in nature. These encounters, however, are often constrained by the capacity of the searcher to detect and/or recognize its target. This can be due to limited binding and perception abilities of the searcher or hiding and avoiding mechanisms used by the target. Hence detection failure upon passage over the target location turns the process into an n-passage problem, with n being random. Here we provide a general description of this detection problem for arbitrary dimensions and arbitrary detection constraints. The mean detection time (MDT) for a random searcher embedded in a sea of homogeneously distributed targets is obtained as a function of the target density ρ, the size domain L, and the effective detection distance a. While the scaling with ρ and L is found to be universal and equivalent to that found for the corresponding first-passage problem, the scaling of the MDT on a depends on the specific detection mechanism considered.