Suppose an astronomer is equipped with a device capable of detecting emissions (whether they be electromagnetic, gravitational, or neutrino) from transient sources distributed throughout the cosmos. Because of source rate density evolution and variation of cosmological volume elements, the sources first detected when the machine is switched on are likely to be ones in the high-redshift Universe; as observation time increases, rarer, more local, events will be found. We characterize the observer's evolving record of events in terms of a 'probability event horizon' (PEH), converging on the observer from great distances at enormous speed, and illustrate it by simulating neutron star (NS) birth events distributed throughout the cosmos. As an initial application of the concept, we determine the approach of this horizon for gamma-ray bursts (GRBs) by fitting to redshift data. The event rates required to fit the model are consistent with the proposed link between core-collapse supernovae (SNe) and a largely undetected population of faint GRBs.