Are all fast radio bursts repeating sources?

We present Monte Carlo simulations of a cosmological population of repeating fast radio burst (FRB) sources whose comoving density follows the cosmic star formation rate history. We assume a power-law model for the intrinsic energy distribution for each repeating FRB source located at a randomly chosen position in the sky and simulate their dispersion measures and propagation effects along the chosen lines-of-sight to various telescopes. In one scenario, an exponential distribution for the intrinsic wait times between pulses is chosen, and in a second scenario we model the observed pulse arrival times to follow a Weibull distribution. For both models we determine whether the FRB source would be deemed a repeater based on the telescope sensitivity and time spent on follow-up observations. We are unable to rule out the existence of a single FRB population based on comparisons of our simulations with the longest FRB follow-up observations performed. We however rule out the possibility of FRBs 171020 and 010724 repeating with the same rate statistics as FRB 121102 and also constrain the slope of a power-law fit to the FRB energy distribution to be -2.0 < γ < -1.0. All-sky simulations of repeating FRB sources imply that the detection of singular events correspond to the bright tail end of the adopted energy distribution due to the combination of the increase in volume probed with distance, and the position of the burst in the telescope beam.