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Observationally, fast radio bursts (FRBs) can be divided into repeating and apparently non-repeating (one-off) ones. It is unclear whether all FRBs repeat and whether there are genuine non-repeating FRBs. We attempt to address these questions using Monte Carlo simulations. We define a parameter $T_c$ at which the accumulated number of non-repeating sources becomes comparable to the total number of the repeating sources, which is a good proxy to denote the intrinsic repeater fraction among FRBs. Assuming that both types of sources exist and that their burst energies follow power law distributions, we investigate how the {\em observed} repeater fraction evolves with time for different parameters. If the lifetime of repeaters is sufficiently long so that the evolutionary effect can be neglected within the observational time span, unless $T_c \rightarrow \infty$ (i.e. there is no genuine non-repeating FRB source) the observed repeater fraction should increase with time first, reach a peak, and then decline. The peak time $T_p$ and the peak fraction $F_{\rm r,obs,p}$ depend on $T_c$ and other repeating rate parameters. With the current data, we pose a lower limit $T_c > 0.1$ d for reasonable parameter values. We predict that future continuous monitoring of FRBs with CHIME or similar wide-field radio telescopes would obtain an $F_{\rm r,obs}$ less than $0.04$. The detection of a smaller peak value $F_{\rm r,obs,p}<0.04$ in the near future would disfavor the ansatz that "all FRB sources repeat".