Non-orthogonal multiple access (NOMA) is a major candidate technology for the future radio networks due to its potential to have high spectral efficiency and to support massive connections. It is also an important advantage for the NOMA to be very compatible with the other physical layer techniques. The relay-aided-NOMA which consists of the interplay between cooperative communication and NOMA, offers very large coverage area besides the high spectral efficiency and the massive connectivity. Hence, the relay-aided-NOMA has great recent attention. Most of these studies are devoted to reveal outage probability and the overall system capacity performances of the systems. However, the bit error performances have been ignored. In this paper, theoretical end-to-end (E2E) average bit error probability (ABEP) expressions are derived for the relay-aided-NOMA using a relay with the decode-forward (DF) protocol. The derived expressions are validated through Monte Carlo simulations. The E2E error performance of the system are examined for different system parameters. Moreover, by considering user fairness in terms of error performance, it is discussed how the optimum power allocation should be done. And finally, a sub-optimum power allocation algorithm is presented according to the proportional fairness index.