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Non-orthogonal multiple access (NOMA) has been considered as a promising solution for improving the spectrum efficiency of next-generation wireless networks. In this paper, the performance of a p-persistent slotted ALOHA system in support of NOMA transmissions is investigated. Specifically, wireless users can choose to use high or low power for data transmissions with certain probabilities. To achieve the maximum network throughput, an analytical framework is developed to analyze the successful transmission probability of NOMA and long term average throughput of users involved in the non-orthogonal transmissions. The feasible region of the maximum number of concurrent users using high and low power to ensure successful NOMA transmissions are quantified. Based on the analysis, an algorithm is proposed to find the optimal transmission probabilities for users to choose high and low power to achieve the maximum system throughput. In addition, the impact of power settings on the network performance is further investigated. Simulations are conducted to validate the analysis.