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We study uncoordinated multiple access using the orthogonal frequency division multiple access (OFDMA) technique where multiple users independently and randomly pick a sub-carrier from a set of sub-carriers for transmission in each slot. Signals transmitted on distinct sub-carriers are orthogonal to each other while signals transmitted by different users on the same sub-carrier interfere with each other. We study average throughput and outage probability performance of such a system in the asymptotic limit of large numbers of sub-carriers and users while keeping the ratio of number of users to sub-carriers fixed, for non-fading, flat fading, and frequency-selective fading additive white Gaussian noise channels. The performance is compared to that of a system in which all users spread their signal over all the sub-carriers as in a code division multiple access (CDMA) system. Our results show that at high signal-to-noise ratio (SNR), transmission without spreading results in higher throughput but requires more retransmissions on the average. At low SNRs, the throughput performance is comparable but the system with spreading requires smaller average number of retransmissions.