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For the classical maximum coverage problem, the greedy algorithm achieves a worst-case 1-1/e approximation, which is optimal unless P = NP. The notion of coverage appears in a wide range of optimization tasks, where empirical evaluations indicate approximation ratios close to 1 for the greedy algorithm on real data. Random models have provided average-case justifications for the empirical performance of many well-known algorithms, but little is known about the average-case performance of greedy for maximum coverage. We analyze the expected approximation ratio of the greedy algorithm in a random model, which we call the left-regular random model. We first show that, for all parameter settings of this model, the expected approximation ratio of the greedy algorithm improves by a constant over its worst-case 1-1/e guarantee. We then identify two simple conditions, either of which ensures that the expected approximation ratio is close to 1 for sufficiently large graphs. Finally, we show that there is a regime where greedy does not achieve an expected approximation better than 0.94. To obtain these results, we develop analytical tools, including a novel application of the differential equation method and a connection to maximum matching in Erdős-Rényi graphs, which may be of independent interest for other random models.