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The difference ($\\mathbf{D}$) between the weighted totals of the histograms is, in a sense, refined by the earth mover's distance ($\\mathbf{EMD}$), which measures the amount of work required to equalize the histograms. We were recently surprised, however, by how little $\\mathbf{EMD}$ actually does refine $\\mathbf{D}$ in certain real-world applications, which led to the main problem in this paper: what is the probability that $\\mathbf{EMD} = |\\mathbf{D}|$? 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