Recent discussion in the public sphere about algorithmic cl assification has involved tension between competing notions of what it means for a probabilistic class ification to be fair to different groups. We formalize three fairness conditions that lie at the heart of these debates, and we prove that except in highly constrained special cases, there is no method that can satis fy these three conditions simultaneously. Moreover, even satisfying all three conditions approximat ely requires that the data lie in an approximate version of one of the constrained special cases identified by our theorem. These results suggest some of the ways in which key notions of fairness are incompatible with each other, and hence provide a framework for thinking about the trade-offs between them.