Summary: In an election in which each voter ranks all of the candidates, we consider the head-to-head results between each pair of candidates and form a labeled directed graph, called the margin graph, which contains the margin of victory of each candidate over each of the other candidates. A central issue in developing voting methods is that there can be cycles in this graph, where candidate \(\mathsf{A}\) defeats candidate \(\mathsf{B},\mathsf{B}\) defeats \(\mathsf{C}\), and \(\mathsf{C}\) defeats \(\mathsf{A}\). It is known that such cycles are unlikely to occur. Under the impartial culture assumption, in a random election with three candidates and a very large number of voters there is a 91.23% chance of avoiding a cycle. By studying the geometry of the space of random elections, we give a mathematical explanation of why this is the case. Our main result is that margin graphs that are more cyclic in a certain precise sense are less likely to occur. This connects the probabilistic study of voting methods to Zwicker’s analysis of Condorcet’s paradox in terms of cycles and cuts.