A vast amount of literature in social choice theory deals with quantifying the probability of certain election outcomes. This is in particular the case for so-called ``voting paradoxes''. One way of computing the probability of a specific voting situation is via counting integer points in associated polyhedra. Here, Ehrhart theory can help, but unfortunately the dimension and complexity of the involved polyhedra grows rapidly with the number of candidates. However, if we exploit available polyhedral symmetries, some computations become possible that otherwise seem infeasible. We exemplarily show this in two very well known examples: Condorcet's paradox and in plurality voting vs plurality runoff.