Consider a system of polynomial equations with integer coefficients. For each prime number p, we may reduce modulo p to obtain a system of polynomials over the field of p elements, and then count the number of solutions. It is generally difficult to describe this count as an exact function of p, so instead we take a statistical point of view, treating the count as a random variable and asking for its limiting distribution as we consider increasing large ranges of primes. Conjecturally, this distribution can be described in terms of the conjugacy classes of a certain compact Lie group. We illustrate this in three examples: polynomials in one variable, where everything is explained in terms of Galois theory by the Chebotarev density theorem; elliptic curves, where the dichotomy of outcomes is predicted by the recently proved Sato-Tate conjecture; and hyperelliptic curves of genus 2, where even the conjectural list of outcomes was only found still more recently.