The aim is to understand the statistics of various deterministic dynamics of n hard spheres of diameter a with random initial data in the Boltzmann-Grad scaling as a tends to zero and n tends to infinity. Details are given for a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with background particles, which do not interact among each other. In the Boltzmann-Grad scaling, we derive the validity of a linear Boltzmann equation for arbitrary long times under moderate assumptions on higher moments of the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. The convergence of the empiric dynamics to the Boltzmann dynamics is shown using semigroup methods to analyse Kolmogorov equations for associated probability measures on collision histories. Based on work with George Stone and Florian Theil.