Abstract: | Three-body gravitational interactions are a prototypical example of a highly-chaotic dynamical system. A binary-single encounter consists of a single star being scattered on a tight binary. Any such encounter terminates when one of the three stars is ejected to infinity, leaving behind a remnant binary; the problem of binary-single star-scattering consists of finding the probability distribution of the orbital parameters of the remnant binary, as a function of the total energy and the total angular momentum. We model the encounter as a series of close, non-hierarchical, triple approaches, interspersed with hierarchical phases, in which the system consists of an inner binary and a star that orbits it – this turns the evolution of the entire encounter to a random walk between consecutive hierarchical phases. We generalise the model to account for dissipative processes, such as tides and gravitational-wave emission, allowing one, *inter alia*, to compute the probability of forming gravitational-wave sources dynamically, via such a process. Our solution can reproduce the results of the extensive body of past numerical simulations. |