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We present a statistical approximate solution of the bound, non-hierarchical three-body problem, and extend it to a general analysis of encounters between hard binary systems and single stars. 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. Here, 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 use the solution of the bound, non-hierarchical three-body problem to find the walker's transition probabilities, which we generalise to situations in which tidal interactions are important. Besides tides, any dissipative process may be incorporated into the random walk model, as it is completely general. Our approximate solution can reproduce the results of the extensive body of past numerical simulations, and can account for different environments and different dissipative effects. Therefore, this model can effectively replace the need for direct few-body integrations for the study of binary-single encounters in any environment. Furthermore, it allows for a simply inclusion of dissipative forces typically not accounted for in full N-body integration schemes.