In this paper, we investigate the asymptotic dynamics of inflationary cosmological models that are based in scalar-tensor theories of gravity. Our main aim is to explore the global structure of the phase space in the framework of single-field inflation models. For this purpose, we emphasize the adequate choice of the variables of the phase space. Our results indicate that, although single-field inflation is generic in the sense that the corresponding critical point in the phase space exists for a wide class of potentials, along given phase space orbits - representing potential cosmic histories - the occurrence of the inflationary stage is rather dependent on the initial conditions. We have been able to give quantitative estimates of the relative probability (RP) for initial conditions leading to slow-roll inflation. For the nonminimal-coupling model with the φ2 potential, our rough estimates yield an almost vanishing relative probability: 10-13% RP 10-8%. These bonds are greatly improved in the scalar-tensor models, including the Brans-Dicke theory, where the relative probability 1% RP≤100%. Hence, slow-roll inflation is indeed a natural stage of the cosmic expansion in Brans-Dicke models of inflation. It is confirmed as well that the dynamics of vacuum Brans-Dicke theories with arbitrary potentials are nonchaotic.