Bubbles bursting at the ocean surface are an important source of sea-spray aerosol. Indeed, a bubble bursting at the surface of a liquid produces a jet that then breaks up leading to several droplets. Here we simulate the bursting of a single bubble by numerical simulation of the axisymmetric two phases air-water Navier-Stokes equations, in the presence of small initial perturbations in the liquid phase. We describe the impact of this random noise on the size and velocity of the drops for a wide range of control parameters. We then demonstrate how the process of jet drop production is robust to perturbations in the initial conditions, which allows an accurate prediction of the size and speed of jet droplets over the entire parameter space. We propose a mechanism to explain why the existence of some droplets, called "primary droplets" are robust to initial conditions; and discuss why the generation of secondary droplets, much smaller than the primary ones, are sensitive to the same initial conditions. The production of these secondary droplets is sensitive to perturbations in the initial conditions, through a mechanism of pinch-off escape. This can explain previously published experimental observations of bi-modal distributions of jet drops.