In this paper we estimate the probability that cold electrons can be accelerated by an ambient electric field into the runaway regime, and discuss the implications for negative streamer formation. The study is motivated by the discovery of ms duration bursts of γ-rays from the atmosphere above thunderstorms, the so-called Terrestrial Gamma-Ray Flashes. The radiation is thought to be bremsstrahlung from energetic (MeV) electrons accelerated in a thunderstorm discharge. The observation goes against conventional wisdom that discharges in air are carried by electrons with energies below a few tens of eV. Instead the relativistic runaway electron discharge has been proposed which requires a lower threshold electric field; however, seed electrons must be born with energies in the runaway regime. In this work we study the fundamental problem of electron acceleration in a conventional discharge and the conditions on the electric field for the acceleration of electrons into the runaway regime. We use particle codes to describe the process of stochastic acceleration and introduce a novel technique that improves the statistics of the relatively few electrons that reach high energies. The calculation of probabilities for electrons to reach energies in the runaway regime shows that even with modest fields, electrons can be energized in negative streamer tips into the runaway regime, creating a beamed distribution in front of the streamer that affects its propagation. The results reported here suggest that theories of negative streamers and spark propagation should be reexamined with an improved characterization of the kinetic effects of electrons.