Lightning streamers and leaders need thermal electrons to initiate, but free electrons are extremely rare in the wet air of a thundercloud. Here we analyze the probabilities that high electron densities occur in extensive air showers.We argue that relevant air showers are created by cosmic particles with energies between 1015 and 1017 eV impinging onto our atmosphere.We simulate a large number of air showers and perform a stochastic analysis of their results.We present the available densities of thermal electrons as a function of altitude, time interval, and considered area, while neglecting effects of local electric fields.We find that free electron densities at altitudes between 5 and 13 km can reach values of order 103 cm−3, but only in shower cores with a radius on a centimeter scale. Above 6 km, the availability of extreme free electron densities decreases significantly with increasing altitude. Recent measurements by Rison et al. (2016, https://doi.org/10.1038/ncomms10721) indicate that several streamers must have been triggered simultaneously during discharge inception, and we suggest that an extensive air shower could have been the trigger. Rison's measurements show further that the streamers are laterally separated by several tens of meters, so they must have been triggered by electron densities as low as 1 cm−3. Such low electron densities demand a stochastic approach to streamer initiation near hydrometeors.