  We present discharge probability studies performed with a single Thick Gas
Electron Multiplier (THGEM) irradiated with alpha particles in Ar-CO$_2$ and
Ne-CO$_2$ mixtures. We observe a clear dependency of the discharge stability on
the noble gas and quencher content pointing to lighter gases being more stable
against the development of streamer discharges. A detailed comparison of the
measurements with Geant4 simulations allowed us to extract the critical charge
value leading to the formation of a spark in a THGEM hole, which is found to be
within the range of 3-7$\times10^6$ electrons, depending on the gas mixture.
  Our experimental findings are compared to previous GEM results. We show that
the discharge probability of THGEMs exceeds the one measured with GEMs by
orders of magnitude. This can be explained with simple geometrical
considerations, where primary ionization is collected by a lower number of
holes available in a THGEM structure, reaching higher primary charge densities
and thus increasing the probability of a spark occurrence. However, we show
that the critical charge limits are similar for both amplification structures.
