  The full statistical distribution of the superfluid fraction characterizing
one-dimensional Bose gases in random potentials is discussed. Rare
configurations with extreme fluctuations of the disorder potential can fragment
the condensate and reduce the superfluid fraction to zero. The resulting
bimodal probability distribution for the superfluid fraction is calculated
numerically in the quasi-1D mean-field regime of ultracold atoms in laser
speckle potentials. Using extreme-value statistics, an analytical scaling of
the zero-superfluid probability as function of disorder strength, disorder
correlation length and system size is presented. It is argued that similar
results can be expected for point-like impurities, and that these findings are
in reach for present-day experiments.
