  We derive exact statistical properties of a class of recursive fragmentation
processes. We show that introducing a fragmentation probability 0<p<1 leads to
a purely algebraic size distribution in one dimension, P(x) ~ x^{-2p}. In d
dimensions, the volume distribution diverges algebraically in the small
fragment limit, P(V)\sim V^{-\gamma} with \gamma=2p^{1/d}. Hence, the entire
range of exponents allowed by mass conservation is realized. We demonstrate
that this fragmentation process is non-self-averaging. Specifically, the
moments Y_\alpha=\sum_i x_i^{\alpha} exhibit significant fluctuations even in
the thermodynamic limit.
