  We investigate the problem of ballistically controlled reactions where
particles either annihilate upon collision with probability $p$, or undergo an
elastic shock with probability $1-p$. Restricting to homogeneous systems, we
provide in the scaling regime that emerges in the long time limit, analytical
expressions for the exponents describing the time decay of the density and the
root-mean-square velocity, as continuous functions of the probability $p$ and
of a parameter related to the dissipation of energy. We work at the level of
molecular chaos (non-linear Boltzmann equation), and using a systematic Sonine
polynomials expansion of the velocity distribution, we obtain in arbitrary
dimension the first non-Gaussian correction and the corresponding expressions
for the decay exponents. We implement Monte-Carlo simulations in two
dimensions, that are in excellent agreement with our analytical predictions.
For $p<1$, numerical simulations lead to conjecture that unlike for pure
annihilation ($p=1$), the velocity distribution becomes universal, i.e. does
not depend on the initial conditions.
