  We study diffusion-controlled two-species annihilation with a finite number
of particles. In this stochastic process, particles move diffusively, and when
two particles of opposite type come into contact, the two annihilate. We focus
on the behavior in three spatial dimensions and for initial conditions where
particles are confined to a compact domain. Generally, one species outnumbers
the other, and we find that the difference between the number of majority and
minority species, which is a conserved quantity, controls the behavior. When
the number difference exceeds a critical value, the minority becomes extinct
and a finite number of majority particles survive, while below this critical
difference, a finite number of particles of both species survive. The critical
difference $\Delta_c$ grows algebraically with the total initial number of
particles $N$, and when $N\gg 1$, the critical difference scales as
$\Delta_c\sim N^{1/3}$. Furthermore, when the initial concentrations of the two
species are equal, the average number of surviving majority and minority
particles, $M_+$ and $M_-$, exhibit two distinct scaling behaviors, $M_+\sim
N^{1/2}$ and $M_-\sim N^{1/6}$. In contrast, when the initial populations are
equal, these two quantities are comparable $M_+\sim M_-\sim N^{1/3}$.
