What determines the anyon fusion outcome (if several are possible)?

Given the fusion rule (for anyons) "A x B = 1 + C" it is possible for the anyons A and B to fuse to the vacuum "1" or to fuse to the anyon C.

What determines what will happen if they fuse? Is the outcome probabalistic?

Morally speaking, the fusion rules tell you that if you take particle A and bring it close to Particle B they will interact and fuse together. The resulting state will be a super-position of no particles -- corresponding to the identity -- and the C particle. If we denote the quantum dimensions of particle $X$ by $d_X$, then the resulting state will be $|\psi \rangle = \sqrt{\frac{1}{d_{A}d_B}} |vac \rangle + \sqrt{\frac{d_C}{d_A d_B}} |C\rangle$.
(note that the fusion rules tell you that $d_A d_B = 1+ d_C$ so the state is indeed normalized).