Probabilistic finiteness properties for profinite groups

We introduce various probablistic finiteness conditions for profinite groups related to positive finite generation (PFG). We investigate completed group rings which are PFG as modules, and use this to answer a question of Kionke and the second author on positively finitely related groups. Using the theory of projective covers, we define and characterise a probabilistic version of the $\mathrm{FP}_n$ property for profinite groups, called $\mathrm{PFP}_n$. Finally, we prove how these conditions are related to previously defined finiteness conditions and each other.