  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.
