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Stochastic descriptions have for a long time been part of biology. Usually a claim of biological randomness accepts two understandings, a weak but trivial and a strong but demanding. The weak says we are dealing with biological randomness when a biological description involves a random process. But it is very likely that these only two features – random and biological – would be considered far from being enough to claim biological randomness. For, it might be the case that the description can be formulated (i) within a biological non-stochastic framework or (ii) within a non-biological stochastic framework, typically one of chemistry or physics. So (i) and (ii) separately have often been considered at the base of a strong claim of biological randomness. I propose to combine those two to get a definition of epistemic biological randomness, expressed as a stochastic biological description satisfying two conditions, together necessary and sufficient: irreducibility of randomness within biology and irreducibility to non-biological randomness. I show that such a definition includes an unfortunate loophole, and that once the definition is consequently corrected, it can be argued to not be satisfiable by any biological stochastic description. I give support of the single premise of the argument, based on the probability's results known in mathematics and physics. I discuss possible impacts of the argument in debates about ontological indeterminism accessed through biology and interpretations of biological probabilities. I conclude by promoting methodological non-reductionism with respect to biological randomness.
