  The field of algorithmic randomness studies what it means for infinite binary
sequences to be random for some given uncertainty model. Classically,
martingale-theoretic notions of such randomness involve precise uncertainty
models, and it is only recently that imprecision has been introduced into this
context. As a consequence, the investigation into how imprecision alters our
view on martingale-theoretic random sequences has only just begun. In this
contribution, where we allow for non-computable uncertainty models, we
establish a close and surprising connection between precise and imprecise
uncertainty models in this randomness context. In particular, we show that
there are stationary imprecise models and non-computable non-stationary precise
models that have the exact same set of random sequences. We also give a
preliminary discussion of the possible implications of our result for a
statistics based on imprecise probabilities, and shed some light on the
practical (ir)relevance of both imprecise and non-computable precise
uncertainty models in that context.
