  We use the martingale-theoretic approach of game-theoretic probability to
incorporate imprecision into the study of randomness. In particular, we define
several notions of randomness associated with interval, rather than precise,
forecasting systems, and study their properties. The richer mathematical
structure that thus arises lets us, amongst other things, better understand and
place existing results for the precise limit. When we focus on constant
interval forecasts, we find that every sequence of binary outcomes has an
associated filter of intervals it is random for. It may happen that none of
these intervals is precise -- a single real number -- which justifies the title
of this paper. We illustrate this by showing that randomness associated with
non-stationary precise forecasting systems can be captured by a constant
interval forecast, which must then be less precise: a gain in model simplicity
is thus paid for by a loss in precision. But imprecise randomness can't always
be explained away as a result of oversimplification: we show that there are
sequences that are random for a constant interval forecast, but never random
for any {\comp} (more) precise forecasting system. We also show that the set of
sequences that are random for a non-vacuous interval forecasting system is
meagre, as it is for precise forecasting systems.
