  There are many randomness notions. On the classical account, many of them are
about whether a given infinite binary sequence is random for some given
probability. If so, this probability turns out to be the same for all these
notions, so comparing them amounts to finding out for which of them a given
sequence is random. This changes completely when we consider randomness with
respect to probability intervals, because here, a sequence is always random for
at least one interval, so the question is not if, but rather for which
intervals, a sequence is random. We show that for many randomness notions,
every sequence has a smallest interval it is (almost) random for. We study such
smallest intervals and use them to compare a number of randomness notions. We
establish conditions under which such smallest intervals coincide, and provide
examples where they do not.

Probability