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

The Smallest Probability Interval a Sequence Is Random for: A Study for Six Types of Randomness