In that case it seems to me impossible to tell from only the information of a single sequence.
But if you have multiple sequences where one is the binary coding of a Mozart symphony and another is the binary encoding of Shakespeare sonnet and another is the binary encoding of days of the week or a DNA sequence if you have enough data points like that I think you can start to say Well they're probably not random because we see coherent patterns.
The second aspect is coherent pattern depends on perspective. So what seems random to us maybe a very common physical constant to an alien civilization. so if we see a binary sequence that represents pi, maybe we say oh well that's not random because it's accurately pi to like a thousand places... but if we see another sequence that looks random maybe it's a physical constant to an alien civilization that's accurate to a thousand places as well. But we don't know that so it looks random...to us. So randomness doesn't just depend on entropy, I think in a realistic point of view, probability is not enough, it also depends on something which Maybe it's a little bit harder to measure perhaps which is perspective. Random depends on the context.