Are there different degrees of "random" or is it exclusively a binary result?

Question

I'd like to understand if the randomness of a sequence is boolean or if there are different degrees of randomness.

Perhaps the randomness (confidence) test of a series can be based upon

• the algorithm used to determine randomness (theoretic proof)
• the limit of numbers (if applicable) that the randomness test can apply to within a certain degree of error.

Answer 1

Randomness is a on-or-off thing. However, there is an "almost random" called "peusdorandom" that can be seen as an intermediate property. There isn't a real sense of "$\epsilon$-random" or something as far as I know.

Answer 2

First you need to carefully define randomness. Consider a binary sequence where the first bit has $0.5$ chance to be $0$ or $1$, then each successive bit has $0.6$ chance to match the previous one and $0.4$ chance to be different. This sequence would fail many of the classical tests for randomness because sequential bits would be too likely to match. One definition of randomness is that the sequence is not compressible, and this one fails. The entropy per character is $(0.6\log \frac 1{0.6}+0.4 \log \frac 1{0.4})/\log 0.5 \approx 0.97$ bits, so we can compress the string a bit ($3\%$) but it is random within that.