[PeterisP]

One aspect of Kolmogorov approach is that implicitly models things like biased probabilities through compressibility.

The shortest representation of rolls of fair dice or coins is their exact results, but if there's "less randomness" in some way (biased coin/die, sum of two dice which means non-uniform probabilities, combination of some predictable pattern with random noise) then there are more compact representations of that information, and all of that gets captured by the Kolmogorov approach without any explicit handling of the various possibilities.