How would you get a non-uniform, non-random probability distribution without intelligent bias?
Why do you assume the only kind of bias is an intelligent one?
I'm not sure what you mean by non-random here, but it's trivial to get non-uniform distributions. Beta decay is exponential in time, not uniform. Ortho and para hydrogen form in various mixtures depending on temperature, 75% to 25% at room temperature, not uniform.