If you want to go down this road, then we will have to switch to a nuclear decay based coin or something like that. In the case of a coin toss there never was a

If you want to go down this road, then we will have to switch to a nuclear decay based coin or something like that. In the case of a coin toss there never was any real randomness, as you say we were just ignorant of the initial conditions. Given a distribution over the possible initial conditions, we can determine the probabilities for heads and tails. The possibly huge number of degrees of freedom and deterministic chaos will of course make this an unpleasant exercise.