I guess the subject here is chaos, which arises even in classical mechanics, with a very nicely define classical Hamiltonian. "Deterministic chaos" is one of the wonderful oxymorons modern science can produce. In system having exponential dependence to initial conditions, evolution from two "infinitely" close initial states will diverge, and you have a prediction horizon.

If you can reliably predict, say, weather to 3 days and after that all your predictions fail, and you now measure initial conditions 10 times more accurately, you may be able to predict up to 4 days; again 10 times more accurate (100 times from original situation), and you're only able to predict to 5 days. It is likely that you will never be able to predict more than 14 days, whatever precise measures you make.

Flipping a coin raises the same problem.

I guess the subject here is chaos, which arises even in classical mechanics, with a very nicely define classical Hamiltonian. "Deterministic chaos" is one of the wonderful oxymorons modern science can produce. In system having exponential dependence to initial conditions, evolution from two "infinitely" close initial states will diverge, and you have a prediction horizon.

If you can reliably predict, say, weather to 3 days and after that all your predictions fail, and you now measure initial conditions 10 times more accurately, you may be able to predict up to 4 days; again 10 times more accurate (100 times from original situation), and you're only able to predict to 5 days. It is likely that you will never be able to predict more than 14 days, whatever precise measures you make.

Flipping a coin raises the same problem.

Edit after discussion in comment section: I'm talking of a coin thrown and landing on a table, which will stabilize after hitting the surface with its edge, bouncing and spinning, which is a chaotic process.