Flipping a coin with same initial conditions

Question

Today, in my physics class my teacher was talking about how we can never predict the outcome of a coin flip. So I thought: Will the outcome of a coin flip be the same if we do not change the initial conditions (such as launch angle, force position where force is applied,etc.)? Intuitively, I feel that the answer would be yes. But is there something related to quantum mechanics that may produce a different answer?

Answer 1

Today, in my physics class my teacher was talking about how we can never predict the outcome of a coin flip

Your teacher was most likely not talking about this from a QM perspective of how experiments have probabilistic outcomes due to the inherent nature of QM (as we currently understand it).

Your teacher was most likely making a comment about how it is nearly impossible to know all of the relevant initial conditions, system parameters, etc. to accurately predict the result of a coin toss. However, on the spatial and temporal scales a coin toss resides on, it is safe to say we are in the classical mechanics regime. Quantum effects likely play no significant role in any of this. Therefore, you are correct in saying that if we could exactly reproduce the initial conditions of the entire system, then we would most certainly expect the same outcome each time.

In other words, your teacher was talking about inability to predict the outcome based on lack of sufficient information of the system, not because of any underlying quantum mechanical probabilities.

Answer 2

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.

Answer 3

All things in nature are fundamentally quantum mechanical. However in general (i.e. there are some exceptions) as objects scale up in size from atom to molecule... etc. the effect of quantum decoherence takes place making them to behave less quantum probabilistic and more classically macroscopic, obeying classical Newtonian mechanics. In the last case the system becomes deterministic or at least chaotic (too many variables and initial conditions) for a control experiment to replicate 100% the same result each time.

But yes, for your macroscopic coin experiment. If ideally you could replicate all initial conditions and isolate from all unstable factors, you would replicate the same outcome of the tossing coin each time.

Answer 4

According to deterministic classical physics: If you know the initial conditions perfectly then you can predict the outcome with 100% accuracy.

So in short, yes. If a coin was flipped with the exact same initial conditions then it would always land on the same side.

The above statements are "in principle". The other answers (and your teacher's comment) refer to various "in practice" challenges with learning and actually implementing an experiment that recreates the same initial conditions, but this doesn't address your question which I interpret as an "in principle" question.

That said, even in practice I don't think it would be challenging to build a tunable coin flipping machine that can be tuned between hitting heads or tails with very high fidelity.

The biggest uncertainty probably comes from air resistance so if you build a flipping machine in an evacuated vacuum chamber the results should become pretty reproducible as long as you strike the coin in close to the same location each trial and with close the same force. You could probably tune the flipper to have the coin land squarely on heads or squarely on tails each time. Now, if you tune the flipper so that the coin hits the floor "on edge" then you would like see an increase in the variance of the results again, more like when flipping a coin by hand. Would the variance reach the variance flipping it by hand? I'm not sure.

Quantum mechanically speaking the answer gets a little more muddled. I'd say the answer depends on exact details about how the problem is posed. I would say that in practice the answer for any realistic coin flipping experiment will be the same as the classical expectation (because coins are so big). So even taking into account quantum mechanics, the "in practice" answer is going to be again, yes, you get the same results with the same initial conditions.

It's hard to answer the question "in principle" because I'm not sure what assumptions go into the problem and experiment setup, so I won't even hazard an "in principle" quantum answer.