What is the probability that a tossed coin came up heads? A coin was tossed and covered over.  What is the probability that it came up heads? A: $\frac{1}{2}$
B: Either $1$ or $0$
I recently discussed with a mathematician who said answer "B" is the only acceptable answer, and that "A" is wrong.  I favour "A" but don't necessarily think "B" is wrong - just not as informative.
Seeing as we're being precise here, let's assume it didn't land on its edge.
Just to be clear here, this is a random event which has already been decided, but the result of that decision is unknown to us.  It appears to boil down to whether we insist that a random variable ceases to be random the moment it is decided, or when the result becomes known.
It would seem to me that if you insist $A$ is wrong, you make it impossible to calculate the probability of any outcome which has been decided but is as yet unknown, which would be missing an opportunity.
A: Interpreted Bayesian-style, where probability is a number assigned to a degree of belief, it's still $\frac{1}{2}$: you have received no information which would cause you to update your previous $\frac{1}{2}$ belief, so you never perform a Bayes-rule update and so your belief has not changed.
Interpreted frequentist-style, I don't know the answer to your question.
This highlights an important point that "probability" is not an absolute concept. Nothing has a probability; things only have a probability relative to other knowledge. The two of us can coherently assign different probabilities to the same event: there is no absoluteness here.
(A comment by Robert Frost points out that additionally, the probability of a coin-toss coming up heads is not $\frac12$ anyway. There's a nonzero chance of its landing on its side, and a nonzero chance of its vanishing in mid-air, for instance.)
A: Alright I'm going to make my comments an answer so that people can respond to it specifically if they wish.

It would seem to me that if you insist A is wrong, you make it impossible to calculate the probability of any outcome which has been decided but is as yet unknown, which would be missing an opportunity.

This paragraph is not correct. Even if one cannot calculate the probability of a future event, one can still compute the likelihood according to some probabilistic model, and make corresponding appropriate choices. That in no way changes the fact that, if you guarantee the coin to be flipped tomorrow and that it will land heads or tails on that flip, then the probability that it lands heads on that flip is either 1 or 0. But when we make decisions now we might decide to do so according to the average expected outcome.
One must clearly distinguish between probability theory and how we 'apply' it to the real world by interpreting the purely mathematical theorems as statements about reality.
The Bayesian approach is to interpret a probability as our personal confidence level of the truth of some statement. This enables us to compute our confidence in some statement based on our confidence in other related statements. Perfectly fine. The only catch is that this probability has a priori nothing to do with the actual fact of the matter. Our confidence level might be 100% but we may still be wrong. However, we have no choice but to use the Bayesian approach whenever there is something we do not know about and yet want to make decisions based on what we believe.
Under this approach, the answer is that we have $\frac12$ confidence in the coin having come up heads.
An alternative approach is to treat probability literally, in which case any event that has a fixed outcome will have probability 0 or 1 and not anything in-between, whether or not it is in the past or future. This leaves open the possibility that the universe has true randomness, in which case some events will be random in the mathematical sense.
Under this approach the answer is that the probability of the coin having come up heads is either $0$ or $1$ but we don't know which.
The obvious advantage of the first approach is that it allows us to include our beliefs in the analysis. The advantage of the second approach is that it is not subjective. So one has to pick the approach that is relevant to the desired goal of inquiry.
Note that in both approaches, we can observe empirically the law of large numbers in our past experience, and hence extrapolate that to future events where we expect aggregate outcomes to usually follow the same law, despite being often unable to predict individual outcomes, and even if we believe they are all fixed. Based on this we can in fact justify probability theory! Contrast this with the fact that the law of large numbers is a mathematical theorem and justified by probability theory.
