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JGWeissman comments on Avoiding doomsday: a "proof" of the self-indication assumption - Less Wrong
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What do you mean by "knowledge of chances"? There is no inherent chance or probability in a coin flip. The result is deterministically determined by the state of the coin, its environment, and how it is flipped. The probability of .5 for heads represents your own ignorance of all these initial conditions and your inability, even if you had all that information, to perform all the computation to reach to logical conclusion of what the result will be.
I'm just talking about the difference between, e.g., knowing that a coin is fair, versus not having a clue about the properties of the coin and its propensity to produce various outcomes given minor permutations in initial conditions.
By "a coin is fair", do you mean that if we considered all the possible environments in which the coin could be flipped (or some subset we care about), and all the ways the coin could be flipped, then in half the combinations the result will be heads, and in the other half the result will be tails?
Why should that matter? In the actual coin flip whose result we care about, the whole system is not "fair", there is one result that it definitely produces, and our probabilities just represent our uncertainty about which one.
What if I tell you the coin is not fair, but I don't have any clue which side it favors? Your probability for the result of heads is still .5, and we still reach all the same conclusions.
For one thing, it'll change how we update. Suppose the coin lands heads ten times in a row. If we have independent knowledge that it's fair, we'll still assign 0.5 credence to the next toss. Otherwise, if we began in a state of pure ignorance, we might start to suspect that the coin is biased, and so have difference expectations.
That is true, but in the scenario, you never learn the result of a coin flip to update on. So why does it matter?