> I think you haven't thought about this deeply enough yet.
On the contrary, I've thought about it quite deeply. Or at least deeply enough to talk about it in this context.
> You take it as self evident that P(X) = 0.5 is false for that event, but how do you prove that?
By definition a fair coin is one for which P(H) = P(T) = 1/2. See e.g. https://en.wikipedia.org/wiki/Fair_coin. Fair coins flips are also by definition independent, so you have a series of independent Bernoulli trials. So P(H^k) = P(H)^k = 1/2^k. And P(H^k) != 1/2 unless k = 1.
> Assuming you flip a coin and you indeed get 100 heads in a row, does that invalidate the calculated probability? If not, then what would?
Why would that invalidate the calculated probability?
> If not, then what would?
P(X) = 0.5 is a statement about measures on sample spaces. So any proof that P(X) != 0.5 falsifies it.
I think what you're really trying to ask is something more like "is there really any such thing as a fair coin?" If you probe that question far enough you eventually get down to quantum computation.
But there is some good research on coin flipping. You may like Persi Diaconis's work. For example his Numberphile appearance on coin flipping https://www.youtube.com/watch?v=AYnJv68T3MM