Clearly we are not talking of an unbiased coin. Hence, $p(H) = x$ itself is a random variable which can take a value between $ [0,1]$. Thus, my question is - what sort of distribution this $x$ can follow?
What it looks like, naively, that as $x \in \mathbb{R}$, point probabilities gets to 0. If I would have asked "what is the probability that $ a \le x \le b $ with $a,b \in [0,1]$ then, perhaps one standard way of answering would be $b-a$.
Taking this further, no matter what sort of distribution this $x$ follows, I can see that when $a \to b$, the $p(H) \to 0$.
Hence, my question is, would be precise enough to state this point probability as 0 ?