I appreciate your attempt but you did not convince me the slightest bit. Let's take the 1,333,337 coin tosses all heads. This result has no bearing on the proba

I appreciate your attempt but you did not convince me the slightest bit. Let's take the 1,333,337 coin tosses all heads. This result has no bearing on the probability of the coin at all. It may make you strongly doubt that the coin is indeed a fair coin but - and that is the point I am trying to get at - there is nothing that prevents a fair coin from coming up heads 1,333,337 times in a row. Whatever your experiment shows, it could always be a statistical fluke. And the infinite case does not changes much, at least not in a way obvious to me. Back with those super powers I toss the coin infinitely often and get heads 50 % of the time. That was fun, let me do that again tomorrow. Well, again heads 50 % of the time. This is the way it goes for a long time but then something strange happens, one day all tosses come up heads. The very next day everything is back to normal. What is now the probability of heads, we got two different answers for your way of defining he probability? And all it took was an extreme statistical outlier on single day.