Answer by zzzbbx for Is the claim "this coin is fair" falsifiable?

From a scientific point of view you can only rely on statistics. For example, if you were to toss a coin we all agree that it would not be possible to toss the coin an infinite number of times. The purpose of statistics is, precisely, to determine a sufficient number of experiments to infer some kind of information about a phenomenon under observation.
For example, in the case of the coin, you may want to use the Bayes' Theorem to determine whether it is likely or not that a fair coin would produce the sequence of heads and tails you are observing. From statistics you can determine the number of tosses (or experiments) you need to make sure the probability you compute is sufficiently reliable. As your demands, in terms of reliability, get closer and closer to 100%, the number of tosses you need to make increases to infinity.
Your argument about the fact that an all head sequence is always possible with some positive probability is correct. In fact, it does happen with probability one simply because you may model the system as a Markov chain with recurrent states; indeed, any sequence of finite length will eventually occur. However, this is not the point. The point is, precisely, what you said: different events do have some probabilities to occur. This is the key observation statisticians make use of to determine the size of a sufficiently large sample they can possibly use to derive the statistics they need (in your case, the probability that the coin is fair given a sequence of coin tosses).
We know that, in principle, any prediction might be wrong even if the confidence is 99.9% but in practice, statistics is all you can use.