The title of this question is almost a retorical question. My point is that there is no way to define probability in a non circular manner.

Let's say the probality of getting a tail when tossing a coin is $\frac{1}{2}$, e.i the coin is fair. What does it mean? It means that the relative frequency of tails tends to $\frac{1}{2}$ as $N$ (the number of trials) tends to infinity. But what does the term "tends" means in this context? It means that the probability of getting any fixed relative frequency distinct from $\frac{1}{2}$ (say $\frac{1}{2}+0.0001$) tends to $0$ as $N$ tends to infinity. But we fall then in a circular definition!

Actually there is no way of knowing if the coin is fair, except by a physical exploration of the coin itself. (what, by quantum mechanics, won't really tell you with certainty if the coin is fair).

The only thing we can say, after tossing the coin a lot of times, is that the probability that the coin is fair enough is very high.

So my question is "What does it mean that the probablity of getting a tail is $\frac{1}{2}$?"

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    $\begingroup$ A fair coin is a mathematical concept and not a physical one. Therefore plain and simple: a fair coin is by definition a coin with equal probability of heads and tails. $\endgroup$ – Klaus Jan 27 at 0:54
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    $\begingroup$ So your question isn't really about what fair coin means, but rather what probability really means. Right? $\endgroup$ – Arthur Jan 27 at 0:54
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    $\begingroup$ You’re wrong that it’s impossible to define probability in a non-circular way; it’s just defined in terms of set theory, using a sample space of events, each of which is, by definition, associated with a probability between 0 and 1. However, like any mathematical model, it cannot perfectly mirror reality, which is what I think you’re trying to say - but that doesn’t really matter, since applications of these models have shown them to be practically useful. $\endgroup$ – Frpzzd Jan 27 at 0:54
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    $\begingroup$ An alternative viewpoint is that probabilities represent subjective degrees of belief. You could check out Probability: The Logic of Science by Jaynes for one presentation of this viewpoint. $\endgroup$ – littleO Jan 27 at 1:06
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    $\begingroup$ You're correct; there is no way to determine if a physical coin is fair. There is also no way to determine if a physical rod is exactly one meter long. Do you also claim there is no mathematical concept of length? $\endgroup$ – saulspatz Jan 27 at 1:27

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