This came up in a discussion a few years ago, where people were arguing about the meaning of probability: is it long-run frequency, is it subjective belief, is it betting odds, etc? I wrote:

Probability is a mathematical concept. I think Martha Smith’s analogy to points, lines, and arithmetic is a good one. Probabilities are probabilities to the extent that they follow the Kolmogorov axioms. (Let me set aside quantum probability for the moment.) The different definitions of probabilities (betting, long-run frequency, etc), can be usefully thought of as models rather than definitions. They are different examples of paradigmatic real-world scenarios in which the Kolmogorov axioms (thus, probability). Probability is a mathematical concept. To define it based on any imperfect real-world counterpart (such as betting or long-run frequency) makes about as much sense as defining a line in Euclidean space as the edge of a perfectly straight piece of metal, or as the space occupied by a very thin thread that is pulled taut. Ultimately, a line is a line, and probabilities are mathematical objects that follow Kolmogorov’s laws. Real-world models are important for the application of probability, and it makes a lot of sense to me that such an important concept has many different real-world analogies, none of which are perfect.

We discuss some of these different models in chapter 1 of BDA.

P.S. There’s been some discussion and I’d like to clarify my key point, why I wrote this post. My concern is that I’ve read lots of articles and books that claim to give the single correct foundation of probability, which might be uncertainty, betting, or relative frequency, or coherent decision making, or whatever. My point is that none of these frameworks is the foundation of probability; rather, probability is a mathematical concept which applies to various problems, including long-run frequencies, betting, uncertainty, decision making, statistical inference, etc. In practice, probability is not a perfect model for any of these scenarios: long-run frequencies are in practice not stationary, betting depends on your knowledge of the counterparty, uncertainty includes both known and unknown unknowns, decision making is open-ended, and statistical inference is conditional on assumptions that in practice will be false. That said, probability can be a useful tool for all these problems.