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The "definition" via axioms provides a mathematical structure that we
interpret either as "relative frequency" or as "degrees of belief".
Indeed, I think that any phenomenon which satisfies the axioms can serve
as an "interpretation". As they say, "If it walks like a duck, ...".
As far as the probability of rain tomorrow, I always explained to my
students that the language is so imprecise that the numerical value has
only rhetorical utility. We need to know:
1) How much rain in cm. ?
2) In which locations?
3) During what time span?
Does 70% probability mean that it rains in 70% of the locations or 70%
of the time or what?
Your instincts are correct. That example is severely flawed because we
have not made the experiment clear.
Continue to question the simple examples. You will learn from it.
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