What is a Probability?

It's very common to lean on the idea that a probability is just "chance", but in mathematics, probability is a much more general and wildly used tool. So, what is a probability measure? And, what are some examples (some based in chance and some not). "Everything is 50 50" can even make sense in the more general setting of what a probability is actually defined to be.

00:00 Intro
00:28 What is a Probability
01:30 A physical and possibly unfair coin
02:40 A string of successes until a failure
07:15 Everything is 50 50?
10:10 Making probabilities from integrals
12:40 Where do probabilities come up?

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[NikolajKuntner]

There is a 30's axiomatization of the probability concept via random sequences Richard by von Mises, to rival Kolmogorov approach. While Mises' Wikipedia article mentions this in a sentence, it today has faded into obscurity.
But it's not too difficult to connect the two to a good degree, and it might formalize a sense to move back to chance in your sense: If Q are the rationals and X resp. S denote the event space resp. sigma algebra, then any sequence s:N->X can be transformed to the sequence avg_s:N->(S->Q) defined as
(avg_s)_n := A \mapsto | {k<=n | s_k in A} | / n
In words, the n'th element of avg_s is a map out of the algebra that counts how often the given s already ended up in A.
Now given a probability P, the subset of sequences
{s:N->X | lim_{n\to\infty} avg_s = P}
exactly holds the possible sequences of events such that in the long run they behave as sample sequences in X using P.
I think Mises approach amounted to find axioms for the set of "random sequences w.r.t. some probability". To be contrasted with the measure theoretic maps into [0, 1]. There's a historical account by Lambalgen from 96, which argues why this eventually didn't catch on.

Now the question in your title is something I ask myself far too often. I'd like to read up more on credence and the theories there. I think Standford Encyclopedia has an overview over math papers in that direction also. Now when I hear the word probability, my mind usually goes to the more common probability concept. A pet peeve of mine is to delineate "Bayes's theorem", which e.g. might pop out of combinatorical and also measure theoretic approaches, from "Bayes' Rule", which I take to mean what Bayesianist's always use, whether or not they work with a frequentist or Kolmogorov-axiomized framework. Humanity is likely not at its wits end, when it comes to chance.
I don't know to what extent we can justify the notion of Kolmogorov probability for one-time events and such, even if, but that theory always struck me as annoyingly costly in terms of the overhead to set it up. When I'm old and have the time, I should try invest the time and carve out what the point is (in terms of probability theory concepts) where you really need general powersets (as opposed to e.g. working with classes).
If you look at the Reverse Mathematics Wikipedia page, for example, it's always shocking to me how little you really need to prove most theorems.

PS I think one of the last videos didn't start with "Hey it's Nathan." I was perplexed!