Any one off event is an example. But I assume you know that, so can you clarify what you mean by "a probability that can not be analyzed in terms of frequencies or other proportions"?
Any one off event is an example. But I assume you know that, so can you clarify what you mean by "a probability that can not be analyzed in terms of frequencies or other proportions"?
Let us start with coin flips. You repeatedly flip a coin and the number of heads will come out to be about half the number of trials.
Where does that come from? It is not some intrinsic property of the coin, it comes from varying initial conditions. If you had enough precision when controlling your hand movements, you could in principle force an outcome with high probability.
But assuming you can not or at least do not do that, there is a certain set of initial states, some will lead to heads, some to tails, and each toss will start from a randomly selected initial state. So given my ignorance of the exact initial state, the coin will land heads with a probability equal to the number of initial states leading to heads divided by the number of initial states compatible with my observations of the initial state. [1]
Repeatedly tossing a coin will sample the set of initial states and the result will match the proportion of the number of states. At least as long as I am not wrong about the set of initial states.
The same applies to something like an election. I have imperfect knowledge about the state of the world but there is a set of states compatible with my knowledge about the world and certain subsets of them will lead to certain candidates to win.
[1] Maybe adjusted by some probability distribution over the initial states if they are not equally likely to be picked.