interpretations of probability (frequentist vs subjective)

The axioms tell you how probability numbers behave, but not what a number like 'P = 0.3' actually means about the world. There are two leading answers. The frequentist (relative-frequency) view says a probability is the long-run fraction of times an event happens if you could repeat the experiment over and over: P(heads) = 0.5 means that across many, many tosses the share of heads settles near one half. The subjective (Bayesian) view says a probability is a degree of belief — a coherent measure of how confident a person is — so P = 0.3 expresses how strongly you would bet, even for a one-off event.

Each interpretation shines where the other struggles. The frequentist reading is concrete and testable for repeatable experiments — coins, dice, manufacturing defects — but it has nothing to say about genuinely unique events, since you cannot repeat them. 'The probability this particular candidate wins the election is 0.3' makes easy sense as a degree of belief but awkward sense as a long-run frequency, because there is only one such election. The Bayesian reading handles one-off statements naturally and lets you update beliefs as evidence arrives, at the cost of depending on a starting opinion (a prior) that different people might set differently.

The reassuring fact is that both camps obey the very same Kolmogorov axioms; they disagree about meaning, not about the arithmetic. Whether you read P(A) as a frequency or a belief, the complement rule, additivity, and inclusion-exclusion all hold identically. So the interpretations are not rival mathematics — they are different stories about what the shared mathematics is describing, and competent practitioners often switch between them depending on whether a problem is about repeatable chances or about uncertainty in a single unknown.