The mathematics of probability is well understood. Probability is a non-negative, additive set function whose maximum value is unity. What is harder to understand is the application of the formal notion to the actual world. One point of application is statistical: when kinds of event or trials (such as the tossing of a coin) can be described, and the frequency of occurrence of particular outcomes (such as the coin falling heads) is measurable, then we can begin to think of the probability of that kind of outcome in that kind of trial. One account of probability is therefore the frequency theory, associated with Venn and Richard von Mises (1883–1953), that identifies the probability of an event with such a frequency of occurrence. A second point of application is the description of an hypothesis as probable when the evidence bears a favoured relationship to it. If this relation is conceived of as purely logical in nature, as in the work of Keynes and Carnap, probability statements are not empirical measures of frequencies, but represent something like ‘partial entailments’ or measures of possibilities left open by the evidence and by the hypothesis. Formal confirmation theories and range theories of probability are developments of this idea. The third point of application is in the use probability judgements have in regulating the confidence with which we hold various expectations. The approach sometimes called subjectivism or personalism, but more commonly known as Bayesianism, associated with de Finetti and Ramsey, sees probability judgements as expressions of a subject’s degree of confidence in an event or kind of event, and attempts to describe constraints on the way we should have degrees of confidence in different judgements that explain those judgements having the mathematical form of judgements of probability (see Dutch book, exchangeability, representation theorem). For Bayesians, probability or chance is not an objective or real factor in the world, but rather a reflection of our own states of mind. However, those states of mind need to be governed by empirical frequencies, so this is not an invitation to licentious thinking. See also principal principle.