Foundations of Probability

Probability theory is that part of mathematics that is concerned with the description and modeling of random phenomena, or in a more general — but not unanimously accepted — sense, of any kind of uncertainty. Probability is assigned to random events, expressing their tendency to occur in a random experiment, or more generally to propositions, characterizing the degree of belief in their truth. Probability is the fundamental concept underlying most statistical analyses that go beyond a mere description of the observed data. In statistical inference, where conclusions from a random sample have to be drawn about the properties of the underlying population, arguments based on probability allow to cope with the sampling error and therefore control the inference error, which is necessarily present in any generalization from a part to the whole. Statistical modeling aims at separating regularities (structure explainable by a model) from randomness. There, the sampling error and all the variation that is not explained by the chosen optimal model are comprised in an error probability as a residual category.