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Historically, attempts to quantify probabilistic reasoning date back to antiquity. There was a particularly strong interest starting in the 12th century, with the work of the Scholastics, with the invention of the half-proof (so that two half-proofs are sufficient to prove guilt), the elucidation of moral certainty (sufficient certainty to act upon, but short of absolute certainty), the ...
The probability of an event A is written as (), [29] (), or (). [30] This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure.
Bayesian probability (/ ˈ b eɪ z i ə n / BAY-zee-ən or / ˈ b eɪ ʒ ən / BAY-zhən) [1] is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation [2] representing a state of knowledge [3] or as quantification of a personal belief.
Despite the ongoing debate about the cognitive processes involved in human reasoning, recent research has shown that multiple approaches can be useful in modeling human thinking. For instance, studies have found that people's reasoning is often influenced by their prior beliefs, which can be modeled using Bayesian probability theory. [21]
Bayesian epistemology is a formal approach to various topics in epistemology that has its roots in Thomas Bayes' work in the field of probability theory. [1] One advantage of its formal method in contrast to traditional epistemology is that its concepts and theorems can be defined with a high degree of precision.
This is not the only way probabilistic statements are used in ordinary human language: when people say that "it will probably rain", they typically do not mean that the outcome of rain versus not-rain is a random factor that the odds currently favor; instead, such statements are perhaps better understood as qualifying their expectation of rain ...
[27] [28] Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory". [28] In mathematics, the theory of stochastic processes is an important contribution to probability theory, [29] and continues to be an active topic of research for both theory and applications. [30] [31] [32]
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .