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I wonder if there are some other attitudes to the probability than those related to Kolmogorov? Did somebody try to do those things differently?

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    $\begingroup$ Yes. One variation on Kolmogorov's axioms is to have finite additivity $P(\cup_i A_i) = \sum_i P(A_i)$ for a finite collection of mutually disjoint events $A_i,$ but not to include an axiom that guarantees this for a countably infinite sequence of disj sets. Many familiar practical thms can still be proved, but some measure-theoretic complications are avoided. // Book "How to Gamble If You Must" by Dubins and Savage explores this alternate formulation (more seriously and deeply than the flippant title might lead you to suspect). 1960s. Google and look beyond commercial stuff on first few pgs. $\endgroup$ – BruceET Sep 28 '18 at 6:18
  • $\begingroup$ Along similar lines, I think, is Bruno de Finetti, Theory of Probability: A Critical Introductory Treatment, 2 vols., 1970, English translation Wiley 1974. $\endgroup$ – Calum Gilhooley Sep 28 '18 at 11:46
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There are many variations to the axioms of probability. The main distinctions are

  1. Finite additivity instead of countable additivity. This was favored by early statisticians like Bernoulli.

  2. Bayesian axioms of probability defined using conditional probabilities. Kolmogorov's axioms are a special case of this.

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