On this page, we discuss the so-called axiomatic approach to defining probability (introduced by Andrey Kolmogorov in 1933).

Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

  • Understand the meaning of the three Kolomogorov axioms that any probability model must satisfy, and commit each to memory.

  • Apply the axioms and their simple consequences to find the probability of events in simple situations.

Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

  • Use the Kolmogorov axioms and set algebra to prove simple laws of probability.

  • Understand that the intention of the axiomatic approach to probability is to put the theory of probability on firm mathematical foundations and avoid the pitfalls and paradoxes that come with reasoning intuitively about probability.

To prepare for class

  • Watch this video (by Brendan Cordy, 8m18s) which introduces the three Kolmogorov axioms of probability.

    Any sample space \(\Omega\), together with a method of assigning probabilities to events that obeys the three axioms, is called a probability model. These are the fundamental objects of study in probability.

  • Watch this video starting (by probabilitycourse.com, 13m28s) which shows how to use the three Kolmogorov axioms to prove various other simple (but important) laws of probability (you can skip the first 5 minutes of the video):

  • Watch the following video (by Spoonful of Maths, 6m08s) which shows some practical examples of finding the probability of a union of two events, both when the events are mutually exclusive, or not:

After class

  • Watch this video (by Carneades.org, 6m54s) which explains the meaning of an “axiom” as opposed to a “definition” or “theorem” (in philosophy or mathematics):


Brendan Cordy Avatar Brendan Cordy
Gabriel Indurskis Avatar Gabriel Indurskis


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