P&S L2 C2 S5
S5: The Kolmogorov Axioms
The axioms are simple, and almost obvious. There is one more axiom needed when the population is infinite, but we will not use this axiom in our elementary probability course. WHY is Kolmogorov so famous for thinking up these axioms?
The answer lies in history. Probability theory was a field separate from mathemaics, and it had its own rules and methods, which did not fit into standard mathematical methods. Kolmogorov found a way to bring probability within the standard mathematical theories, using these axioms
These are the Kolmogorow axioms for Finite Populations. If the Population is infinite, we must add one more axiom - for disjoint sets:
This is written just for show/completeness - we will not be using this axiom in our course on elementary probability theory.
The important thing to note about Kolmogorov Axioms is that they apply only to the MODELS, and not to REALITY. For example, we can model a coin flip as a random draw from a population with 2 Elements {Heads, Tails}. The Kolmogorov Axioms apply to the model. In Reality, suppose we flip the coin and it rolls away from us and down into a river. This outcome is not within the model. But this is not a serious problem in applying the model to reality -- it all depends on the PURPOSE for which the model is to be used.
Khan Academy: Lesson on Addition Rule for Probability — NOTE carefully that analysis is done in MODEL WORLD -- we assume all balls are equally likely -- this assumption may fail for MANY reasons in a REAL WORLD REPLICATION of the model