Probability Axioms
Lesson 3 Chapter 2
Let's roll a dice and ask the following informal question: What is the chance of getting six as the outcome? It is equivalent to another more formal question: What is the probability of getting a six in rolling a dice? Informal answer: The same as getting any other number most probably. Formal response: 1/6. How do we interpret the calculation of 1/6? Well, it is clear that when you roll a dice, you get a number in the range of {1,2,3,4,5,6}, and you do NOT get any other number. We can call {1,2,3,4,5,6} the outcome space that nothing outside of it may happen. To mathematically define those chances, some universal definitions and rules must be applied, so we all agree with.

To this aim, it is crucial to know what governs the probability theory. We start with axioms. The definition of an axiom is as follows: "a statement or proposition which is regarded as being established, accepted, or self-evidently true." Before stepping into the axioms, we should have some preliminary definitions.
Sample and Event Space
Probability theory is mainly associated with random experiments. For a random experiment, we cannot predict with certainty which event may occur. However, the set of all possible outcomes might be known.
Sample Space
After defining the sample space, we should define an event.
Event
Now, let's discuss some operations on events.

The sets A and B are inside the sample space .
Axioms
Andrey Kolmogorov, in 1933, proposed Kolmogorov Axioms that form the foundations of Probability Theory. The Kolmogorov Axioms can be expressed as follows: Assume we have the probability space of . Then, the probability measure
is a real-valued function mapping
as satisfies all the following axioms:
Outcomes
Using the axioms, we can conclude some fundamental characteristics as below: