Probability Axioms

Lesson 2 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

**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.**

*outcome space*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.

## 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: