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

We call the set of all possible outcomes as the sample space and we denote it by $\Omega$ .

After defining the sample space, we should define an event.

Event

An event $E$ is a set embracing some possible outcomes. Any event $E$ is a subset of the sample space $\Omega$ . The empty set $\varnothing$ is called the impossible event as it is null and does not represent any outcome.

Now, let's discuss some operations on events.

Union: For any set of events $\{E_{1},E_{2},\ldots,E_{n}\}$ , the union event $\bigcup_{i=1}^{n}E_i$ consists of all outcomes that occurred in any of E_{i} events at least once. Ex: The $A \cup B$ indicates that if $A$ or $B$ occurred.
Intersection: For any set of events $\{E_{1},E_{2},\ldots,E_{n}\}$ , the intersection event $\bigcap_{i=1}^{n}E_i$ consists of all outcomes that occurred in all of E_{i} events at least once. Ex: The $A \cap B$ indicates if $A$ and $B$ both occurred.
Mutually Exclusive: Two events $A$ and $B$ are mutually exclusive, if they cannot occur concurrently. In other words, $A \cap B = \varnothing$ . Ex: (A) throwing a fair coin and (B) rolling a dice. A and B are clearly mutually exclusive.
Complement Set: For any event $E$ , we denote $E^c$ as the complement of $E$ and stands for all outcomes in the sample space $\Omega$ that are not in $E$ . Basically $E \cap E^c = \varnothing$ and $E \cup E^c = \Omega$ .

The sets A and B are inside the sample space $\Omega$ .

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 $(\Omega, \mathcal{A}, \mathbb{P})$ . Then, the probability measure $\mathbb{P}$ is a real-valued function mapping $\mathbb{P}: \mathcal{A} \rightarrow \mathbb{R}$ as satisfies all the following axioms:

1
For any event $E \in \Omega$ , $P(E) \geq 0$ (the probability of occurrence is non-negative).
2
$P(\Omega) = 1$ .
3
$P(\bigcup_{i=1}^{n}E_i) = \sum_{i=1}^{n} P(E_i)$ for any set of mutually exclusive events $\{E_{1},E_{2},\ldots,E_{n}\}$ .

Outcomes

Using the axioms, we can conclude some fundamental characteristics as below:

If event $A$ is a subset of event $B$ ( $A \subseteq B$ ), then $P(A) \leq P(B)$ .
If $A$ is an event and $A^{c}$ is the complementary set (all other events except $A$ in the event space $\Omega$ ), then $P(A^{c}) = 1 - P(A)$ .
The probability of the empty set is zero ( $P(\varnothing) = 0$ ) as the empty set is the complementary set of the sample space $\Omega$ .
For any event $E$ , we have the probability bound of $0 \leq P(E) \leq 1$ .

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