Axiomatic Approach To Probability

Axiomatic Approach To Probability: Introduction

Axiomatic Probability is an extension of the Classical Probability theory. Axiomatic probability has its inherent roots in the philosophy of science. The axioms are used to construct a probability model that is consistent and complete. This axiomatic probability can be applied for solving problems in any field of science.  

In the history of probability, mathematicians have represented the underlying assumptions and axioms of probability in three ways. The axiomatic approach to probability is a unifying theory of all these probabilistic approaches that allows one to formalize the presentation.

The concept of probability can only be applied to experiments in which the total number of outcomes is known, so we cannot apply the concept of probability until we have that information.

Axiomatic Approach To Probability

We need to know how many possible experiment outcomes apply probability in day-to-day situations. An axiomatic approach to probability refers to the probability of an event based on additional evidence. The word itself refers to a method of assigning probabilities based on some predefined axioms. Essentially, this is done to quantify the event and to make it easier to calculate whether the event occurred or not.

It is not uncommon for us to rely on numerous words to describe the probability of things happening. Statistical probability theory quantifies the chances of events happening or not happening. Previous classes discussed several methods to assign probabilities to an event associated with an experiment, knowing the number of outcomes. In addition to the axiomatic approach, other methods define probability. There are a few rules or axioms that are presented in this approach to assign a probability

Axiomatic Probability Conditions

Putting it in simpler terms, axiomatic probability is another way of describing the likelihood of an event. An event’s probability is a number between 0 and 1, where 0, roughly speaking, indicates that the event is impossible and 1, that it is certain. Probabilities indicate the likelihood of an event occurring. 

The Kolmogorov Axioms are three conditions set up by Andrey Kolmogorov for axiomatic probability.

We define ‘S’ as the sample space, ‘E’ as the event, ‘*’ as the possible outcomes, ‘n’ as the number of subsets, & ‘F’ as the event space.

S = {ω1,ω2,…ωn}

First Axiom

When an event occurs, its probability is a positive number, 

P(E) ∈R , P(E) ≥ 0,∀ E ∈ F

Or

0≤P(ωi)≤1 for each i=1nP(ωi)=S 

Second Axiom

In the sample space, the probability of the sum of all subsets is 1.

P(S) = 1 (OR) P(ω1)+P(ω2)+…+P(ωn)=1

Third Axiom

A mutually exclusive event is E1 and E2.

P(E1∪E2)=P(E1)+P(E2)

The same applies to P(E∪ϕ) .

Therefore, P(E∪ϕ) = P(E) + P(ϕ) = P(E)

Here, P(ϕ) is a null set (or) P(ϕ) = 0

Axiomatic Probability Applications

• Risk assessment and modeling are everyday applications of probability theory. Based on actuarial science, calculating prices and making decisions in the insurance industry and the stock market. 
• Probability may also be used to compare trends in biology and ecology.
• The probability is used to design games with historical references, market surveys, and player feedback.
• Probability theory is also applied in natural language processing, such as the cache language model and other statistical language models. 

Probability of Event 

The probability of an event exists when all possible outcomes are divided by the number of favorable outcomes. Probability is defined as the likelihood of all outcomes occurring equally.

Probability of an event = number of favorable outcomes/ total number of outcomes

In intuition, a probability is simply a measure of likelihood or chance of an event occurring. Using the simplest examples, the probability of a particular outcome A occurring in an experiment can be calculated by dividing the number of ways that A can occur by the total number of outcome possibilities.

What Is Conditional Probability?

The conditional probability predicts the outcome of one event based on the outcome of another. Assuming A has already been observed, conditional probability is the chance that event B will be observed assuming A has already occurred. Given that event A has already occurred, the conditional probability that event B will happen is P(B/A) = P(A ∩ B)/P(A).

What is Experimental Probability?

Calculating experimental probability requires observing the results and values of the probability experiments. Experimental probability is defined as the ratio between the number of times a trial has taken place and the number of trials taken. Considering that experimental probabilities are based on real-life examples, they may differ from theoretical probabilities.

Conclusion

Among the three axioms Kolmogorov proposed for establishing the likelihood of any event are the following. First, an event that has never occurred is represented by 0, which is sure to occur by 1. An event that has never occurred is represented by 0, which is sure to occur by 1.

To calculate the probability of each outcome, all the probabilities of each possible outcome are added together and equal to 1. If you flip a coin, either a head or a tail is the possible outcome. The outcome of a coin flip has a 1 in 2 chance of being heads and 1 in 2 chances of being tails. Adding the probabilities together should result in a single outcome since there are only two possible outcomes.