Axiomatic Definition of Probability, Solved Questions & Application

• Axiomatic probability is a unifying probability theory in Mathematics.

• The axiomatic approach to probability sets down a set of axioms that apply to all of the approaches of probability which includes frequentist probability and classical probability. 

• These rules are generally based on Kolmogorov's Three Axioms. 

• Axiomatic probability set starting points for mathematical probability.

Axiomatic Definition of Probability-

One important thing we need to know about probability is that probability can be applied only to experiments where we know the total number of outcomes of the given experiment.

In simpler words, unless and until we know the total number of outcomes of an experiment, we cannot apply the concept of probability.

Thus, we should know the total number of possible outcomes of the experiment in order to apply probability in day to day situations. Axiomatic Probability is just one more way of describing the probability of an event (E). As, we get to know from the word itself, in this approach, some axioms are predefined before assigning probabilities. This is done to ease the calculation of occurrence or non-occurrence of the event and quantize the event.

Three Axioms of Kolmogorov’s  

The axiomatic approach to probability was introduced by Russian mathematician Andrey Nikolaevich Kolmogorov, who lived from 1903 to 1987. He said that there exist three axioms that can be applied to determine the probability of any event (E).

Kolmogorov's Three Axioms are as follows:-

Let’s know all the three axioms:-

The First Axiom:

• The first axiom of axiomatic probability states that the probability of any event must lie between 0 and 1.

• Here 0 represents that the event will never happen and 1 represents that the event will definitely happen.

• The probability of any event cannot be negative. The smallest value for the probability of any event P (A) is zero and if probability P (A) =0, then event A will never happen.

The Second Axiom:

• The second axiom of the axiomatic probability of the whole sample space is equal to one (100 per cent).

• This is because the sample space S consists of all possible outcomes of our random experiment or if the experiment is performed anytime, something happens. So, the outcome of each trial always belongs to the sample space of experiment S.

• Therefore, the event S always occurs and P(S) =1. 

• Let us take an example: if we roll a die, Sample space(S) = {1,2,3,4,5,6}, and since the outcome of the event will always lie among the numbers 1 to 6, then P(S)=1.

The Third Axiom:

• The third axiom of probability is the most interesting one.

• The basic idea of this axiom is that if some of the events are disjoint (that is there is no overlap between the events), then the probability of the union of two events must be equal to the summations of their probabilities.

• Let us take an example if A1 and A2 are mutually exclusive events or outcomes, then P (A1 ∪ A2) = P (A1) + P (A2). 

• Here, ∪ stands for ‘union’.

Solved Questions

Question 1) In an election, there are four candidates. Let the four candidates be A, B, C, and D. Based on the polling analysis, it is estimated that A has a 20 per cent chance of winning the election this time, while candidate B has a 40 per cent chance of winning the election. What is the probability that candidate A or B will win the election?

Solution) We notice that the events that {A wins election}, {B wins election}, {C wins election}, and {D wins election} are disjoint events since more than one of the events cannot occur at the same time. For example, if candidate A wins, then candidate B cannot win the elections. We know that the third axiom of probability states that,

Therefore, Probability P (A wins election or B wins election) = P ({A wins the election} ∪ {B wins the election}) = P ({A wins election}) +P ({B wins election})

=P ({A wins election}) +P ({B wins election})

=(20/100)+(40/100)

=0.2+0.4

= 0.6

Therefore, the probability that candidate A or candidate B will win the election is equal to 0.6

Applications Of Axiomatic Probability

• Applied in modelling and in risk assessment. The markets and insurance companies rely on this for determining price and decision making.

• In biology and ecology, it can be used to analyze trends.

• With the feedback from players and references from old games, we can use probability for designing games.

The Approach And Conditions For Axiomatic Probability

The conditions for axiomatic probability definition is an equation satisfying the event. The applications of axiomatic probability are already specified above. However, the real deal of applying this helps us give the following benefits. They are

• Introspection

• For casual observation

• Understanding economic models

Understanding Axiomatic System

If you're wondering if the term axiomatic system makes it look like a big concept, it’s wrong! The axiomatic system is just used for deriving theorems from a set of axioms. So in brief, it shows that every theorem does have an axiomatic system containing a few sets of axioms to prove the conditions and events. A true statement that doesn’t require any evidence or proof is what the axiom is. The reasoning starts from the axioms. The example for an axiom is that all right angles are said to be equal to each other. As this is the truth, we don’t need any proof for arguing with the same. 

So an axiomatic system is the collection of such truths with no need for proofs or axioms. 

Conclusion

The article is useful for the students as it will develop a clear concept about Axiomatic Probability. The article discusses the definition and three axioms of Kolmogorov’s and solved questions etc.