What is The Axiomatic Definition of Probability?
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.
FAQs on Axiomatic Definition of Probability, Solved Questions & Application
1. What is the axiomatic definition of probability, and how does it differ from other approaches?
Axiomatic probability is defined by a set of formal rules, called axioms, introduced by Kolmogorov. Unlike classical or frequentist approaches, it does not rely on equally likely outcomes or repeated experiments. Instead, it assigns a probability to events in a sample space, ensuring mathematical consistency through three core axioms.
2. What are Kolmogorov's three axioms of probability as per the CBSE Class 12 Maths syllabus?
The three axioms are:
- Non-negativity: For any event E, the probability is always non-negative, i.e., P(E) ≥ 0.
- Normalization: The probability of the entire sample space S is 1; P(S) = 1.
- Additivity: For any two mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B).
3. Why can the probability of any event never exceed 1 in the axiomatic system?
Since the probability of the total sample space is set as 1 (Normalization Axiom), and every event is a subset of the sample space, no event can have a probability greater than 1. This ensures logical consistency within the probability function.
4. How is a sample space different from an event in probability theory?
A sample space (S) lists all possible outcomes of an experiment, while an event (E) is any subset of those outcomes. For example, when rolling a die, S = {1,2,3,4,5,6}, and E could be {2,4,6} for 'even numbers'.
5. How does the axiomatic model handle cases when events are not mutually exclusive?
If events have common outcomes (not mutually exclusive), use the addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). This formula subtracts the overlap to prevent double counting, as covered in the Class 12 Maths topics.
6. Why is the axiomatic approach considered essential for modern mathematical probability?
The axiomatic approach provides a general and flexible foundation, working for all types of experiments—finite or infinite, equally likely or not. It resolves inconsistencies in older definitions and allows probability to apply accurately in complex real-world fields like finance and science.
7. Can the probability of an event be an irrational number? Explain with an example.
Yes, probabilities can be irrational numbers. For instance, the probability of randomly landing inside a circle within a square (in geometric probability) might be π/4, which is irrational. The axioms only require probabilities to be between 0 and 1, not necessarily rational.
8. What misconceptions do students often have about mutually exclusive events under the axiomatic framework?
One common misconception is believing mutually exclusive events can occur together or that their probabilities must add up to 1. In reality, mutually exclusive means they cannot occur simultaneously, but the sum of their probabilities can be less than 1 if other outcomes are possible.
9. How is the axiomatic system applied in real-life problem-solving or risk assessment?
The axiomatic system provides a robust framework for assigning probabilities in risk assessment, insurance, and market analysis. It ensures calculations are consistent, which is vital for applications where outcomes are not equally likely or datasets are large and complex.
10. What happens if any of Kolmogorov's axioms are violated in a probability problem?
If any axiom is violated, the calculated probabilities are not valid. This can lead to errors such as assigning negative probabilities or probabilities greater than 1, making results meaningless or incorrect within the rules of probability theory.
