AXIOMATIC OR MODERN APPROACH TO PROBABILITY - Quantitative Techniques for management

Axiomatic Approach to Define Probability

The axiomatic approach to probability which closely relates the theory of probability with the modern metric theory of functions and also set theory was proposed by A.N. Kolmogrov. The axiomatic definition of probability includes both the classical and the statistical definition as particular cases and overcomes the deficiencies of each of them.

On this basis, it is possible to construct a logically perfect structure of the modern theory of probability and at the same time to satisfy the enhanced requirements of modern natural science. The axiomatic development of mathematical theory of probability relies entirely upon the logic of deduction.

Sample Space

The set of all possible outcomes of a random experiment is called the sample space of the experiment. A sample space is often defined based on the objectives of the analysis. The sample space of a random experiment is denoted by S and its element are denoted by ei, where i = 1, 2, ...... n. Thus, a sample space having n elements can be written as:

S = {e1, e2, ......, en}.

If a random experiment consists of rolling a six faced die, the corresponding sample space consists of 6 elementary events. Thus, S = {1, 2, 3, 4, 5, 6}.

Similarly, in the toss of a coin S = {H, T}.

The elements of S can either be single elements or ordered pairs. For example, if two coins are tossed, each element of the sample space would consist of the set of ordered pairs, as shown below:

S = {(H, H), (H, T), (T, H), (T, T)}

Finite and Infinite Sample Space

A sample space consisting of finite number of elements is called a finite sample space, while if the number of elements is infinite, it is called an infinite sample space. Finite sample spaces are conceptually and mathematically simpler. Still, sample spaces with infinite number of elements are quite common. As an example of infinite sample space, consider repeated toss of a coin till a head appears. Various elements of the sample space would be:

S = {(H), (T, H), (T, T, H), ...... }.

Discrete and Continuous Sample Space

A sample space is discrete if it consists of a finite or countable infinite set of outcomes. A sample space is continuous if it contains an interval (either finite or infinite) of real numbers.

Event

An event is any subset of a sample space. In the experiment of roll of a die, the sample space is S = {1, 2, 3, 4, 5, 6}. It is possible to define various events on this sample space, as shown below:

Let A be the event that an odd number appears on the die. Then A = {1, 3, 5} is a subset of S. Further, let B be the event of getting a number greater than 4. Then B = {5, 6} is another subset of S. Similarly, if C denotes an event of getting a number 3 on the die, then C = {3}.

It should be noted here that the events A and B are composite while C is a simple or elementary event.

Occurrence of an Event

An event is said to have occurred whenever the outcome of the experiment is an element of its set. For example, if we throw a die and obtain 5, then both the events A and B, defined above, are said to have occurred.

It should be noted here that the sample space is certain to occur since the outcome of the experiment must always be one of its elements.

Definition of Probability (Modern Approach)

Let S be a sample space of an experiment and A be any event of this sample space. The probability of A, denoted by P(A), is defined as a real value set function which associates a real value corresponding to a subset A of the sample space S. In order that P(A) denotes a probability function, the following rules, popularly known as axioms or postulates of probability, must be satisfied.

Definition of Probability (Modern Approach)

The first axiom implies that the probability of an event is a non-negative number less than or equal to unity. The second axiom implies that the probability of an event that is certain to occur must be equal to unity. Axiom III gives a basic rule of addition of probabilities when events are mutually exclusive.

The above axioms provide a set of basic rules that can be used to find the probability of any event of a sample space.

Probability of an Event

Let there be a sample space consisting of n elements, i.e., S = {e1, e2, ...... en}. Since the elementary events e1, e2, ...... en are mutually exclusive, we have, according to axiom III

Probability of an Event

It is obvious from the above that the probability of an event can be determined if the probabilities of elementary events, belonging to it, are known.

The Assignment of Probabilities to various Elementary Events

The assignment of probabilities to various elementary events of a sample space can be done in any one of the following three ways :

  1. Using Classical Definition: We know that various elementary events of a random experiment, under the classical definition, are equally likely and, therefore, can be assigned equal probabilities. Thus, if there are n elementary events in the sample space of an experiment and in view of the fact that

    Assignment of Probabilities to various Elementary Events

    (axiom II), we can assign a probability equal to 1/n to every elementary event or, using symbols, we can write P(e1) 1/n for i = 1, 2, …,Further, if there are m elementary events in an event A, we have,
    m elementary events in an event
    We note that the above expression is similar to the formula obtained under classical definition.
  2. Using Statistical Definition: Using this definition, the assignment of probabilities to various elementary events of a sample space can be done by repeating an experiment a large number of times or by using the past records.
  3. Subjective Assignment: The assignment of probabilities on the basis of the statistical and the classical definitions is objective. Contrary to this, it is also possible to have subjective assignment of probabilities. Under the subjective assignment, the probabilities to various elementary events are assigned on the basis of the expectations or the degree of belief of the statistician. These probabilities, also known as personal probabilities, are very useful in the analysis of various business and economic problems.

It is obvious from the above that the Modern Definition of probability is a general one which includes the classical and the statistical definitions as its particular cases. Besides this, it provides a set of mathematical rules that are useful for further mathematical treatment of the subject of probability.

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