Probability Theory Review
Page 5
Assigning & Interpreting Probabilities - 1
The Axiomatic/Classical Interpretation

 

Approaches to Assigning and Interpreting Probabilities
The question now arises: How exactly does one assign a probability to an event? In other words, if P(A) = x , where x  is a number such that 0 = x  = 1, how does one determine the precise numerical value of x ? A related and equally important question is: Just what does such a number mean?

There are three principal ways to assign and interpret probabilities: the classical approach (also referred to in this Website as axiomatic), the frequency approach, and the subjective approach.

Technically, the axiomatic and classical interpretations are different. Classical probability assumes a priori (presumptively) that all outcomes are equally likely (equiprobable) whereas axiomatic probability makes no prior assumption about the likelihood of events. In most other respects, however, the classical and axiomatic approaches are practically indistinguishable. Given the strong similarities, they are treated here jointly as a single approach, primarily to enhance our understanding of the concept of probability.

1. The Axiomatic/Classical Interpretation
The axiomatic approach relies on the axioms of probability, along with essential definitions, and the logical structure of the given problem to determine the probabilities of events. No empirical data (such as obtained by statistical sampling) or intuitive judgments regarding likelihoods are required. Probability assessments are obtained by logical analysis of the problem situation. The classical approach does the same with the added condition that all outcomes (elementary events) are assumed equiprobable. It is called classical because it follows from the original interpretation given to the notion of probability in relation to games of chance.

[The classical approach was initially based on combinatorial analysis of discrete events as implemented empirically by Gerolamo Cardano in the 1560s, who also wrote Liber de Ludo Aleae. Cardano handsomely supplemented his income with considerable earnings from gambling and chess. The first known work that discusses games of chance, however, is Fra Luca Paccioli's Summa de Arithmetica, Geometria, Proportioni e Proportionalita (1494). The classical approach was first treated formally by Pierre-Simon de Laplace in 1814 in his A Philosophical Essay on Probabilities. The modern axiomatic basis of probability was finally formalized in 1933 when Andrei Kolmogorov published his Foundations of the Theory of Probability.]

The classical approach assumes that all outcomes are equally likely to occur, that is, that the underlying process (activity) giving rise to the outcomes is in effect random. Thus, if there are n  equally likely possible outcomes from a given process and one of the outcomes must occur, the following probability assignments hold true:

1. Each outcome has a probability (of occurring) equal to 1/n .

2. If an event A is defined as consisting of m  such outcomes, then P(A) = m/n .

The first statement should be intuitively obvious, in accordance with axioms 1 and 2. The second statement follows from the fact that, since the outcomes are mutually exclusive elementary events, the generalized version of axiom 3 applies. This shows the steadfast relationship that exists between the classical approach and the axioms of probability.

Example 13: In the roll of a fair die, the probability of any one of the six outcomes is 1/6.

Example 14: In the roll of a fair die, the event E = {2, 4, 6} has P(E) = 3/6 = 1/2.

Example 15: In the draw of a card from a well-shuffled deck, the event A = {ace} = {A?, A?, A?, A?} has P(A) = 4/52 = 1/13.

The essence of the notion of classical probability can now be discerned: a probability is simply a quantitative measure describing the degree of possibility that an uncertain event has of occurring. This measure is computed by counting the number of outcomes from a given process that instantiate the event of interest and dividing by the total number of possible outcomes, given that all outcomes are equiprobable (equally likely). In the classical view, the theoretical abstraction called probability is the rate of occurrence (the incidence) expected to hold for the event of interest in the conduct of a given process or activity, obtained by logical analysis of the process itself. In other words, the assignment of a classical probability is an exercise in logical deduction performed with respect to an actual physical process.
 
One would then expect the axiomatic approach to reinterpret the classical probability of a real-world event as the proportion of theoretical outcomes in U  that constitute the theoretical event in U  relative to the total number of theoretical outcomes in U — where U is an abstract mathematical space, not the actual physical process. In other words, an axiomatic probability would be a purely mathematical concept stating the percentage of equiprobable outcomes that instantiate the occurrence of the event of interest in the event space —a purely theoretical entity— which would then be taken as the measure of the likelihood (the probability) of the corresponding event in the physical world. The theory would thereby be entirely formalized and rendered independent from any particular application, such as the archetypal games of chance. Euclid would have been pleased, some would say.

Unfortunately, things are not that simple. There is a serious logical error in the above axiomatic "reinterpretation," namely, that the outcomes are posited equiprobable. If the outcomes must a priori be equiprobable in order to determine probabilities of events, what criterion would one use to ascertain the “equal probability” of the outcomes (which are, after all, elementary events) to begin with? Circular arguments are not acceptable in logic and mathematics. Perhaps Immanuel Kant could have shed some light on this knotty problem with his epistemology of synthetic a priori judgments.

Naturally, this illogical outcome caused much consternation in learned circles. Fortunately for the edifice of reason, Andrei Kolmogorov cut the Gordian knot by banishing the vexing “equiprobable outcomes” from his axioms, thus giving rise to the modern axiomatic theory of probability. The Kolmogorov axioms limit themselves to arbitrary events about which nothing is said of their a priori likelihood.

Is this a satisfactory solution, defining away the problem? Insofar as axioms are concerned, apparently so. Consider Euclid’s geometric points. They are dimensionless, which means they take up no space. Physically, therefore, they cannot possibly exist. They are purely axiomatic fiction. Still, mathematically, they accomplish their function perfectly. If an axiom system allows for a logically coherent and empirically validated and fruitful theory, then surely the axioms must be valid for that theory.

Yes, but what about praxis? How do we deal with the primitive outcomes of dice and coins and playing cards that populate the real world? Well, I’d suggest you use your common sense. If the device looks symmetrically balanced and you think the process in which it is used is legitimately random, why, I’d say the outcomes are equiprobable, logical circularity notwithstanding. And frankly, so would the rest of the planet. You would thus be employing the classical approach to assign probabilities to the primitive outcomes as well as any subsequently defined compound events.

On the other hand, if after a conscientious examination of both device and process there is no reason to doubt that the device is symmetrically balanced and there is no reason to doubt that the process in which it is used is conducted without bias or distortion favoring any particular outcome, then by logical analysis one would be correct in concluding that the elementary events in the problem’s event space —the abstract representation of the physical process— are to be construed as equiprobable. You would then be employing the axiomatic approach to assign probabilities. Note that contrary to the classical approach, the axiomatic approach leaves open the possibility that outcome probabilities not be equiprobable. (Classical probability has nothing to say if the outcomes are not equally likely.) Note also that the frame of reference for axiomatic probabilities is the event space U —a theoretical construct— not the real-world process per se. Still, the key distinction is that no assumption is made a priori about the likelihood of the events. Such assessments are made by logically analyzing the structure of the given problem. (Keep in mind, however, that not all problems need lend themselves to logical analysis.)

In the classical variant, the outcomes are posited equally likely. This may (or may not) hold true for games of chance, but in the real world of managerial decision making, the events of interest are, by and large, not equally likely at all! This is due to the fact that the underlying processes giving rise to many of the events of managerial interest, though uncertain to some extent, are generally not purely random. Economic activity is decidedly intentional in nature: people work more or less systematically in order to achieve certain ends. Consequently, classical probabilities present serious limitations to decision analysts in practical applications. The fact is they are seldom used in actual managerial decision problems, save for comparison or reference purposes.

Modern axiomatic probability theory (not to be confused with the axiomatic approach) simply ignores the problem of initially assigning probabilities to outcomes in U. Thus, it offers no guidance as to how to jumpstart the probability assessment procedure. This again limits its usefulness in the analysis of real-world decision problems. The value of axiomatic probability theory for decision analysis lies, basically, in that it furnishes a sound theoretical basis upon which to conduct practical probabilistic analyses. The actual conduct of the practical analyses, however, will require a practicable (as opposed to a purely theoretical) approach. We shall now turn to this.

Summarizing, axiomatic probabilities, as the term is used in this Website, refer to probabilities derived by means of the axioms of probability and the logical structure of the problem being analyzed. Empirical data and subjective assessments are not involved in the axiomatic paradigm. If the outcomes under consideration are a priori deemed equiprobable, the axiomatic approach is tantamount to the classical approach. Axiomatic probability theory provides a rigorous foundation on which to ground decision analysis.

 

Probability Interpretations

Interpretations of Probability