Probability
probability
Likelihood; the approach of a mental judgment to conformity with its object; approximation to truth. Probability produces in the mind the state of opinion. It occupies the whole range between doubt, the absence of assent, and certitude, which is complete and unreserved assent. Intrinsic probability is based upon the nature of the object in question; extrinsic probability, upon the authority of those capable of forming a scientific judgment.
Fuente: New Catholic Dictionary
Probability
In general
Chance, possibility, contingency, likelihood, likehness, presumption. conjecture, prediction, forecast, credibility, relevance;
the quality or state of being likely true or likely to happen;
a fact or a statement which is likely true, real, operative or provable by future events;
the conditioning of partial or approximate belief or assent;
the motive of a presumption or prediction;
the conjunction of reasonable grounds for presuming the truth of a statement or the occurrence of an event;
the field of knowledge between complete ignorance and full certitude;
an approximation to fact or truth;
a qualitative or numerical value attached to a probable inference, and
by extension, the systematic study of chances or relative possibilities as forming the subject of the theory of probability.
A. The Foundation of Probability. We cannot know everything completely and with certainty. Yet we desire to think and to act as correctly as possible hence the necessity of considering methods leading to reasonable approximations, and of estimating their results in terms of the relative evidence available in each case. In D VI-VII (infra) only, is probability interpreted as a property of events or occurrences as suchwhether necessary or contingent, facts are simply conditioned by other facts, and have neither an intelligence nor a will to realize their certainty or their probability. In other views, probability requires ultimately a mind to perceive it as such it arises from the combination of our partial ignorance of the extremely complex nature and conditions of the phenomena, with the inadequacy of our means of observation, experimentation and analysis, however searching and provisionally satisfactory. Thus it may be said that probability exists formally in the mind and materially in the phenomena as related between themselves. In stressing the one or the other of these two aspects, we obtain (1) subjectize probability, when the psychological conditions of the mind cause it to evaluate a fact or statement with fear of possible error; and (2) objective probability, when reference is made to that quality of facts and statements, which causes the mind to estimate them with a conscious possibility of error. Usually, methods can be devised to objectify technically the subjective aspect of probability, such as the rules for the elimination of the personal equation of the inquirer. Hence the methods established for the study and the interpretation of chances can be considered independently of the state of mind as such of the inquirer. These methods make use of rational or empirical elements. In the first case, we are dealing with a priori or theoretical probability, which considers the conditions or occurrences of an event hypothetically and independently of any direct experience. In the second case, we are dealing with inductive or empirical probability. And when these probabilities are represented with numerals or functions to denote measures of likelihood, we are concerned with quantitative or mathematical probability. Methods involving the former cannot be assimilated with methods involving the latter, but both can be logically correlated on the strength of the general principle of explanation, that similar conjunctions of moral or physical facts demand a general law governing and justifying them.
B. The Probability-Relation. Considering the general grounds of probability, it is pertinent to analyze the proper characteristics of this concept and the valid conditions of its use in inferential processes. Probability presents itself as a special relation between the premisses and the conclusion of an argument, namely when the premisses are true but not completely sufficient to condition the truth of the conclusion. A probable inference must however be logical, even though its result is not certain, for its premisses must be a true sign of its conclusion. The probability-relation may take three aspectsit is inductive, probable or presumptive. In strict induction, there is an essential connection between the facts expressed in the premisses and in the conclusion, which almost forces a factual result from the circumstances of the predication. This type of probability-relation is prominent in induction proper and in statistics. In strict probability, there is a logical connection between the premisses and the conclusion which does not entail a definite factual value for the latter. This type of probability-relation is prominent in mathematical probability and circumstantial evidence. In strict presumption, there is a similarity of characteristics between the fact expressed in the conclusion and the real event if it does or did exist. This type of probability-relation is prominent in analogy and testimony. A presumptive conclusion should be accepted provisionally, and it should have definite consequences capable of being tested. The results of an inductive inference and of a probable inference may often be brought closer together when covering the same field, as the relations involved are fundamental enough for the purpose. This may be done by a qualitative analysis of their implications, or by a quantitative comparison of their elements, as it is done for example in the methods of correlation. But a presumptive inference cannot be reduced to either of the other two forms without losing its identity, because the connection between its elements is of an indefinite character. It may be said that inductive and probable inferences have an intrinsic reasonableness, while presumptive inferences have an extrinsic reasonableness. The former involve determinism within certain limits, while the latter display indeterminacy more prominently. That is why very poor, misleading or wrong conclusions are obtained when mathematical methods are applied to moral acts, judiciary decisions or indirect testimony The activity of the human will has an intricate complexity and variability not easily subjected to calculation. Hence the degree of probability of a presumptive inference can be estimated only by the character and circumstances of its suggested explanation. In moral cases, the discussion and application of the probability-relation leads to the consideration of the doctrines of Probabilism and Probabiliorism which are qualitative. The probability-relation as such has the following general implications which are compatible with its three different aspects, and which may serve as general inferential principle
Any generalization must be probable upon propositions entailing its exemplification in particular cases;
Any generalization or system of generalizations forming a theory, must be probable upon propositions following from it by implication;
The probability of a given proposition on the basis of other propositions constituting its evidence, is the degree of logical conclusiveness of this evidence with respect to the given proposition;
The empirical probability (p = S/E) of a statement S increases as verifications accrue to the evidence E, provided the evidence is taken as a whole; and
Numerical probabilities may be assigned to facts or statements only when the evidence includes statistical data or other numerical information which can be treated by the methods of mathematical probability.
C. Mathematical Probability. The mathematical theory of probability, which is also called the theory of chances or the theory of relative possibilities, is concerned with the application of mathematical methods to the determination of the likelihood of any event, when there are not sufficient data to determine with certainty its occurrence or failure. As Laplace remarked, it is nothing more than common sense reduced to calculation. But its range goes far beyond that of common sense for it has not only conditioned the growth of various branches of mathematics, such as the theory of errors, the calculus of variations and mathematical statistics, but it has also made possible the establishment of a number of theories in the natural and social sciences, by its actual applications to concrete problems. A distinction is usually made between direct and inverse probability. The determination of a direct or a priori probability involves an inference from given situations or sets of possibilities numerically characterized, to future events related with them. By definition, the direct probability of the occurrence of any particular form of an event, is the ratio of the number of ways in which that form might occur, to the whole number of ways in which the event may occur, all these forms being equiprobable or equally likely. The basic principles referring to a priori probabilities are derived from the analysis of the various logical alternatives involved in any hypothetical questions such as the following(a) To determine whether a cause, whose exact nature is or is not known, will prove operative or not in certain circumstances; (b) To determine how often an event happens or fails. The comparison of the number of occurrences with that of the failures of an event, considered in simple or complex circumstances, affords a baisis for several cases of probable inference. Thus, theorems may be established to deal with the probability of success and the probability of failure of an event, with the probability of the joint occurrence of several events, with the probability of the alternative occurrence of several events, with the different conditions of frequency of occurrence of an event; with mathematical expectation, and with similar questions. The determination of an a posteriori or inverse probability involves an inference from given situations or events, to past conditions or causes which rnay have contributed to their occurrence. By definition, an inverse probability is the numerical value assigned to each one of a number of possible causes of an actual event that has already occurred; or more generally, it is the numerical value assigned to hypotheses which attempt to explain actual events or circumstances. If an event has occurred as a result of any one of n several causes, the probability that C was the actual cause is Pp/E (Pnpn), when P is the probability that the event could be produced by C if present, and p the probability that C was present before the occurrence of that event. Inverse probability is based on general and special assumptions which cannot always be properly stated, and as there are many different sets of such assumptions, there cannot be a coercive reason for making a definite choice. In particular, the condition of the equiprobability of causes is seldom if ever fulfilled. The distinction between the two kinds of probability, which has led to some confusion in interpreting their grounds and their relations, can be technically ignored now as a result of the adoption of a statistical basis for measuring probabilities. In particular, it is the statistical treatment of correlation which led to the study of probabilities of concurrent phenomena irrespective of their direction in time. This distinction may be retained, howe, for the purpose of a general exposition of the subject. Thus, a number of probability theorems are obtained by using various cases of direct and inverse probability involving permutations and combinations, the binomial theorem, the theory of series, and the methods of integration. In turn, these theurems can be applied to concrete cases of the various sciences.
D. Interpretations of Probability. The methods and results of mathematical probability (and of probability in general) are the subject of much controversy as regards their interpretation and value. Among the various theories proposed, we shall consider the following
Probability as a measure of belief,
probability as the relative frequency of events,
probability as the truth-frequency of types of argument,
probability as a primitive notion,
probability as an operational concept,
probability as a limit of frequencies, and
probability as a physical magnitude determined by axioms.
I. Probability as a Measure of BeliefAccording to this theory, probability is the measure or relative degree of rational credence to be attached to facts or statements on the strength of valid motives. This type of probability is sometimes difficult to estimate, as it may be qualitative as well as quantitative. When considered in its mathematical aspects, the measure of probable inference depends on the preponderance or failure of operative causes or observed occurrences of the case under investigation. This conception involves axioms leading to the classic rule of Laplace, namelyThe measure of probability of any one of mutually exclusive and apriori equiprobable possibilities, is the ratio of the number of favorable possibilities to the total number of possibilities. In probability operations, this rule is taken as the definition of direct probability for those cases where it is applicable. The main objections against this interpretation are
that probability is largely subjective, or at least independent of direct experience;
that equiprobability is taken as an apriori notion, although the ways of asserting it are empirical;
that the conditions of valid equiprobability are not stated definitely;
that equiprobability is difficult to determine actually in all cases;
that it is difficult to attach an adequate probability to a complex event from the mere knowledge of the probabilities of its component parts, and
that the notion of probability is not general, as it does not cover such cases as the inductive derivation of probabilities from statistical data.
II. Probability as a Relative Frequency. This interpretation is based on the nature of events, and not on any subjective considerations. It deals with the rate with which an event will occur in a class of events. Hence, it considers probability as the ratio of frequency of true results to true conditions, and it gives as its measure the relative frequency leading from true conditions to true results. What is meant when a set of calculations predict that an experiment will yield a result A with probability P, is that the relative frequency of A is expected to approximate the number P in a long series of such experiments. This conception seems to be more concerned with empirical probabilities, because the calculations assumed are mostly based on statistical data or material assumptions suggested by past experiments. It is valuable in so far as it satisfies the practical necessity of considering probability aggregates in such problems. The main objections against this interpretation are
that it does not seem capable of expressing satisfactorily what is meant by the probability of an event being true;
that its conclusions are more or less probable, owing to the difficulty of defining a proper standard for comparing ratios;
that neither its rational nor its statistical evidence is made clear;
that the degree of relevance of that evidence is not properly determined, on account of the theoretical indefinite ness of both the true numerical value of the probability and of the evidence assumed, and
that it is operational in form only, but not in fact, because it involves the infinite without proper limitations.
III. Probability as Truth-Frequency of Types of ArgumentsIn this interpretation, which is due mainly to Peirce and Venn, probability is shifted from the events to the propositions about them; instead of considering types and classes of events, it considers types and classes of propositions. Probability is thus the ability to give an objective reading to the relative tiuth of propositions dealing with singular events. This ability can be used successfully in interpreting definite and indefinite numerical probabilities, by taking statistical evaluations and making appropriate verbal changes in their formulation. Once assessed, the relative truth of the propositions considered can be communicated to facts expressed by these propositions. But neither the propositions nor the facts as such have a probability in themselves. With these assumptions, a proposition has a degree of probability, only if it is considered as a member of a class of propositions; and that degree is expressed by the proportion of true propositions to the total number of propositions in the class. Hence, probability is the ratio of true propositions to all the propositions of the class examined, if the class is finite, or to all the propositions of the same type in the long run, if the class is infinite. In the first case, fair sampling may cover the restrictions of a finite class; in the second case, the use of infinite series offers a practical limitation for the evidence considered. But in both cases, probability varies with the class or type chosen, and probability-inferences are limited by convention to those cases where numerical values can be assigned to the ratios considered. It will be observed that this interpretation of probability is similar to the relative frequency theory. The difference between these two theories is more formal than material in both cases the probability refers ultimately to kinds of evidence based on objective matter of fact. Hence the Truth-Frequency theory is open to the sime objections as the Relative-Frequency theory, with proper adjustments. An additional difficulty of this theory is that the pragmatic interpretation of truth it involves, has yet to be proved, and the situation is anything but improved by assimilating truth with probability.
IV. Probability as a Primitive NotionAccording to this interpretation, whicn is due particularly to Keynes, probability is taken as ultimate or undefined, and it is made known through its essential characteristics. Thus, probability is neither an intrinsic property of propositions like truth, nor an empty concept, but a relative property linking a proposition with its partial evidence. It follows that the probability of the same proposition varies with the evidence presented, and that even though a proposition may turn out to be false, our judgment that it is probable upon a given evidence can be correct. Further, since probability belongs to a proposition only in its relation to other propositions, probability-inferences cannot be the same as truth-inferences as they cannot break the chain of relations between their premisses, they lack one of the essential features usually ascribed to inference. That is why, in particular, the conclusions of the natural sciences cannot be separated from their evidence, as it may be the case with the deductive sciences. With such assumptions, probability is the group name given to the processes which strengthen or increase the likelihood of an analogy. The main objection to this interpretation is the arbitrary character of its primitive idea. There is no reason why there are relations between propositions such that p is probable upon q, even on the assumption of the relative character of probability. There must be conditions determining which propositions are probable upon others. Hence we must look beyond the primitive idea itself and place the ground of probability elsewhere.
V. Probability as an Operattonal ConceptIn this interpretation, which is due particularly to Kemble, probability is discussed in terms of the mental operations involved in determining it numerically. It is pointed out that probability enters the postulates of physical theories as a useful word employed to indicate the manner in which results of theoretical calculations are to be compared with experimental data. But beyond the usefulness of this word, there must be a more fundamental concept justifying it; this is called primary probability which should be reached by an instrumentalist procedure. The analogy of the thermometer, which connects a qualitative sensation with a number, gives an indication for such a procedure. The expectation of the repetition of an event is an elementary form of belief which can be strengthened by additional evidence. In collecting such evidence, a selection is naturally made, by accepting the relevant data and rejecting the others. When the selected data form a pattern which does not involve the event as such or its negative, the event is considered as probable. The rules of collecting the data and of comparing them with the theoretical event and its negative, involve the idea ol correspondence which leads to the use of numbers for its expression. Thus, probability is a number computed from empirical data according to given rules, and used as a metric and a corrective to the sense of expectation, and the ultimate value of the theory of probability is its service as a guide to action. The main interest of this theory lies in its psychological analysis and its attempt to unify the various conceptions of probability. But it is not yet complete; and until its epistemological implications are made clear, its apparent eclecticism may cover many of the difficulties it wishes to avoid. — T.G.
VI. Probability as a Limit of Frequencies. According to this view, developed especially by Mises and by Wald, the probability of an event is equal to its total frequency, that is to the limit, if it exists, of the frequency of that event in n trials, when n tends to infinity. The difficulty of working out this conception led Mises to propose the notion of a collective in an attempt to evolve conditions for a true random sequence. A collective is a random sequence of supposed results of trials when (1) the total frequency of the event in the sequence exists, and (2) the same property holds with the same limiting value when the sequence is replaced by any sequence derived from it. Various methods were devised by Copeland, Reichenbach and others to avoid objections to the second conditionthey were generalized by Wald who restricted the choice of the “laws of selection” defining the ranks of the trials forming one of the derived sequences, by his postulate that these laws must form a denumerable set. This modification gives logical consistency to this theory at the expense of its original simplicity, but without disposing of some fundamental shortcomings. Thus, the probability of an event in a collective remains a relative notion, since it must be known to which denumerable set of laws of selection it has been defined relatively, in order to determine its meaning, even though its value is not relative to the set. Controversial points about the axiomatization of this theory show the possibility of other alternatives.
VII. Probability as a Physical Magnitude determined by Axioms.. This theory, which is favoured mainly by the Intuitionist school of mathematics, considers probability as a physical constant of which frequencies are measures. Thus, any frequency is an approximate measure of one physical constant attached to an event and to a set of trialsthis constant is the probability of that event over the set of trials. As the observed frequencies differ little for large numbers of trials from their corresponding probabilities, some obvious properties of frequencies may be extended to probabilities. This is done without proceeding to the limit, but through general approximation as in the case of physical magnitudes. These properties are not constructed (as in the axiomatization of Mises), but simply described as such, they form a set of axioms defining probability. The classical postulates involved in the treatises of Laplace, Bertrand or Poincare have been modified in this case, under the joint influence of the discovery of measure by Borei, and of the use of abstract sets. Their new form has been fully stated by Kolmogoroff and interpreted by Frechet who proposes to call this latest theory the ‘modernized axiomatic definition’ of probability. Its interpretation requires that it should be preceded by an inductive synthesis, and followed by numerical verifications.
Bibliography. The various theories outlined in this article do not exhaust the possible definitions and problems concerning probability, but they give an idea of the trend of the discussions. The following works are selected from a considerable literature of the subject.
Laplace, Essai sur les Probabilites.
Keynes, A Treatise on Probability.
Jeffreys, Theory of Probability.
Uspensky, Introduction to Mathematical Probability.
Borel, Traite de Calcul des Probabilites (especially the last volume dealing with its philosophical aspects).
Mises, Probability, Statistics and Truth.
Reichenbach, Les Fondements Logiques du Calcul dcs Probabilites.
Frechet, Recherches sur le Calcul des Probabilites.
Ville, Essai sur la Theorie des Collectifs.
Kolmogoroff, Grundbegriffe der Wahrscheinhchkeitsrechnung.
Wald, Die Widerspruchsfreiheit des Kollektivbegriffes.
Nagel, The Theory of Probability.
— T.C.