PROBABILITY, CALCULUS OF. — Science concerned, from both the theoretical and practical points of view, with the study of causal or random events, that is, those physical phenomena whose occurrence or non-occurrence is attributed to the intervention of “chance” (v. MATEMATICA, XIII). It finds applications in mathematical statistics, physics, actuarial mathematics, etc.
The calculus of probability originated in the study of games of chance (one of them, the game of zara, is also mentioned in the Divina Commedia, Purg., c. VI, vv. 1–9), which provided the first examples of probabilistic schemes: for example, the scheme of an urn containing a given number of balls, all practically identical, numbered from 1 to s, of which
b are white (those bearing the numbers 1... b; b < s) and the others black; when a ball is drawn “at random” from the urn, that is, when, after mixing the balls in the urn, any one of them is drawn without examining it before the drawing, one is faced with various “random events”: that the ball drawn is white; that the ball drawn is black; that the ball drawn bears the number 1; that the ball drawn bears an even number, etc. Similarly, when two drawings are made, specifying whether the second drawing takes place after the ball drawn first has or has not been replaced in the urn, and the pair of drawings is considered as a single trial, one is faced with other random events: that both balls are white; that the second is black; etc.
Two events are called incompatible when they cannot both occur in the same trial (in drawing one ball, the event that it is white and the event that it is black are incompatible), and compatible otherwise (in drawing two balls, the event that the first is white and the second black is compatible).
Now, with reference to the “random” events of a probabilistic scheme, it is not possible, for each one of them, to predict before conducting a trial whether or not it will occur. It is possible, however, to judge the greater or lesser likelihood that it will occur in a trial. An evaluation of this likelihood, obtained experimentally, is given by the relative frequency with which the event has already occurred in a certain group of n trials, that is, by the ratio v/n between the number v of times the event in question has occurred and the number n of trials conducted, all under the same conditions. Experience shows that, when one proceeds through various groups of trials, the relative frequencies of each of the random events that may be considered—for example, the event that the ball drawn in a trial is white—are approximately equal to one another when the number of trials is large, and, in the example given, approximately equal to the ratio b/s between the number of white balls and the total number of balls in the urn. Calling, in this example, possible cases the mutually incompatible events constituted by drawing, in a trial, a ball bearing respectively the number 1, 2, ... s, experience also shows that, in a large number of trials, they occur with relative frequencies approximately equal to one another, and therefore approximately equal to 1/s; consequently, they are called equally possible or equally probable cases. In this and other schemes, this assertion may also be made subjectively, but with unanimous agreement, on the basis of an examination of the conditions, without resorting to observations. It follows that an a priori evaluation can be made of the likelihood that the event in question will occur in a trial, supplied by the ratio b/s between the number of cases favorable to the event and the number of (equally) possible cases; and that this a priori evaluation, called the probability of the event, is a practical prediction of the a posteriori evaluation supplied by the relative frequency in a large number of trials.
These considerations also apply to other probabilistic schemes (for example, a regular die whose center of gravity coincides with its geometrical center, which is thrown in order to observe the number appearing on the upper face, etc.) in which the intervention of “chance” is recognized—chance understood as resulting from the presence of a large number of causes acting upon the phenomenon, causes that cannot all be taken into consideration, both because of their number and because some of them are inadequately determined; or else as resulting from the fact that imperceptible differences in the initial situation may lead to considerable variations in the final result, so that the result cannot be predicted on the basis of the imperfect knowledge one has of the initial situation. The preceding considerations extend to schemes regarded as a continuous infinity of possible cases (for example, the scheme of a needle rotating about its own center, whose position after a large number of rotations, following a push, is considered); and the constancy of the frequencies of a given event is also recognized when the cases are not equally possible (for example, the event that “heads” results when a “bent” coin is tossed onto a plane, subject to certain precautions intended to ensure that the trials take place “under the same conditions”).
Accordingly, with reference to the schemes just mentioned, and always from an experimental point of view, one may state: the probability of a random event in a given trial is a constant, lying between zero and 1, capable of predicting the relative frequency of the event in a large number of trials, all conducted under the same conditions. When the presence of a finite number of equally possible cases is recognized, some of them favorable to the event, the probability of the event is given by the ratio between the number of cases favorable to it and the number of possible cases; this definition is suitably extended to continuous cases. The approximation of the relative frequency to the probability is ordinarily the closer, the greater the number of trials. This assertion, suggested by experience, constitutes the well-known empirical postulate of chance.
If a given event is impossible in a trial, its probability is zero; if it is certain, its probability is equal to unity. The probability that, when a regular coin is tossed “at random,” “heads” will result is 1/2. Therefore, if, when a coin is tossed a large number of times, one finds that the relative frequency of the event “heads” differs considerably from 1/2, one may assert—and with increasing certainty if the fact continues to occur as the number of trials increases—that the coin is not regular, or that the tosses are not being carried out regularly.
What is the probability that, in the game of lotto, a given pair will appear in a drawing on one wheel? The number of possible fives that can be drawn from the urn (containing 90 balls numbered from 1 to 90), representing the (equally) possible cases, is 90 × 89 × 88 × 87 × 86 : (1 × 2 × 3 × 4 × 5) = 43,949,268, whereas the number of favorable cases is given by the number of fives containing the given pair, that is, by the number of possible triples that can be formed with the other 88 numbers, namely 88 × 87 × 86 : (1 × 2 × 3) = 109,736; hence the probability of a pair is equal to the quotient of the second number by the first, that is, 1/400.5. The probability of a triple is 1/11,748, and that of a quadruple is 1/511,038.
The probabilities of events belonging to the same probabilistic scheme are linked by the two principles of total and compound probabilities. The first states that, when two or more events are incompatible, the probability of the event, called the “logical sum” of the given events, consisting in the occurrence of any one of the given events, is equal to the sum of the probabilities of the individual events. The second states that, given two or more compatible events, the probability of their logical product, that is, of the event consisting in the occurrence, in a trial, of all the events under consideration, is equal to the product of the probabilities of the individual events, each determined on the assumption that the preceding events have occurred; in particular, if, with reference to a certain ordering of the events, each of them is independent of the logical product of the preceding events, its probability is equal to the product of the probabilities of the individual events. The two principles of total and compound probabilities, with reference to the schemes indicated, are experimentally verified.
From an abstract point of view, a probabilistic scheme consists of a set of entities, called events, among which the relations of incompatibility and compatibility are defined, and to each of which there corresponds a number, between zero and one, called the probability of the event; the probabilities of the various events are linked to one another by the principles of total and compound probabilities, which, when admitted for a countably infinite number of events, are called extended principles. From this theoretical standpoint, the c. d. p. constitutes a branch of mathematics. The connection between this abstract theory and the realistic schemes that may be considered is founded on the empirical postulate of chance: whenever a set of abstract events is made to correspond to a set of random events, with the probabilities satisfying the empirical postulate of chance, then, in determining, on the basis of the two principles indicated, the probability of any event deduced from the events initially considered by means of the operations of logical addition or multiplication, that probability likewise has a physical meaning, that is, it represents a prediction of the relative frequency with which the event itself will occur. The evaluation of probability by means of the ratio of the number of favourable cases to the number of possible cases, when these are recognized as equally possible on the basis of a large number of trials, all under the same conditions, is called objective, since there is unanimous agreement; for other events that likewise occur in reality (the event that a particular horse wins a race; the event that a given party obtains a majority in elections), one also speaks in practice of probability, but in this case the evaluations of probability are called “subjective” in order to distinguish them from the preceding cases, since different persons, on the basis of their different knowledge of the phenomenon and also of the differing characteristics of human nature, are led to evaluate that probability differently, and it is in no way possible to test these evaluations experimentally through a large number of trials under the same conditions. The c. d. p. is applicable in these cases as well, but it serves only as a rule of conduct ensuring the necessary consistency between subjective evaluations of the probabilities of given events and those of events deduced from the former through operations of logical addition and multiplication.
In the c. d. p., a prominent place is occupied by the study of random or chance variables, that is, of those quantities connected with a system of random events that are incompatible with one another and such that one of them is certain to occur in a trial, so that the value assumed by each of them in a trial depends on the event that occurs in that trial. The mean value of a random variable whose possible values are finite in number is the sum of the products of the values it may assume by their respective probabilities; the concept extends to general random variables. The mean value is a prediction of the arithmetic mean of the values that the random variable will assume in a large number of trials. The deviation of a random variable from its mean value is the difference between the values it may assume and its mean value; the deviation has zero mean. Examples of random variables are provided by games of chance: the gain, positive or negative, that a player may realize in a game is represented by a random variable; its mean value is called the mathematical expectation and thus represents a prediction of the average gain per game that the player will realize in a large number of games, all under the same conditions. It follows that a game is fair if the mathematical expectation is zero.
For example, a person who plays a pair in the lottery receives, in the event of winning, 250 times the stake; consequently, the gain, positive or negative, that the bank may realize for each lira played on a given pair is a random variable that may assume one of the values, in lire, 1 or −249, according as the pair does not appear or does appear, incompatible events with probabilities, respectively, 399.5/400.5 and 1/400.5. Therefore, the mathematical expectation of the bank is L. (1 × 399.5/400.5 − 249 × 1/400.5) = L. 0.376: that is, on average the bank wins approximately 38 centesimi for every lira played on pairs, and even more on triples or quadruples. To justify this fact, one must bear in mind the bank’s expenses, as well as the necessity of a margin to cover any unfavourable deviations.
The mean square deviation is the square root of the mean value of the square of the deviation; it is an index suitable for judging the greater or lesser concentration, around the mean value, of the values that the random variable will assume in the trials carried out.
The ordinary operations of analysis can be extended to more general random variables; in particular, one considers the sum and the product of two or more random variables belonging to the same probabilistic scheme. The mean value of the sum of two or more random variables, independent or dependent, is equal to the sum of the mean values of the individual random variables (Cantelli, 1911); the mean value of their product is equal to the product of the individual mean values, provided, for example, that each is independent of the product of the preceding random variables.
Among the various theorems of the calculus of probabilities, in addition to Bernoulli’s theorem (v. MATEMATICA, XIII), which, with its generalization to random variables (Tchebycheff, 1867), is called the “Law of large numbers,” the uniform law of large numbers (Cantelli, 1917) is fundamental. In the particular case of the Bernoulli scheme involving an event of constant probability in each trial, it states that, with a probability as close to one as desired, the successive relative frequencies of the event, in an unlimited succession of trials, tend, in the sense of analysis, toward the probability of the event; it therefore makes it possible to clarify the meaning of the empirical postulate of chance. Another famous theorem, due to Bayes (1764), concerning the probability of causes, has found applications in seeking to reach a judgment on the unknown probability of an event whose frequency is known. Criticism of these applications has led to the development of important fields of mathematical statistics, concerning both the theory of dispersion and the study of samples. The former, when applied to the study of human mortality, makes it possible to assert that the phenomenon of mortality among individuals possessing homogeneous characteristics (the same age, the same social category, all in initially good health, etc.) behaves, albeit with the necessary approximations, like a random event of constant probability in each trial. It is therefore formally possible to speak of the probability of death and to apply the calculus of probabilities, always from a normal standpoint, to problems of statistics and actuarial mathematics. Other applications of the calculus of probabilities concern the theory of “errors of observation,” which, in general, when measurements are one-dimensional, are distributed according to Gauss’s law (v. MATEMATICA, XIII), whereas two-dimensional errors are distributed according to Bravais’s law (1847); the study of the dispersion of artillery shots against a target belongs to this field. Other fundamental applications of the calculus of probabilities concern the kinetic theory of gases, wave mechanics, etc.