PROBABILITY, CALCULATION OF. - Science that deals, both from the theoretical and from the practical point of view, with the study of causal or random events, that is, of those physical phenomena whose occurrence or non-occurrence is attributed to the intervention of "chance" (see also MATHEMATICS, XIII). It finds applications in mathematical statistics, in physics, in actuarial mathematics, etc.
The calculus of probability originated from the study of games of chance (one of them, that of zara, is also spoken of in the Divine Comedy, 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 to one another, numbered from 1 to s, of which b are white (those bearing the numbers 1... b; b < s) and the others black; when one draws "at random" a ball from the urn, that is, when, after having mixed the balls in the urn, one draws any one of them, without examining it before the draw, one is faced with various "random events": that the ball drawn be white; that the ball drawn be black; that the ball drawn bear the number 1; that the ball drawn bear an even number, etc. Similarly, when two draws are made, specifying whether the second draw takes place after having put back or not into the urn the ball drawn first, and one considers the set of the two draws as a single trial, one is faced with other random events: the two balls be both white; the second be black; etc.
Two events are called incompatible when they cannot both occur in the same trial (in drawing one ball, the event that it turns out white and the event that it turns out black are incompatible), compatible in the contrary case (in drawing two balls, the event that the first turns out white and the second black are compatible).
Now, with reference to the "random" events of a probabilistic scheme, one cannot for each of them foresee, before carrying out a trial, whether it will occur or not. One can judge, however, the greater or lesser facility that it has of occurring in a trial. An evaluation of such facility, 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 the times that the event considered has occurred and the number n of the trials carried out, all under the same conditions. Experience shows that, when one proceeds to various groups of trials, the relative frequencies of each of the random events that can be considered, for example, of the event that the ball drawn in a trial be white, turn out approximately equal to one another, when the number of trials is large, and in the example indicated approximately equal to the ratio b/s between the number of white balls and the total number of balls in the urn. Calling, for this example, possible cases the events, incompatible with one another, constituted by the drawing, in a trial, of a ball bearing respectively the number 1, 2,... s, experience also shows that they present themselves, in a large number of trials, with relative frequencies approximately equal to one another, and therefore approximately equal to 1/s; hence they are called equally possible or equally probable cases: such an assertion, in this and in other schemes, can also be made subjectively, but with unanimous consent, on the basis of the examination of the conditions, without resorting to observations. It follows that one can carry out an a priori evaluation of the facility of occurring in a trial of the event considered, provided 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 provided by the relative frequency in a large number of trials.
These considerations also refer to other probabilistic schemes (for example, a given, regular die, whose center of gravity coincides with the geometric center, which is thrown to observe the number that appears on the upper face, etc.) in which one recognizes the intervention of "chance," which is asserted as due to the presence of a large number of causes acting on the phenomenon, causes that it is not possible to consider all, both because of their number and because they are in part poorly determined; or due again to the fact that imperceptible differences in the initial situation can lead to strong variations in the final result, so that the result is not foreseeable on the basis of the imperfect knowledge that one has of the initial situation. The preceding considerations extend to schemes considered as a continuous infinity of possible cases (for example, the scheme of a needle, which rotates about its own center, of which one considers the position that it can assume, after having carried out a large number of turns, following a push): and the constancy of the frequencies of a given event is recognized also when the cases are not equally possible (for example, the event that "heads" turns up in tossing a "loaded" coin onto a plane, under certain precautions apt to guarantee that the trials take place "under the same conditions").
Therefore, with reference to the said schemes, and always from an experimental point of view, one can assert: the probability of a random event of a given trial is a constant, comprised between zero and 1, apt to predict the relative frequency of the event in a large number of trials, all carried out under the same conditions. When one recognizes the presence of a finite number of equally possible cases, some of which favorable to the event, the probability of the event is provided 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 usually stronger the greater the number of trials. This assertion, suggested by experience, constitutes the well-known empirical postulate of chance.
If it is possible that a given event occur in a trial, its probability is null; if it is certain, its probability is equal to unity. The probability that, in tossing "at random" a regular coin, "heads" turns up is 1/2. Therefore, if, in tossing a coin a large number of times, one finds that the relative frequency of the event "heads" differs notably from 1/2, one can assert, and this always with greater 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 carried out in a regular manner.
What is the probability that, in the game of the lotto, in one draw on one wheel, a determined ambo turns up? The number of possible five-number combinations that can be drawn from the urn (containing 90 balls from 1 to 90) which represent the (equally) possible cases is 90 × 89 × 88 × 87 × 86 : (1 × 2 × 3 × 4 × 5) = 43,949,268, while the number of cases favorable is given by the number of five-number combinations containing the given ambo, that is, by the number of possible triples that can be made with the other 88 numbers, namely by 88 × 87 × 86 : (1 × 2 × 3) = 109,736; therefore the probability of an ambo is equal to the quotient between the second and the first number, that is, 1/400.5. The probability of a terno is 1/11,748 and that of a quaterno is 1/511,038.
The probabilities of events belonging to the same probabilistic scheme are linked by the two principles of total probabilities and of 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 single trial of all the events considered, is equal to the product of the probabilities of the individual events, each determined under the hypothesis 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, that is, if its probability that all the events considered occur 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, prove to be 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 among themselves by the principles of total and compound probabilities, which, when they are admitted for a countable infinity of events, are called extended principles. From this theoretical aspect, the c. d. p. [calculus of probabilities] constitutes a branch of mathematics. The link between this abstract theory and the realistic schemes that can 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, satisfying the probabilities of the empirical postulate of chance, then, determining, on the basis of the two principles indicated, the probability of any event, deduced from the events considered initially, by means of the operations of logical sum or product, this 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 favorable 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, will be called objective, insofar as there is unanimous consensus; even for other events that nonetheless present themselves in reality (the event that a given horse wins a race; the event that a given party reaches a majority in elections) one speaks practically of probability, but in that case the evaluations of probability are called "subjective" to distinguish them from the preceding cases, insofar as different persons, on the basis of different knowledge of the phenomenon and also on the basis of the different characteristics of human nature, are led to evaluate this probability in different ways, and it is in no way possible to verify these evaluations experimentally through a large number of trials under the same conditions. Even in these cases the c. d. p. is applicable, but it is only a norm of conduct for the necessary coherence between the subjective evaluations of the probabilities of given events and those of events deduced from the preceding ones on the basis of operations of logical sum and product.
In the c. d. p. a preeminent place is occupied by the study of aleatory or random variables, that is, of those quantities connected to a system of random events incompatible among themselves and such that one of them certainly occurs in a trial, so that the value that each of them assumes in a trial depends on the event that occurs in the trial. The mean value of a random variable, whose possible values are finite in number, is the sum of the products of the values that it can assume, by the 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 that it can assume and its mean value; the deviation has a mean value of zero. Examples of random variables are provided by games of chance: the gain (positive or negative) that a player can realize in a game is represented by a random variable; its mean value is called 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, whoever plays an ambo in the lottery collects, in case of a win, 250 times the stake; therefore the gain, positive or negative, that the bank can realize for each lira played on a given ambo is a random variable that can assume one of the values, in lire, 1, or -249, according to whether the ambo does not come out or does come out, incompatible events with probabilities respectively 399.5/400.5; 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, the bank on average wins about 38 centesimi for each lira played on ambi, and even more for terni or quaterni. To justify this fact one must bear in mind the expenses of the bank as well as the necessity of a loading to cover possible unfavorable 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 that are 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 that, for example, each is independent of the product of the preceding random variables.
Among the various theorems of the calculus of probabilities, besides that of Bernoulli (v. MATEMATICA, XIII), which, with its generalization to random variables (Tchebycheff, 1867), is called the "Law of large numbers", fundamental is the uniform law of large numbers (Cantelli, 1917), which, in the particular case of Bernoulli's scheme of an event of constant probability in every trial, states that, with a probability as close to one as one wishes, 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 specify the meaning of the empirical postulate of chance. Another famous theorem, of Bayes (1764), on the probability of causes, has found applications in the search for a judgment on the unknown probability of an event whose frequency is known. Criticisms of these applications have developed important fields of mathematical statistics, concerning both the theory of dispersion and the study of samples. The former, applied to the study of human mortality, makes it possible to affirm that the phenomenon of mortality among individuals having requisites of homogeneity (same age, same social category, all in good initial conditions of health, etc.) behaves, albeit with the necessary approximations, as a random event of constant probability in every trial. Therefore one can speak, formally, of a probability of death and can apply the calculus of probabilities, always from a normal point of view, 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 the measurements are one-dimensional, are distributed according to Gauss's law (v. MATEMATICA, XIII), while two-dimensional errors are distributed according to Bravais's law (1847): the study of the dispersion of artillery fire against a target falls within this subject. Other fundamental applications of the calculus of probabilities concern the theory of gases, wave mechanics, etc.