PROBABILITY, CALCULUS OF. – The science that studies, from both a theoretical and practical standpoint, events that are causal or aleatory, that is, physical phenomena whose occurrence or non-occurrence is attributed to the intervention of “chance” (cf. also MATHEMATICS, XIII). It finds applications in mathematical statistics, physics, actuarial mathematics, etc.
The calculus of probability originated from the study of games of chance (one of these, the game of dice, is mentioned in the *Divine Comedy*, *Purg.*, c. VI, vv. 1–9), which provided the first examples of probabilistic schemes: for instance, 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 to *b*; *b* < *s*) and the rest black. When a ball is drawn “at random” from the urn—that is, after the balls have been mixed and one is drawn without prior inspection—the result is various “random events”: that the ball drawn is white; that it is black; that it bears the number 1; that it bears an even number, etc. Similarly, when two draws are made, specifying whether the second draw occurs after the first ball has been replaced in the urn or not, and considering the pair of draws as a single trial, other random events arise: both balls are white; the second is black; etc.
Two events are said to be incompatible when they cannot both occur in the same trial (drawing one ball, the event that it is white and the event that it is black are incompatible), and compatible otherwise (drawing two balls, the event that the first is white and the second black are compatible).
Now, with reference to the “random” events of a probabilistic scheme, it is not possible to predict, before conducting a trial, whether any given event will occur. However, one can judge the greater or lesser likelihood of its occurrence in a trial. An assessment of this likelihood, derived 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 *v₁/n* between the number *v₁* of times the event has occurred and the number of trials conducted under identical conditions. Experience shows that, when various groups of trials are conducted, the relative frequencies of each of the random events under consideration—for example, the event that the ball drawn in a trial is white—tend to be 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. If, in this example, we call the possible cases the mutually incompatible events constituted by the drawing, in a single trial, of a ball bearing the numbers 1, 2, 3, …, *s* respectively, experience also shows that these occur, in a large number of trials, with relative frequencies that are approximately equal to one another, and thus approximately equal to 1/*s*; hence they are called equally possible or equally probable cases. This assertion can be made subjectively in this and other schemes, but it is universally accepted on the basis of an examination of the conditions, without recourse to observation. It follows that one can make an a priori assessment of the likelihood of the event occurring in a trial, given by the ratio *b/s* between the number of favorable cases and the number of (equally) possible cases, and that this a priori assessment, called the probability of the event, is a practical prediction of the a posteriori assessment provided 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 geometric center, which is thrown to observe the number on the upper face, etc.), in which the intervention of “chance” is recognized as due either to the presence of a large number of causes acting on the phenomenon—causes that cannot all be considered, either because of their number or because some are ill-defined—or to the fact that imperceptible differences in the initial conditions can lead to significant variations in the final outcome, making the result unpredictable based on the imperfect knowledge of the initial situation. The foregoing considerations extend to schemes involving a continuous infinity of possible cases (for example, the scheme of a needle rotating about its center, the position of which is observed after a large number of rotations following a push): 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 a biased coin lands “heads” when tossed on a plane under certain precautions to ensure that the trials are conducted “under identical conditions”).
Therefore, with reference to the schemes described above, and always from an experimental standpoint, it can be affirmed: the probability of a random event in a given trial is a constant between zero and one, capable of predicting the relative frequency of the event in a large number of trials conducted under identical conditions. When a finite number of equally possible cases is recognized, some of which are favorable to the event, the probability of the event is given by the ratio between the number of favorable cases and the number of possible cases; this definition is appropriately 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 an event is possible in a trial, its probability is greater than zero; if it is certain, its probability is equal to one. The probability that, when tossing a fair coin “at random,” “heads” results is 1/2. Therefore, if, after tossing a coin a large number of times, the relative frequency of the event “heads” differs noticeably from 1/2, it can be affirmed—with increasing certainty as the number of trials grows—that the coin is biased or that the tosses are not performed regularly.
What is the probability that, in the Italian lottery, in a single drawing on a single wheel, a specific number is drawn? The number of possible five-number combinations that can be drawn from the urn (containing 90 balls numbered 1 to 90) representing the (equally) possible cases is 90 × 89 × 88 × 87 × 86 / (1 × 2 × 3 × 4 × 5) = 43,049,268, while the number of favorable cases is given by the number of five-number combinations containing the given number, that is, by the number of possible triplets that can be formed with the other 88 numbers, namely 88 × 87 × 86 / (1 × 2 × 3) = 109,736; hence the probability of a single number is equal to the quotient of the second number divided by the first, that is, 1/400.5. The probability of a “terno” (three-number match) is 1/11,748, and that of a “quaterna” (four-number match) is 1/511,038.
The probabilities of events belonging to the same probabilistic scheme are governed by the two principles of total probability and compound probability. The first states that, when two or more events are mutually exclusive, 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, the event consisting in the occurrence in a trial of all the considered events—is equal to the product of the probabilities of the individual events, each determined on the hypothesis that the preceding events have occurred; in particular, if, with reference to a certain ordering of the events, each is independent of the logical product of the preceding events—that is, if its probability of occurring together with all the considered events is equal to the product of the probabilities of the individual events. The two principles of total and compound probability, with reference to the indicated schemes, are experimentally verified.
From an abstract point of view, a probabilistic scheme consists of a set of entities called events, among which relations of mutual exclusivity and compatibility are defined, and to each of which corresponds a number between zero and one, called the probability of the event; the probabilities of the various events are related to one another by the principles of total and compound probability, which, when admitted for a countable infinity of events, are called extended principles. From this theoretical standpoint, the so-called calculus of probability 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, satisfying the probabilities to the empirical postulate of chance, then, by determining, on the basis of the two indicated principles, the probability of any event deduced from the initially considered events by means of logical sum or product operations, this probability also 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 as 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, is said to be objective insofar as there is unanimous agreement; even for other events that occur in reality (the event that a given horse wins a race; the event that a given party achieves a majority in elections), one speaks practically of probability, but in such cases the evaluations of probability are called “subjective” to distinguish them from the previous cases, insofar as different persons, based on differing knowledge of the phenomenon and also on differing characteristics of human nature, are led to evaluate this probability differently, and there is no way to experimentally verify these evaluations through a large number of trials under the same conditions. Even in these cases, the calculus of probability is applicable, but it serves only as a rule of conduct for the necessary coherence between the subjective evaluations of the probabilities of given events and those of events deduced from the previous ones by means of logical sum and product operations.
In the calculus of probability, the study of random variables occupies a prominent place—that is, those quantities linked to a system of mutually exclusive random events such that one of them certainly occurs in a trial, so that the value each 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 it can 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 assumes in a large number of trials. The deviation of a random variable from its mean value is the difference between the values 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 achieve 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 achieves 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 playing the lottery on a “ambo” receives, in case of a win, 250 times the stake; therefore the gain, positive or negative, that the bank can achieve for each lira wagered on a given ambo is a random variable that can assume one of the values, in lire, 1 or –249, depending on whether the ambo does not come up or does come up, mutually exclusive events with probabilities 399.5/400.5 and 1/400.5 respectively. Hence the mathematical expectation of the bank is lire (1 × 399.5/400.5 – 249 × 1/400.5) = lire 0.376: that is, the bank on average wins about 38 cents for every lira wagered on ambo, and even more for terno or quaterno. To justify this fact, one must take into account the bank’s expenses as well as the need for a margin to cover possible unfavorable deviations.
The standard deviation is the square root of the mean value of the square of the deviation; it is an index suited to judge the degree of clustering, 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, the sum and product of two or more random variables belonging to the same probabilistic scheme are considered. 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 so-called calculus of probability (c. d. p.), besides that of Bernoulli (v. MATEMATICA, XIII), which, with its generalization to random variables (Tchebycheff, 1867), is termed the "Law of Large Numbers," the uniform law of large numbers (Cantelli, 1917) is fundamental. This law, in the particular case of the Bernoulli scheme for an event with constant probability in each trial, states that, with a probability as close to one as desired, the successive relative frequencies of the event, in an unlimited sequence of trials, tend, in the sense of analysis, toward the probability of the event. It thus allows for a precise formulation of the meaning of the empirical postulate of chance.
Another famous theorem, that of Bayes (1764), concerning the probability of causes, has found applications in the assessment of 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, allows one to affirm that the phenomenon of mortality among individuals possessing homogeneous characteristics (same age, same social category, all initially in good health, etc.) behaves, albeit with due approximations, like a random event with constant probability in each trial. Thus, one can formally speak of the probability of death and apply the c. d. p., always from a normal standpoint, to problems of statistics and actuarial mathematics. Other applications of the c. d. p. pertain to the theory of observational errors, which, in general, when measurements are one-dimensional, are distributed according to the law of Gauss (v. MATEMATICA, XIII), while two-dimensional errors are distributed according to the law of Bravais (1847); this subject includes the study of the dispersion of artillery fire against a target. Other fundamental applications of the c. d. p. concern the theory of gases and wave mechanics, among others.