GEOMETRICAL PROBARTLITY.
In what precedes it has been assumed except in one case that the number of equally probable events governing the event E is finite. The definition may be extended to include the case in which the number of events is infinite pro vided appropriate measures for the totalities of cases can be devised. Such measures are available when each case corresponds to a set of values or one or more continuous variables. Problems requiring the adoption of such measures are generally geometrical, and hence this branch of the subject is called geometrical probability. Merely an indication of its nature may be given by one or two very elementary examples.
Although the number of points on any line segment cannot be counted, yet their totality may be measured by the length of the segment. Similarly the totality of points is an area may be measured by that area; and so on. The probability that, all values being equally prob able, a number less than unity is also less than one-half is because the totality of cases is measured by a unit segment, and that of the favorable cases by half a unit segment. Simi larly, if all points within a circle of radius a are equally likely to be selected, the probability that one of them selected, at random, is not Z distant more than 2 — from the centre is f.
Many very interesting problems have been solved, some of them by exceedingly ingenious methods, but this branch of the subject is quite special and cannot receive further attention here.
In what precedes some important parts of the theory have been indicated. A study of its important applications, notably those to the discussion of errors of observations and to problems of insurance, must be sought else where. The growth of the importance of the subject has been remarkable, for in the begin ning it was concerned with problems of gam bling, and it has become the basis of all forms of insurance. It has attracted and held the deep attention of nearly all the ablest mathe maticians from Cardan to Sylvester and Poin cars. Among its devotees have also been Pascal, Fermat, De Moivre, Leibnitz, the Ber noullis, D'Alembert, Euler, Lagrange, Bayes, Condorcet, Laplace, Poisson, De Morgan, Bert rand and Czuber. The theory has been applied freely, and in many cases rashly, too little at tention having been paid to the fact that its applications really lie in the domain of those events whose occurrences may properly be com pared to the drawings of balls from a bag. Hence it is to exercise considerable care in connection with the literature of the subject. See LEAST SQUARE, METHOD OF ; STA TISTICAL METHOD.
Bibliography.— Bertrand, des Prob abilitesY (1889) • De Morgan, An Essay on Probabilities' (1838) ; Laplace, (Theorie Analy tiques des Probabilites' (1812) • Whitworth, 'Choice and Chance' (5th ed., 19151)- William son, (Integral Calculus (London 1906; chapter on Probability).