784 PROBABILITY The rule in such cases is that the sum of squares of the apparent errors is to be made a minimum, as in the case of a single element. To take a very simple example : A substance is weighed, and the weight is found to be W. It is then divided into two portions, whose weights are found to be P and Q. What is the most probable weight of the body ? Taking A and B as the weights of the two portions, the apparent errors are P - A, Q - B, and that of the whole is W - A - B ; hence (P - A) 2 + (Q - B) 2 + ( W - A - B) 2 = minimum there being two independent variables A, B. 2B + A-Q + W dx-=2c. 2 Thus the probability required is p--2c/a. Laplace in solving this question suggests that by making a great number of trials, and counting the cases where the rod falls on a line, we could determine the value of ir from this result. He further considers, for a given value of a, what length 2c should be chosen for the rod so as tp give the least chance of error in a given large number N of throws. In art. 8 we have shown that the chance that the number of successes shall lie between ^>Nr is 9 /~ ra x? -: c/ c dx > where i 2?(1 -p)N For a given probability =-, ra is given. We have then a given chance that the number of successes shall differ from its most pro bable value //NT by an error r which is the least possible fraction of the latter when r/pN, or when 1/apN, or when VpU~-p)/P is trie least possible; that is, when p~ l - I =im/2c- 1 1 is the least 1 If S = number of successes, we have an assigned chance zr that S lies between ;A"ir; that is, the value uf ir Us between _-* , or c -( - -L ) a 8r a S S 2 / Hence the error in n is least when 2cr/S 2 is lea-t. Now roc vXl />)", Scocj), and St-xp nearly; hence Vp(l-p>/P is to be- the least possible. which are the most probable weights of the whole and the two parts. VI. ON LOCAL PROBABILITY. 60. It remains to give a brief account of the methods of deter mining the probabilities of the fulfilment of given conditions by variable geometrical magnitudes, as well as the mean values of such magnitudes. Recent researches on this subject have led to many very remarkable results ; and we may observe that to English mathematicians the credit almost exclusively belongs. It is a new instance, added to not a few which have gone before, of a revival for which we have to thank the eminent men who during the 19th century have enabled the country of Newton to take a place less unworthy of her in the world of mathematical science. At present the investigations on this subject have not gone beyond the theoretical stage ; but they should not be undervalued on this account. The history of the theory of probabilities has sufficiently shown that what at first seems merely ingenious and a [ matter of curiosity may turn out to have valuable applications to : practical questions. How little could Pascal, James Bernoulli, and De Moivre have anticipated the future of the science which they were engaged in creating ? 61. The great naturalist Buffon was the first who proposed and solved a question of this description. It was the following : A floor is ruled with equidistant parallel lines ; a rod, shorter than the distance between each pair, being thrown at random on the floor, to find the chance of its falling on one of the lines. Let x be the distance of the centre of the rod from the nearest line, 6 the inclination of the rod to a perpendicular to the parallels, 2a the common distance of the parallels, 2c the length of rod ; then, as all values of x and Q between their extreme limits are equally probable, the whole number of cases will be represented by dxdO- Now if the rod crosses one of the lines we must Lave so that the favourable cases will be measured by /-., .. /-cco.sf possible, or when c is the greatest possible. Now the greatest value of c is a ; the rod therefore should be equal to the distance between the lines. Laplace s answer is incorrect, though originally given right, (see Todhunter, p. 591 ; also Czuber, p. 90). 62. Questions on local probability and mean values arc of course reducible, by the employment of Cartesian or other coordinates, to multiple integrals. Thus any one relating to the position of two variable points, by introducing their coordinates, can be made to depend on quadruple integrals, whether in finding the sum of the values of a given function of the coordinates, with a view to obtaining its mean value, or in finding the number of the favour able cases, when a probability is sought. The intricacy and difficulty to be encountered in dealing with such multiple integrals and their limits is so great that little success could be expected in attacking such questions directly by this method ; and most of what has been done in the matter consists in turning the difficulty by various considerations, and arriving at the result by evading or simplifying the integrations. AVe have a certain analogy here in the variety of contrivances and artifices used in arriving at the values of definite integrals without performing the integrations. We will now select a few of such questions. 63. If a given space S is included within a given space A, the chance of a point P, taken at random on A, falling on S, is j-S/A. But if the space S be variable, and M(S) be its mean value (66). For, if we suppose S to have n equally probable values Sj, S 2 , S 3 . . . . , the chance of any one S x being taken, and of P falling on S lf is fc-n- Sj/A; now the whole probability p = Pi + 2>->+2 } a + > which leads at once to the above expression. The chance of two points falling on S is, in the same way, and so on. In such a case, if the probability be known, the mean value follows, and vice versa. Thus, we might find the mean value of the ntii power of the distance XY between two points taken at random in a line of length Z, by considering the chance that, if n more points are so taken, they shall all fall between X and Y. This chance is for the chance that X shall be one of the extreme points, out of the whole (?i + 2), is 2(?i + 2)~ 1 ; and, if it is, the chance that the other extreme point is Y is (n + 1)- 1 . Therefore M(XY) = 2Z(n + 1) -
+ 2) - 1 .
64. A line I is divided into n segments by n - 1 points taken at random ; to find the mean value of the product of the n segments. Let a, b, c, . . . be the segments in one particular case. If n new points are taken at random in the line, the chance that one falls on each segment is 1.2.3... nabc . . . /l n ; hence the chance that this occurs, however the line is divided, is Now the whole number of different orders in which the whole 2n - 1 points may occur is 1 2/( - 1 ; out of these the number in which one of the first series falls between every two of the second is easily found by the theory of permutations to be | n | n - 1 . Hence the required mean value of the product is 65. If M be the mean value of any quantity depending on the positions of two points (e.g. , their distance) which are taken, one in a space A, the other in a space B (external to A) ; and if M be the same mean when both points are taken indiscriminately in the whole space A + B ; M a , M& the same mean when both points are taken in A and both in B respectively ; then (A + B) 2 M - 2 ABM + A 2 M a + B 2 M 6 . If the space A = B, 4M if, also, M B = M6, 2M - 66. The mean distance of a point P within a given area from a fixed straight line (which does not meet the area) is evidently the distance of the centre of gravity G of the area from the line. Thus, if A, B are two fixed points on a line outside the area, the mean value of the area of the triangle APB = the triangle AGB. From this it will follow that, if X, Y, Z are three points taken
at random in three given spaces on a plane (such that they cannot