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 ^
 
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 PUBLISHED BY 
 
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 MATHEMATICAL MONOGRAPHS. 
 
 tniTED BV 
 
 MANSFIELD MERRIMAN and ROBERT S. WOODWARD. 
 
 No. 7. 
 
 PROBABILITY 
 
 AND 
 
 THEORY OF ERRORS. 
 
 BY 
 
 ROBERT S. WOODWARD, 
 
 President Carnegie Institution of Washington. 
 
 NEW YORK: 
 
 JOHN WILEY & SONS. 
 
 London: CHAPMAN & HALL, Limited. 
 
 630526
 
 Copyright, 1896, 
 
 liV 
 
 MANSFIELD MERRIMAN and ROBERT S. WOODWARD 
 
 UNDER THE T I LE 
 
 HIGHER MATHEMATI-CS. 
 
 First Edilion, September, i8g6. 
 Second Edition, January, 1898. 
 Third Edition, August, 1900. 
 Fourth Edition, January, 1906, 
 
 4/22
 
 EDITORS' PREFACE. 
 
 The volume called Higher Mathematics, the first edition 
 of which was published in 1896, contained eleven chapters by 
 eleven authors, each chapter being independent of the others, 
 but all supposing the reader to have at least a mathematical 
 training equivalent to that given in classical and engineering 
 colleges. The pubhcation of that volume is now discontinued 
 and the chapters are issued in separate form. In these reissues 
 it will generally be found that the monographs are enlarged 
 by additional articles or appendices which either amplify the 
 former presentation or record recent advances. This plan of 
 publication has been arranged in order to meet the demand of 
 teachers and the convenience of classes, but it is also thought 
 that it may prove advantageous to readers in special lines of 
 mathematical literature. 
 
 It is the intention of the pubhshers and editors to add other 
 monographs to the series from time to time, if the call for the 
 same seems to warrant it. Among the topics which are under 
 consideration are those of elHptic functions, the theory of num- 
 bers, the group theory, the calculus of variations, and non- 
 Euclidean geometry; possibly also monographs on branches of 
 astronomy, mechanics, and mathematical physics may be included. 
 It is the hope of the editors that this form of pubhcation may 
 tend to promote mathematical study and research over a wider 
 field than that which the former volume has occupied. 
 
 December, 1905. 
 
 Ill
 
 AUTHOR'S PREFACE. 
 
 In republishing this short treatise in book form the author 
 solicits criticism but offers no apology. The type of the book 
 he has sought to imitate is that shown in the " mathematical 
 tracts " of the late Sir George B. Air}'. The brevity and the con- 
 crete illustrations of these " tracts " have served very effectively 
 in introducing students to a number of the more difficult fields 
 of applied mathematics; and it is hoped that this treatise will 
 serve a similar end. 
 
 The theor\' of probabihty and the theory of errors now con- 
 stitute a formidable body of knowledge of great mathematical 
 interest and of great practical importance. Though developed 
 largely through appHcations to the more precise sciences of as- 
 tronomy, geodesy, and physics, their range of applicabihty extends 
 to all of the sciences; and they are plainly destined to play an 
 increasingly important role in the development and in the appli- 
 cations of the sciences of the future. Hence their study is not 
 only a commendable element in a liberal education, but some 
 knowledge of them is essential to a correct understanding of 
 daily events. 
 
 No special novelty of presentation is claimed for this work; 
 but the reader may find it advantageous to know that a definite 
 plan has been followed. This plan consists in presenting each 
 principle, first, by means of a simple, concrete example; passing, 
 secondly, to a general statement by means of a formula; and, 
 thirdly, illustrating appHcations of the formula by concrete 
 examples. Great pains have been taken also to secure clear and 
 correct statements of fundamental facts. If these latter are 
 duly understood, the student needs Httle additional aid; if 
 they arc not duly understood, no amount of aid will forward him. 
 
 The passage from the elementary concrete to the advanced 
 abstract may appear to be abrupt to the reader in some cases. 
 It is hoped, however, that any large gaps may be easily bridged 
 and that any serious difficulties may be easily overcome by means 
 of the references given to the literature of the subject. In any 
 event the student will find that in this, as in all of the more ardu- 
 ous sciences, his greatest pleasure and his highest discipline will 
 come from bridging such gaps and from surmounting such diffi- 
 culties. 
 
 Washington, D. C, December, 1905.
 
 CONTENTS. 
 
 Art. I. Introduction Page 7 
 
 2. Permutations 11 
 
 3. Combinations 13 
 
 4. Direct Probabilities 16 
 
 5. Probability of Concurrent Events 19 
 
 6. Bernoulli's Theorem 22 
 
 7. Inverse Probabilities 24 
 
 8. Probabilities of Future Events 27 
 
 9. Theory of Errors 30 
 
 ic. Laws of Error 3^ 
 
 11. Typical Errors of A System 33 
 
 12. Laws of Resultant Error 34 
 
 13. Errors of Interpolated Values 37 
 
 14. Statistical Test of Theory 44
 
 PROBABILITY AND THEORY OF ERRORS. 
 
 Art. 1. Introduction, 
 
 It is a curious circumstance that a science so profoundly 
 mathematical as the theory of probability should have origi- 
 nated in the games of chance which occupy the thoughtless and 
 the profligate.* That such is the case is sufficiently attested 
 by the fact that much of the terminology of the science and 
 many of its familiar illustrations are drawn directly from the 
 vocabulary and the paraphernalia of the gambler and the trick- 
 ster. It is somewhat surprising, also, considering the antiquity 
 of games of chance, that formal reasoning on the simpler 
 questions in probability did not begin before the time of Pascal 
 and Fermat. Pascal was led to consider the subject during the 
 year 1654 through a problem proposed to him by the Chevalier 
 de Mere, a reputed gamblcr.f The problem in question is 
 known as the problem of i)oints and may be stated as follows: 
 two pla3^ers need each a given number of points to win at a 
 certain stage of their game; if they stop at this stage, how should 
 the stakes be divided ? Pascal corresponded with his friend 
 Fermat on this question ; and it appears that the letters which 
 passed between them contained the earliest distinct formulation 
 of principles falling within the theory of probability. These 
 
 * The historical facts referred to in this article are drawn mostly from Tod- 
 hunter's History of the Mathematical Theory of Probability from the time of 
 Pascal to thai of Laplace (Cambridge and London, 1865). 
 
 f " Un problcme relatif aux jeux de hasard, propose a un austere janseniste 
 par un homme du monde, a ete I'origine du calcul des probabilites. " Poisson, 
 Recherches surla Probabilite des Jugements (Paris, 1837).
 
 8 PROBABILITY AND THEORY OF ERRORS. 
 
 acute thinkers, however, accompHshed little more than a correct 
 start in the science. Each seemed to rest content at the time 
 with the approbation of the other. Pascal soon renounced 
 such mundane studies altogether ; Fermat had only the scant 
 leisure of a life busy with affairs to devote to mathematics; 
 and both died soon after the epoch in question, — Pascal in 
 1662, and Fermat in 1665. 
 
 A subject which had attracted the attention of such dis- 
 tinguished mathematicians could not fail to excite the interest 
 of their contemporaries and successors. Amongst the former 
 Huygens is the most noted. He has the honor of publishing 
 the first treatise* on the subject. It contains only fourteen 
 propositions and is devoted entirely to games of chance, but it 
 gave the best account of the theory down to the beginning of 
 the eighteenth century, when it was superseded by the more elab- 
 orate works of James Bernoulli, f Montmort,:}; and De Moivre.§ 
 Through the labors of the latter authors the mathematical 
 theory of probability was greatly extended. They attacked, 
 quite successfully in the main, the most difficult problems; 
 and great credit is due them for the energy and ability dis- 
 played in developing a science which seemed at the time to 
 have no higher aim than intellectual diversion. || Their names, 
 undoubtedly, with one exception, that of Laplace, are the most 
 important in the history of probability. 
 
 Since the beginning of the eighteenth century almost every 
 mathematician of note has been a contributor to or an expos- 
 itor of the theory of probability. Nicolas, Daniel, and John 
 Bernoulli, Simpson, Euler, d'Alembert, Bayes, Lagrange, Lam- 
 bert, Condorcet, and Laplace are the principal names which 
 figure in the history of the subject during the hundred years 
 
 * De Ratiociniis in Ludo Aleae, 1657. 
 
 t Ars Conjectandi, 1713. 
 
 lEssai d'Analyse sur les Jeux de Hazards, 1708. 
 
 §The Doctrine of Chances, 1718. 
 
 ITodhunter says of Montmort, for example, "In 1708 he published his 
 worlc on Chances, where with the courage of Columbus he revealed anew world 
 to Mathematicians."
 
 INTRODUCTION. 9 
 
 ending with the first quarter of the nineteenth century. Of 
 contributions from this brilliant array of mathematical talent, 
 the Theorie Analytique des Probabilites of Laplace is by far 
 the most profound and comprehensive. It is, like his M^- 
 canique Celeste in dynamical astronomy, still the most elabo- 
 rate treatise on the subject. An idea of the grand scale of the 
 work in its present form* maybe gained by the facts that the 
 non-mathematical introductionf covers about one hundred and 
 fifty quarto pages; and that, in spite of the extraordinary 
 brevity of mathematical language, the pure theory and its ac- 
 cessories and applications require about six hundred and fifty 
 pages. 
 
 From the epoch of Laplace down to the present time the 
 extensions of the science have been most noteworthy in the 
 fields of practical applications, as in the adjustment of obser- 
 vations, and in problems of insurance, statistics, etc. Amongst 
 the most important of the pioneers in these fields should 
 be mentioned Poisson, Gauss, Bessel, and De Morgan. Nu- 
 merous authors, also, have done much to simplify one or an- 
 other branch of the subject and thus bring it within the range 
 of elementary presentation. The fundamental principles of 
 the theory are, indeed, now accessible in the best text-books 
 on algebra ; and there are many excellent treatises on the pure 
 theory and its various applications. 
 
 Of all the applications of the doctrine of probability none 
 is of greater utility than tlie theory of errors. In astronomy, 
 geodesy, physics, and chemistry, as in every science which at- 
 tains precision in measuring, weighing, and computing, a 
 knowledge of the theory of errors is indispensable. By the aid 
 of this theory the exact sciences have made great progress dur- 
 
 *The form of the third edition published in 1S20, and of Vol. VII of the 
 complete works of Laplace recently republished under the auspices of the 
 Acad6mie des Sciences by Gauthier-Villars. This Vol. VII bears the date 1886. 
 
 f " Cette Introduction," writes Laplace, " est le developpement d'une Le5on 
 sur les Probabilites, que je donnai en 1795, aux Ecoles Normales, oil je fus ap- 
 pele comme professeur de Mathematiques avec Lagrange, par un decret de la 
 Convention nationale."
 
 10 PROBABILITY AND THEORY OF ERRORS. 
 
 ing the nineteenth century, not only in the actual determination 
 of the constants of nature, but also in the fixation of clear 
 ideas as to the possibilities of future conquests in the same di- 
 rection. Nothing, for example, is more satisfactory and in- 
 structive in the history of science than the success with which 
 the unique method of least squares has been applied to the 
 problems presented by the earth and the other members of the 
 solar system. So great, in fact, are the practical value and 
 theoretical importance of the method of least squares, that it is 
 frequently mistaken for the whole theory of errors, and is 
 sometimes regarded as embodying the major part of the doc- 
 trine of [probability itself. 
 
 As may be inferred from this brief sketch, the theory of 
 probability and its more important applications now constitute 
 an extensive body of mathematical principles and precepts. 
 Obviously, therefore, it will be impossible within the limits of 
 a single condensed monograph to do more than give an out- 
 line of the salient features of the subject. It is hoped, how- 
 ever, in accordance with the general plan of the volume, that 
 such outline will prove suggestive and helpful to those who 
 may come to the science for the first time, and also to those 
 who, while somewhat familiar with the difficulties to be over- 
 tome, have not acquired a working knowledge of the subject. 
 Effort has been made especially to clear up the difficulties of 
 the theory of errors by presenting a somewhat broader view of 
 the elements of the subject than is found in the standard 
 treatises, which confine attention almost exclusively to the 
 method of least squares. This chapter stops short of that 
 method, and seeks to supply those phases of the theory which 
 are either notably lacking or notably erroneous in works 
 hitherto published. It is believed, also, that the elements here 
 presented are essential to an adequate understanding of the 
 well-worked domain of least squares.* 
 
 *The author has given a brief but comprehensive statement of the method 
 of least squares in the volume of Geographical Tables published by the Sniith- 
 sonion Institution, 1894.
 
 PERMUTATIONS. 
 
 n 
 
 Art. 2. Permutations. 
 
 The formulas and results of the theory of permutations 
 and combinations are often needed for the statement and so- 
 lution of problems in probabihties. This theory is now to be 
 found in most works on algebra, and it w ill therefore suffice 
 here to state the principal formulas and illustrate their mean- 
 ing by a few numerical examples. 
 
 The number of permutations of ii things taken r in a group 
 is expressed by the formula 
 
 (;/), = ;/(;/ - i)(;/ - 2) . . . {ii - r -f i). (i) 
 
 Thus, to illustrate, the number of ways the four letters a, b, 
 
 c, d can be arranged in groups of two is 4 . 3 = 1 2, and the groups 
 are 
 
 ab, ba, ac, ca, ad, da, be, cb, bd, db, cd, dc. 
 
 Similarly, the formula gives for 
 w = 3 and r = 2, (3), = 3-2 =6, 
 
 ;/==;" ;-= 3, (7)3 = 7.6.5 = 210, 
 
 n — \o " r = 6, (10), = 10.9.8. 7.6. 5 = 15 1200. 
 
 The results which follow from equation (i) when n and r 
 do not exceed 10 each are embodied in the following table : 
 
 VAi.uiis OF Permutations. 
 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 4 
 
 3 
 
 3 
 
 2 
 
 2 
 
 I 
 
 I 
 
 I 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 2 
 
 90 
 
 72 
 
 5fi 
 
 42 
 
 30 
 
 20 
 
 12 
 
 6 
 
 2 
 
 
 3 
 
 720 
 
 504 
 
 33^' 
 
 210 
 
 120 
 
 60 
 
 24 
 
 6 
 
 
 
 4 
 
 5040 
 
 3024 
 
 16S0 
 
 840 
 
 360 
 
 120 
 
 24 
 
 
 
 
 5 
 
 30240 
 
 15 I 20 
 
 6720 
 
 2S20 
 
 72iJ 
 
 120 
 
 
 
 
 
 6 
 
 1 5 I 200 
 
 60480 
 
 20160 
 
 5040 
 
 720 
 
 
 
 
 
 
 7 
 
 604800 
 
 1S1440 
 
 40320 
 
 5040 
 
 
 
 
 
 
 
 8 
 
 I S 14400 
 
 362S&0 
 
 40320 
 
 
 
 
 
 
 
 
 9 
 
 362SS00 
 
 362S80 
 
 
 
 
 
 
 
 
 
 10 
 
 362S800 
 
 
 
 
 
 
 
 15 
 
 4 
 
 I 
 
 Sp 
 
 9864100 
 
 986409 
 
 109600 
 
 13699 
 
 1956 
 
 325 
 
 64 
 
 The use of this table is obvious. Thus, the number of per- 
 mutations of eight things in groups of five each is found in the 
 fifth line of the column headed with the number 8. It will be
 
 12 PROBABILITY AND THEORY OF ERRORS. 
 
 noticed that the. last two numbers in each column (excepting 
 that headed with i) are the same. This accords with the for- 
 mula, whicli gives for the number of permutations of n things 
 in groups of ;/ the same value as for ;/ things in groups of 
 (;/ — i). It will also be remarked that the last number in each 
 column of the table is the factorial, n\, of the number n at the 
 head of the column. For example, in the column under 7, the 
 last number is 5040 = 1.2.3.4.5.6.7 = 7!. 
 
 The total number of permutations of n things taken singly, 
 in groups of two, three, etc., is found by summing the numbers 
 given by equation (i) for all values of r from i to n. Calling 
 this total or sum Sp, it will be given by 
 
 Sp = '2{7i\. (2) 
 
 To illustrate, suppose n = 3, and, to fix the ideas, let the 
 three things be the three digits i, 2, 3. Then from the above 
 table it is seen that Sp = :^-\-6-\-6= 15 ; or, that the number 
 of numbers (all different) which can be formed from those dig- 
 its is fifteen. These numbers are I, 2, 3 ; 12,13,21,23,31,32; 
 123, 132, 213, 231, 312, 321. 
 
 The values of Sp for ;/ = 1,2,... 10 are given under the 
 corresponding columns of the above table. But when 71 is 
 large the direct summation indicated by (2) is tedious, if not 
 impracticable. Hence a more convenient formula is desirable. 
 To get this, observe that (i) may be written 
 
 if r is restricted to integer values between i and (n — i), both 
 inclusive. This suffices to give all terms which appear in the 
 right-hand member of (2), since the number of permutations 
 for r = {n — 1) is the same as for r = n. Hence it appears 
 that 
 
 ^ ' I I . 2 ' In — ^\\ 
 
 {n - I) 
 \ ' I I . 2 ' (;/ — I ) !/
 
 COMBINATIONS. 13 
 
 But as n increases, the series by which 7i\ is here multiplied 
 approximates rapidly towards the base of natural logarithms ; 
 that is, towards 
 
 ^ = 271 828 1 8 +, log ^ = 0.4342945. 
 Hence for large values of n 
 
 5;i = ;^ ! ^, approximately.* (3) 
 
 To get an idea of the degree of approximation of (3), sup- 
 pose n = 9. Then the computation runs thus (see values in 
 
 the above table) : 
 
 log 
 
 9! = 362880 5-5597630 
 e 0.4342945 
 
 91^ = 986410 5.9940575 
 Sp = 986409 by equation (2). 
 
 The error in this case is thus seen to be only one unit, or 
 about one-millionth of Sp.-f 
 
 Prob. r. Tabulate a list of the numbers of three figures each 
 which can be formed from the first five digits i, . . . 5. How many 
 numbers can be formed from the nine digits ? 
 
 Prob. 2. Is Sf, always an odd number for n odd ? Observe 
 values of Sp in the table above. 
 
 Art. 3. Combinations. 
 
 In permutations attention is given to the order of arrange- 
 ment of the things considered. In combinations no regard is 
 paid to the order of arrangement. Thus, the permutations of 
 the letters a, d, c, d in groups of three are 
 
 {abc) (abd) bac bad acb {acd) cab cad 
 adb adc dab dac bca {bed) cba cbd 
 bda bdc dba dbc cda cdb dca deb 
 
 * See Art. 6 for a formula for computing n\ when n is a large number. 
 
 f When large numbers are to be dealt with, equations (i)' and (3) are easily 
 managed by logarithms, especially if a table of values of log (;/!) is available. 
 Such tables are given to six places in De Morgan's treatise on Probability in 
 the Encyclopaedia Metropolitana, and to five places in Shortrede's Tables 
 (Vol. I, 1849).
 
 14 PROnABlLITV AND THEORY OF ERRORS. 
 
 Rut if the order of arrangement is ignored all of these are 
 seen to be repetitions of the groups enclosed in parentheses, 
 namely, {abc), {abd), [acd), [bed). Hence in this case out of 
 twent}'-four permutations there are only four combinations. 
 
 A general fcjrmula for computing the number of combina- 
 tions of ;/ things taken in groups of r things is easily derived. 
 For the number of permutations of n things in groups of r is 
 b\- (i) of Art. 2 
 
 {,!),. = ;/(;/ - l)(;/ _ 2) ...(;/- r + i) ; 
 
 and since each group of r things gives i . 2 . 3 . . . r = r! per- 
 mutations, the number of combinations must be the cjuotient 
 of (//),. by r\. Denote this number by C[u)^. Then the gen- 
 eral formula is 
 
 This formula gives, for example, in the case of the four let- 
 ters a, b, c, d taken in groups of three, as considered above, 
 
 4.3.2 
 
 Multi[)ly both numerator and denominator of the right-hand 
 member of (i) b}- {n — r) ! The result is 
 
 which shows that the number of combinations of n things in 
 groups of r is the same as the number of combinations of n 
 things in groups of (;; — r). Thus, the number of combina- 
 tions of the first ten letters a, b, c . . .j in groups of three or 
 
 seven is 
 
 10! 
 
 -— — - — 120. 
 
 3'- 7! 
 The following table gives the values C{ii)^ for all values of 
 ;/ and r froin I to 10. 
 
 The mode of using this table is evident. For example, the 
 number (^f combinations of eight things in sets of five each is 
 found on the fifth line of the column headed 8 to be 56.
 
 combinations. 
 Values of Combinations. 
 
 15 
 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 I 
 
 I 
 
 lO 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 I 
 
 2 
 
 45 
 
 36 
 
 28 
 
 21 
 
 15 
 
 10 
 
 () 
 
 3 
 
 I 
 
 
 3 
 
 I20 
 
 84 
 
 56 
 
 35 
 
 20 
 
 10 
 
 4 
 
 I 
 
 
 
 4 
 
 2IO 
 
 126 
 
 70 
 
 35 
 
 15 
 
 5 
 
 I 
 
 
 
 
 
 252 
 
 126 
 
 56 
 
 21 
 
 b 
 
 I 
 
 
 
 
 
 6 
 
 210 
 
 84 
 
 28 
 
 7 
 
 I 
 
 
 
 
 
 
 7 
 
 120 
 
 36 
 
 8 
 
 I 
 
 
 
 
 
 
 
 8 
 
 45 
 
 9 
 
 I 
 
 
 
 
 
 
 
 
 9 
 
 10 
 
 I 
 
 
 
 
 
 
 
 
 
 10 
 
 I 
 
 
 
 
 
 
 
 
 
 
 Sc 
 
 1023 
 
 511 
 
 255 
 
 127 
 
 03 
 
 31 
 
 15 
 
 7 
 
 3 
 
 
 It will be observed that the numbers in any column show 
 a maximum value when n is even and two equal maximum 
 values when // is odd. That this should be so is easily seen 
 from (i)', which shows that C{/i),. will be a maximum for any 
 value of ;/ when ;■ ! (// — ;') ! is a minimum. For ;/ even this is 
 a minimum for r = jyJi ; while for n odd it has equal minimum 
 values for r = l{/i — i) and r — \{ii -\- i). Thus, 
 
 maximum of C{ii)r 
 
 n 
 
 n 
 
 7, for ;/ even, 
 
 n\ 
 
 (2) 
 
 n-]- I , Ji - 1 , 
 
 for ;/ odd. 
 
 2 2 
 
 The total number of combinations of ?/ things taken singly, 
 in groups of two, three, etc., is found by summing the numbers 
 given by (i) for all values of r from I to ;/ both inclusive. 
 Calling this total or sum S^, 
 
 The same sum will also come from (i)' by giving to rail values 
 from I to {?i — i), both inclusive, summing the results, and in- 
 creasing their aggregate by unity. Thus by either process 
 
 I . 2 
 
 1.2.3
 
 IG PROBABILITY AND THEORY OF ERRORS. 
 
 The second member of this equation is evidently equal to 
 
 (i + 0" ~~ I- Hence 
 
 S, = 2C{fi),. = 2" - I. (3) 
 
 The values of S, for values of ;/ and r from i to lo are given 
 under the corresponding columns of the above table. 
 
 Prob. 3. How many different squads of ten men each can be 
 formed from a company of 100 men ? 
 
 Prob. 4 How many triangles are formed by six straight lines 
 each of which intersects the other five ? 
 
 Prob. 5. Examine this statement : " In dealing a pack of cards 
 the number of hands, of thirteen cards each, which can be produced 
 is 635 013 559 600. But in whist four hands are simultaneously held, 
 and the number of distinct deals . . . would require twenty-eight 
 figures to express it." * 
 
 Prob. 6. Assuming combination always possible, and disregarding 
 the question of proportions, find how many different substances 
 could be produced by combining the seventy-three chemical ele- 
 ments. 
 
 Art. 4. Direct Probabilities. 
 
 If it is known that one of two events must occur in any 
 trial or instance, and that the first can occur in a ways and the 
 second in d ways, all of which ways are equally likely to hap- 
 pen, then the probability that the first will happen is expressed 
 mathematically by the fraction a/{a-\~/^), while the probability 
 that the second will happen is /?/{a -\- b). Such events are said 
 to be mutually exclusive. Denote their probabilities hy p and 
 q respectively. Then there result 
 
 the last equation following from the first two and being the 
 mathematical expression for the certainty that one of the two 
 events must happen. 
 
 Thus, to illustrate, in tossing a coin it tnust give " head " or 
 " tail" ; ^ = /; = I, and /^ = r/ = 1/2. Again, if an urn contain 
 ^ =: 5 white and /5 = 8 black balls, the probability of drawing 
 
 * Jevons, Principles of Science, New York, 1S74, p. 217.
 
 blRECT rROlSABlLITlES. 17 
 
 a white ball in one trial is p = 5/13 and that of drawing a 
 black one q = 8/13. 
 
 Similarly, if there are several mutually exclusive events 
 which can occur in a, b, c . . . wa}-s respectively, their probabil- 
 ities/, (], r . . . are given by 
 
 a b c 
 
 ^ "^ ~a-\-b-^c--Y. .' ' ^ ^ a-^b^c-\-. . . ' ^ ^ a^b^c-\-... ' 
 
 (2) 
 / + (7 + ^ + - • •= I- 
 
 For example, if an urn contain ^7=4 white, <^ = 5 black, 
 and (f ^ 6 red balls, the probabilities of drawing a white, black, 
 and red ball at a single trial are ^ = 4/15, q =. 5/15, and 
 r — 6/15, respectively. 
 
 Formulas (i) and (2) may be applied to a wide variety of 
 cases, but it must suffice here to give only a few such. As a 
 first illustration, consider the probability of drawing at random 
 a number of three figures from the entire list of numbers which 
 can be formed from the first seven digits. A glance at the 
 table of Art. i shows that the s\'mbols of formula (1) have in 
 this case the values a =■ 210, and <'r -j- /; = 13699. Hence 
 b = 13489, and / = 210/13699 ; that is, the probability in ques- 
 tion is about 1/65. 
 
 Secondly, what is the probability of holding in a hand of 
 
 whist all the cards of one suit ? Formula (i) of Art. 3 shows 
 
 that the number of different hands of thirteen cards each which 
 
 may be formed from a pack of fifty-two cards is 
 
 52. 5 1. 50... 40 ^ ^ 
 
 - — - — =^ ^— = 635 o 1 3 5 59 600, 
 
 I . 2 . 3 ... 13 ^^ ^ ^^^ 
 
 and the probability required is the reciprocal of this number. 
 The probability against this event is, therefore, very nearly 
 unit)'. 
 
 Thirdly, consider the probabilities presented by the case of 
 an urn containing 4 white, 5 black, and 6 red balls, from which 
 at a single trial three balls are to be drawn. E\'idcntl\' the 
 triad of balls drawn may be all white, all black, all red, partly 
 white and black, partly white and red, partly black and red, or
 
 18 PROIlAIilLlTV AND THEORY OF ERRORS. 
 
 one each of the white, bLick, and red. There are thus seven 
 different probabihties to be taken into account. The theory 
 of combinations shows (see equation (i), Art. 3) that the total 
 
 number of 
 
 4.^.2 
 
 White triads = = 4 = a 
 
 1.2.3 
 
 C A ^ 
 
 Black triads = .; — = 10 = (5 
 
 6 
 
 6.5.4 
 Red triads = — 7 = 20 — c 
 
 I . 
 
 2 . 
 
 3 
 
 5. 
 
 .4. 
 6 
 
 ^ 
 
 6, 
 
 •5. 
 
 4 
 
 
 6 
 
 
 9 
 
 .8. 
 
 7 
 
 
 b 
 
 
 10 
 
 ■9 
 
 .8 
 
 
 6 
 
 
 1 1 
 
 .10.9 
 
 White and black triads =: — — ( 4+10)= 70 = d 
 
 10. 9 . 8 ^ 
 White and red triads = ^ — ( 4-I-20)— 96 = ^ 
 
 Black and red triads 1^ — '—, — '- (io-[-2o)= 135 =y 
 
 White, black, and red triads ^4.5.6 :=I20— ^ 
 
 Sum = 455 
 The total number of these triads is 455, and is, as it should 
 be, the number of combinations in groups of three each of the 
 whole number of balls. Hence formulas (2) give the seven 
 different probabilities which follow, using the initial letters 
 w, d, r to indicate the colors represented in a triad : 
 For a triad ivtmv p = 4/455, 
 
 " " " bbb q = 10/455, 
 
 " " " rrr r = 20/455, 
 
 " " " 7V'ci'b or 7c>bb s = 70/455, 
 
 " " " 7V7vr or tvrr t = 96/455, 
 
 " " " bbr ov brr // = 135/455, 
 
 " " " wbr V = 120/455. 
 
 Prol). 7. Wlien three dice are thrown together, what is the prob- 
 ability that tlie tlirow will be greater than 9 ? 
 
 Prol). 8. Write down a literal fornuila for the probabilities of the 
 several ])ossible triads considered in the above question of the balls, 
 sui)i)osing the numbers of white, black, and red balls to be /, ;«. n, 
 respectively.
 
 PROBABILITY OF CONCURRENT EVENTS. 19 
 
 Art. 5. Probability of Concurrent Events. 
 
 If the probeibilities of two independent events are/, and 
 p.^, respectively, the probability of their concurrence in any 
 single instance is pp^. Thus, suppose there are two urns 
 U^ and U^, the first of which contains a^ white and l\ black 
 balls, and the second a^ white and b^ black balls. Then the 
 probability of drawing a white ball from U^ is/, = aj{(i^ -}~ '^i)> 
 while that of drawing a white ball from U^ is/^ := ^'^Ji^^^i + b^). 
 The total number of different pairs of balls which can be formed 
 from the entire number of balls is {a^ -\- h^){a^ -j" ^^- ^^ these 
 pairs a^a^ are favorable to the concurrence of white in simul- 
 taneous or successive drawings from the two urns. Hence the 
 probability of a concurrence of 
 
 white with white = a^a^/{a^ -j- b^){a^ -\- b^, 
 
 white with black = {aj?^'\-a^b^)/{a, -\- b,){a^ -\- b^, 
 
 black with black = b,b^/{a, + <^,)(^, -f /7J, 
 
 and the sum of these is unity, as required by equations (2) of 
 Article 4. 
 
 In general, if /,, /j, /j . . . denote the probabilities of several 
 independent events, and P denote the probability of theii 
 concurrence, 
 
 P = P,PiP^"' ' (i) 
 
 To illustrate this formula, suppose there is required the 
 probability of getting three aces with three dice thrown simul- 
 taneously. In this case/, = p^ = p^ = 1/6 and 
 P={\/6f = \/2\6. 
 
 Similarly, if two dice are thrown simultaneously the proba- 
 bility that the sum of the numbers shown will be 11 is 2/36; 
 and the probability that this sum i i will appear in two succes- 
 sive throws of the same pair of dice is ^/T)6.t^6. 
 
 The probability that the alternatives of a series of events 
 will concur is evident 1\- given by 
 
 Q = q.q.iz . . . = (i - A)(i -A)(i -A) • • • (2) 
 
 Thus, in the case of the three dice mentioned above, the 
 probability that each will show something other than an ace is
 
 I 
 
 20 PROBABILITY AND THEORY OF ERRORS. 
 
 q^~ q^— q^ = 5/6, and the probability that they vvill concur in 
 this is (2 = 125/216. 
 
 Many cases of interest occur for the apphcation of (i) and 
 (2). One of the most important of these is furnished by suc- 
 cessive trials of the same event. Consider, for example, what 
 may happen in n trials of an event for wliich the probability 
 is /> and against which the probability is q. The probabilit> 
 that the event will occur every time is/". The probability that 
 the event will occur (// — I) times in succession and then fail is 
 /" " Vy. But if the order of occurrence is disregarded this last 
 combination may arrive in ;/ different ways; so that the prob- 
 abilit)' that the event will occur (;/ — I) times and fail once is 
 iip"~\j. Similarly, the probability that the event will happei* 
 (/^ — 2) times and fail twice is \n{}i — •)/ 'V ! ^^^' That is, 
 the probabilities of the several possible occurrences are given b)' „ 
 the corresponding terms in the development of (/ + q)". " 
 
 By the same reasoning used to get equations (2) of Art, 
 3 it may be shown that the maximum term in the expansion 
 of (/H-f/)" is that in which the exponent ;;/, say, of q is 
 the whole number lying between {u -\- \)q — i and {n -\- i)q. ^ 
 In other words, the most probable result in « trials is the 
 occurren'ce of the event {n — m) times and its failure 7n 
 times. When n is large this means that the most probable of 
 all possible results is that in which the event occurs u — nq 
 — ii{\ — q) z= np times and fails 7/(7 times. Thus, if th.e event 
 be that of throwing an ace with a single die the most probable 
 of the possible results in 600 throws is that of 100 aces and 
 500 failures. 
 
 Since q'' is the probability that the event will fail every time 
 in // trials, the [)robability that it will occur at least once in n 
 trials is I — q'\ Calling this probability r,* 
 
 r= I -^"= I -(i -p)\ (3) 
 
 If r in this equation be replaced by 1/2, the corresponding 
 value of n is the number of trials essential to render the 
 
 * See Poisson's Probabilite des Jugements, pp. 40, 41. 
 
 I
 
 PROBABILITY OF CONCURRENT EVEN IS. 21 
 
 chances even that the event whose probabihty is/ will occur 
 
 at least once. Tiiiis, in this case, the value of n is given by 
 
 log 2 
 ;/ = — 
 
 log (I-/)' 
 
 This shows, for example, if the event be the throwing of double 
 sixes with two dice, for which / = 1/36, that the chances are 
 even (r = 1/2) that in 25 throws {ji — 24.614 by the formula) 
 double sixes will appear at least once. 
 
 Equation (3) shows that however small/ may be, so long as 
 it is finite, n may be taken so large as to make r approach in- 
 definitely near to unity ; that is, 11 may be so large as to render 
 it practically certain that the event will occur at least once. 
 
 When 71 is large 
 
 (. M« T ... I '^(^^~ ^) >.^ ;/(;?- l){n-2) 
 
 (I -/)" =1 - np -\ :^—^P TTTs ^ + • • • 
 
 = I — np A h . • . 
 
 ■' ' 1.2 1.2.3' 
 
 = e~"^ approximately. 
 
 Thus an approximate value of r is 
 
 r= I — e- "f, \ogc= 0.4342495. (4) 
 
 This formula gives, for example, for the probability of drawing 
 the ace of spades from a pack of fifty-two cards at least once in 
 104 trials r= i —^~'— 0.865, while the exact formula (3) 
 gives 0.867. 
 
 Similarly, the probability of the occurrence of the event at 
 least / times in ;/ trials will be given by the sum of the terms 
 of (/ 4- (])" from p" up to that in p'g"~* inclusive. This proba- 
 bility must be carefully distinguished from the probability that 
 the event will occur / times only in the n trials, the latter being 
 expressed by the single term in p'q"~^. 
 
 Prob. g. Compare the probability of holding exactly four aces in 
 five liands of whist with the probability c5 'lolding at least four aces 
 in the same number of hands. 
 
 Prob. 10. What is the probability of an event if the chances are 
 even that it occurs at least once in a million trials? See equation (4).
 
 T= '^ ' - ■'<" -:^ ■ • • ^^' + ' W = r^,r<r. (.) 
 
 22 PROBABILITY AND THEORY OF ERRORS. 
 
 Art. 6. Bernoulli's Theorem. 
 
 Denote the exponents of/ and q in the maximum term of 
 (^pj^gY by // and iii respectively, and denote this term by T. 
 
 Then 
 
 I 
 
 As shown in Art. 5, /< in this formula is the greatest whole 
 number in {;/ -}- i^p, and /// the greatest whole number in 
 (^jiJ^ i)^; so that when )i is large, /t and in are sensibly equal 
 to ;// and nq respectively. 
 
 The direct calculation of T by (i) is impracticable when n 
 is large. To overcome this difficulty the following expression 
 is used :* 
 
 .! = ;/V-"v'2^(i+:^ + ^ +...). (2) 
 
 log ^=0.4342495, log 2.T= 0.7981799. 
 
 This expression approaches n"c"' Vmn as a limit with the 
 increase of //, and in this approximate form is known as Stir- 
 ling's theorem. Although a rude approximation to // ! for 
 small values of // this theorem suffices in nearly all cases 
 wherein such probabilities as 7" are desired. Making use of 
 the theorem in (i) it becomes 
 
 V 27tiipq 
 
 That this formula affords a fair approximation even when 
 n is small is seen from the case of a die thrown 12 times. The 
 probability that any particular face will appear in one thrt)w is 
 p =: 1/6, whence q = 5/6; and the most probable result in 12 
 throws is that in which the particular face appears twice and 
 fails to appear ten times. The probability of this result com- 
 puted from (3) is 0.309, while the exact formula (i) gives 0.296. 
 
 The probability that the event will occur a number of times 
 
 * This expression is due to Laplace, Tli6oiie .•\nalytique des Probabilit^s. 
 See also De Morgan's Calci^lus, pp. 600-604.
 
 Bernoulli's theorkm. 33 
 
 comprised between (/< — a) and (// -[- c^) in /i trials is evidently 
 expressed by the sum of the terms in {p + g)" for which the 
 exponent of /> has the specified range of values. Calling this 
 probabilit)' A', jniuing 
 
 1^1 =1 lip -j- //, and in = iiq — //, 
 
 and using Stirling's theorem (which implies that // is a large 
 number),^ 
 
 R = 2-=-\i J^ -r] I 
 
 V2n//pcj^ npl \ nql 
 
 > 
 
 very nearly ; and the summation is with respect to ti from 
 // = — a \.o i( ^^ ^ a. But expansion shows that the natural 
 logarithm of the product of the two binomial factors in this 
 equation is approximately — if /2npq. Hence 
 
 R = :2 — ^ — ^-"'^/a"/? . 
 
 \'2n)ipq 
 
 and. since n is supposed large, this may be replaced by a definite 
 integral, putting 
 
 Thus 
 
 dz — i/Vinpq, and s" = n" /2npq. 
 
 -y a/ ^%npq a/ ^Inpq 
 
 R 
 
 = -^ / c-'-\{z = -^ / i—dz. (4) 
 
 Tliis equation expresses the theorem of James Bernoulli, 
 given in iiis Ars Conjectandi, published in 1713. 
 
 The value of the right-h.and member of (4) varies, as it 
 should, between o and I, and ap[)roaches the latter limit rap- 
 idly as s increases. Thus, writing for brevity 
 
 hi-. 
 
 V? 
 
 
 
 * See Bertrand, Calcul des Probabilites, Paris, 1889, for an extended discus- 
 sion of the questions considered in this Article.
 
 24 
 
 PROBABILITY AND THEORY OF ERRORS. 
 
 the followiiigr table shows the march of the intec;ral 
 
 z 
 
 / 
 
 z 
 
 I 
 
 z 
 
 I 
 
 o.oo 
 
 0.000 
 
 0.75 
 
 0.7II 
 
 1.50 
 
 0.966 
 
 •25 
 
 .276 
 
 1. 00 
 
 .843 
 
 I 75 
 
 .987 
 
 •50 
 
 .520 
 
 1-25 
 
 •923 
 
 2.00 
 
 •995 
 
 To iUustrate the use of (4), suppose there is required the 
 probabiUty that in 6000 throws of a die the ace will appear a 
 number of times which shall be greater than 1/6 X 6000 — 10 
 and less than 1/6 X 6000--]- 10, or a number of times lying 
 between 990 and loio. In this case a = 10, ;/ = 6000,/ ^ 1/6, 
 q — 5/6. Thus, a/V2iipq = 10/V2 . 6000 . 1/6. 5/6 = 0.245. 
 Hence, by (4) and the table, R — 0.27. 
 
 Prob. II. If the ratio of males to females at birth is 105 to 100, 
 what is the probability that in the next 10,000 births the number of 
 males will fall within two per cent of the most probable number? 
 
 Prob. 12. If the chance is even for head and tail in tossing a 
 coin, what is the probability that in a million throws the difference 
 between heads and tails will exceed 1500? 
 
 Art. 7. Inverse Probabilities.* 
 
 If an observed event can be attributed to any one of several 
 causes, what is the probability that any particular one of these 
 causes produced the event ? To put the question in a concrete 
 form, suppose a white ball has been drawn from one of two 
 urns, f/, containing 3 white and 5 black balls, and U^ contain- 
 ing 2 white and 4 black balls ; and that the probability in favor 
 of each urn is required. If f/, is as likely to have been 
 chosen as U^, the probability that U^ was chosen is 1/2. After 
 such choice the probability of drawing a white ball from U^ is 
 3/8. Before drawing, therefore, the probability of getting a 
 white ball from V\ was 1/2 X 3/8 = 3/16, by Art. 5. Similarly, 
 before drawing the probability of getting a white ball from f/ 
 was 1/2 X 2/6 = 1/6. These probabilities will remain un- 
 changed if the number of balls in either urn be increased or 
 
 I 
 I 
 
 See Poisson, Probabilite des Jugements, pp. 81-83.
 
 INVERSE PROBABILITIES. 25 
 
 diininislied so long as the ratio of white to bhick balls is kept 
 constant. Make these numbers the same for the two urns. 
 Thus let the first contain 9 white and 15 black, and the second 
 8 white and 16 black; whence the above probabilities may be 
 written 1/2 X 9/24 and 1/2 X 8/24. It is now seen tliat there 
 are (9 -|- 8) cases favorable to the production of a white ball, 
 each of which has the same antecedent probability, namely, 1/2. 
 Since the fact that a white ball was drawn excludes considera- 
 tion of the black balls, the probability that the white ball came 
 from U^ is 9/17 and that it came from U^ is 8/17; and the sum 
 of these is unity, as it should be. 
 
 To generalize this result, let there be ni causes, (7j, C,, . . . C,,,. 
 Denote their direct probabilities by^/,,^,, . , . q,„\ their antecedent 
 probabilities b)' r,, r,, . . . r„, ; and their resultant probabilities 
 on the supposition of separate existence by p^, p^, ■ . . pm- 
 That is, 
 
 Let D be the common denominator of the right-hand mem- 
 bers in (i), and denote the corresponding numerators of the 
 several fractions hy s., s^, . . . s,„. Then 
 
 p, = sJD, p^ — sJD, . . . /„ = sJD ; 
 
 and it is seen that there are in all (.?, -]- i-, -|- • • • ■^»/) equally 
 possible cases, and that of these s^ are favorable to C,, s^ to 
 C^, . . . Hence, if /*,, P^, . . . P,^ denote the probabilities of 
 the several causes on the supposition of their coexistence, 
 
 Thus in general 
 
 P, = pj^p, P, = pJZip, . . . P,„ = pj^p. (2) 
 
 To illustrate the meaning of these formulas by the above 
 concrete case of the urns it suffices to observe that 
 
 for U„ q, = 3/8 and ;-, = 1/2, 
 
 for U^, q^ — 1/3 and r^ — 1/2 ; 
 whence /, = 3/16, /, = 1/6. /, +/, = 17/48 ; 
 
 and P^ — 9/17, P^ = 8/17. 
 
 As a second illustration, suppose it is known that a white
 
 26 PROBABILITY AND THEORY OF ERRORS. 
 
 ball has been drawn from an urn wliich originally contained ;;/ 
 balls, some of them being black, if all are not white. What is 
 the probabilil)' that the urn contained exactly )i white balls? 
 The facts are consistent with m different and eaually probable 
 h\-potheses (or causes), namel\-, that there were i white and 
 {jn — i) black balls, 2 white and (;// — 2) black balls, etc. 
 Hence in (i), ^7, = ^.^ = . . . = i, and 
 
 /, = i/m, p., = 2/7//, .../'„ = ///w. . . ./,„ = in/m. 
 Thus ^/= (i/2)(;//4- I), 
 
 "2,11 
 
 and P^^=zpJ2p=^ ■ -. 
 
 This shows, as it evidently should, that n = in is the most 
 probable number of white balls in the urn. The probability 
 for this number is P,,^ — 2/{i/i -j- i), which reduces, as it ought, 
 to I for ///! = I. 
 
 Formulas (i) and (2) may also be applied to the problem of 
 estimating the probability of the occurrence of an event from 
 the concurrent testimony of severed witnesses, X^, X^, . . . 
 Denote the probabilities that the witnesses tell the truth by 
 X, A\, . . . Then, supposing them to testify independently, 
 the probabilit}' that they will concur in the truth concerning 
 the event is .i',.i\ . . .', while the probability that they will con- 
 cur in the onl}' other alternativt.-, falsehood, is (i — -i'j)(i —-t\) . . . 
 The two alternatives are equally possible. Hence by equations 
 (i) and (2) 
 
 /, = ,r,.n . . ., /., = (i - ,r,)(i — ,rj . . ., 
 
 (3) 
 
 P^ being the probabilit}' for and /'., that against the events 
 
 To illustrate (3), if the chances are 3 to I that A^, tells the 
 truth and 5 to i that X., tells the truth, .i\ = 3/4, x^ = 5/6, and 
 /' = 15/16; or, the chances are 15 to i that an event occurred 
 if they agree in asserting that it did.* 
 
 * For some interesting applications of equations (3) see note E of Appendix 
 to the Ninth Bridgewater Treatise by Charles Babbage (London, 1838). 
 
 p — 
 
 .r,,t', . . . 
 
 p 
 
 . .-U(l -.r,)(i - X,). . . 
 (i-a-,)(i --1-,)... 
 
 
 . . H-(l - ,r,)(i - x) . . .
 
 PROCABII-ITIES OF FUTURE EVENTS. 27 
 
 It is of theoretical interest to observe that if x,, a\, . . . in 
 (3) are each greater than 1/2, /', ap[)roaches unity as the 
 number of witnesses is indefinite!}' increased. 
 
 Prob. 13. The groups of numbers of one llgure each, two figures 
 each, three figures each, etc., which it is jwssihle to form from the 
 nine digits i, 2, ... 9 are printed on cards and ])laced severally in 
 nine similar urns. What is the probability that the number 777 will 
 be drawn in a single trial by a person unaware of the contents of 
 the urns ? 
 
 Prob. 14. How many witnesses whose credibilities are each 3/4 
 are essential to make /', = o-999 iri equation (3) ? 
 
 Art. 8. Probabilities of Future Events. 
 
 Equations (2) of Art. 7 may be written in the following 
 
 manner: ^ „ „ 
 
 P P P \ 
 
 . • • ~^^ • \ 1 / 
 
 /. /. Pn. ^P ^ ^ 
 
 If /i. A' • • • A" ^''^ found by observation, P^, P^, . . . P,„ will ex- 
 press the probabilities of the corresponding causes or their 
 efTects. When, as in the case of most plu'sical facts, the num- 
 ber of causes and events is indefinitely great, tiie value of any 
 p ox P in (1) becomes indefinitely small, and the value of '^p 
 must be expressed by means of a definite integral. Let x de- 
 note the probability of any particular cause, or of the event to 
 which it gives rise. Then, supposing this and all the other 
 causes mutually exclusive, (i — x) will be the probability 
 against the event. Now suppose it has been observed that in 
 {j)i -f- ii) cases the event in question has occurred /// times and 
 failed n times. The probability of such a concurrence is, by 
 Art. 5,r,i-"'(i — .1')", where <r is a constant. Since a- is unknown, 
 it may be assumed to have any value within the limits o and I ; 
 and all such values are a priori equally possible. Put 
 
 y = cx"'{l — xy. 
 Then evidently the probability that x will fall within any as- 
 signed possible limits a and h is expressed by the fraction 
 
 1 
 
 Jydxj J'ydx-^
 
 28 PROBABILITY AND THEORY OF ERRORS. 
 
 SO that the probability of any particular x is given by 
 
 P = "'"C-")''^. (2) 
 
 /'.r"'(i — xYdx 
 
 This may be regarded as the antecedent probability of the 
 cause or event in question. 
 
 What then is the probability that in the next {r -\- s) trials 
 the event will occur r times and fail s times, if no regard is had 
 of the order of occurrence? If x were known, the answer 
 would be by Arts. 2 and 5 
 
 (r + 5) ! 
 
 ■t r I 
 
 r!5! -^"(I-^)'- (3) 
 
 But since x is restricted only by the condition (2), the required 
 probability will be found by taking the product of (2) and (3) 
 and integrating throughout the range of x. Thus, calling the 
 required probability (9, 
 
 1 
 
 Ar'"+''(i — xY^'dx 
 
 1 
 
 j\"'{\ — xfdx 
 
 r\s\ ^ \t/ 
 
 The definite integrals which appear here are known as Gamma 
 functions. They are discussed in all of the higher treatises on 
 the Integral Calculus. Applying the rules derived in such 
 treatises there results * 
 
 Jf\-s) \ {m-\-ry.{n^sy.{m-\-n^iy. 
 ^ r\s\m\n\ {m -[- n -\- r -\- s +i)\ ' ^5) 
 
 If regard is had to the order of occurrence of the event ; 
 that is, if the probability required is that of the event happen- 
 ing r times in succession and then failing .y times in succession, 
 
 * It is a remarkable fact that formula (?) is true without restriction as to 
 values of m, n, r, s. Tlie formula may be established by elementary considera- 
 tions, as was done by Prevost and Lhuilier, 1795. See Todhuiiter's History oi 
 he Theory of Probability, pp. 453-457.
 
 PROBABILITIES OF FUTURE EVENTS. 29 
 
 the factor {r -\- s)l /rls\ in (3), (4), (5) must be replaced by 
 unity. 
 
 To illustrate these formulas, suppose first that the event 
 has happened j/i times and failed no times. What is the prob- 
 ability that it will occur at the next trial? In this case (4) 
 
 gives 
 
 I I 
 
 Q = i\"'+\lv/ l'x"'dx = {m 4- \)/{m -f- 2). 
 
 
 
 When in is large this probability is nearly unity. Thus, the 
 sun has risen without failure a great number of times m ; the 
 probability that it will rise to-morrow is 
 
 l\l . 2\-' I . 2 
 
 \ ' 7/1' \ ' inj in irr 
 
 which is practically i. 
 
 Secondly, suppose an urn contains white and black balls in 
 an unknown ratio. If in ten trials 7 white and 3 black balls 
 are drawn, what is the probability that in the next five trials 
 2 white and 3 black balls will be drawn? The application of 
 (5) supposes the ratio of the white and black balls in the urn 
 to remain constant. This will follow if the balls are replaced 
 after each drawing, or if the number of balls in the urn is sup- 
 posed infinite. The data give 
 
 m =z 7, ?/ = 3. r = 2, J = 3. 
 
 ;« -[- ^ = 9, 7/ + 5 = 6, r -f- i' = 5 , w -f- ;/ 4" I == 1 1 > 
 in A;- n -{- r A^ s -\- \ = 16. 
 
 Thus by (5) 
 
 5!q!6!ii! 
 ^ ==2T3-!7!iTT6! = 30/91. 
 
 Suppose there are two mutually exclusive events, the first 
 of which has happened ;// times and the second ;/ times in 
 m -{- n trials. What is the probability that the chance of the 
 occurrence of tiie first exceeds 1/2 ? The answer to this ques- 
 tion is given directly by equation (2) by integrating the nume- 
 rator between the specified limits of x. That is.
 
 30 PROBABILITY AND THEORY OF ERRORS. 
 
 rx"'{i — xfdx 
 
 P - -^^^ . (6) 
 
 j' x'\\ — xfdx 
 
 
 
 Thus, if m = I and ;: = o, P = 3/4; or the odds are three to 
 one that the event is more likcl\^ to happen than not. Simi- 
 larly, if the event has occurred ;// times in succession, 
 
 P = I - (i/2)'"+S 
 
 which approaches unity rapidly with increase of n. 
 
 Art. 9. Theory of Errors. 
 
 The theory of errors may be defined as that branch of math- 
 ematics which is concerned, first, with the expression of the re- 
 sultant effect of one or more sources of error to which com- 
 puted and observed quantities arc subject ; and, secondly, with 
 the determination of the relation between the magnitude of 
 an error and the probability of its occurrence. In the case of 
 computed quantities which depend on numerical data, such as 
 tables of logarithms, trigonometric functions, etc., it is usually 
 possible to ascertain the actual values of the resultant errors. 
 In the case of observed quantities, on the other hand, it is not 
 generally possible to evaluate the resultant actu'al error, since 
 the actual errors of observation are usually unknown. In either 
 case, however, it is always possible to write down a symbolical 
 expression which A\'ill show how different sources of error enter 
 and affect the aggregate error; and the statement of such an 
 expression is of fundamental importance in llie theory of errors. 
 
 To fix the ideas, suppose a quantit)' (7 to be a function of 
 several independent quantities x, y, s . . .; that is, 
 
 Q=f{x,y,,...), 
 and let it be required to determine the error in Q due to errors 
 in X, y, ,cr . . . Denote such errors by AQ, Ax, Ay, Ac . . . 
 Then, supposing the errors so small that their squares, prod- 
 ucts, and higher powers may be neglected, Taylor's series gives
 
 LAWS OF ERROR. 31 
 
 This equation may be said to express the resultant actual error 
 x)f the function in terms of the component actual errors, since 
 the actual value of JQ is known when the actual errors of 
 X, y, ,c , . . are known. It should be carefully noted that the 
 quantities x, y, s . .. are supposed subject to errors which are 
 independent of one another. The discovery of the independent 
 sources of error is sometimes a matter of difficulty, and in general 
 requires close attention on the part of the student if he would 
 avoid blunders and misconceptions. Every investigator in work 
 of precision should have a clear notion of the error-equation of 
 the type (i) appertaining to his work; for it is thus only that 
 he can distinguish between the important and unimportant 
 sources of error. 
 
 Prob. 15. Write out the error-ecjuation in accordance with (i) 
 for the function Q - xyz -\- x" log (;'A). 
 
 Prob. 16. In a plane triangle a/b = sin A/^\\\ B. Find the error 
 in (? due to errors in /', .-V, and B. 
 
 Prob. 17. Suppose in place of the data of problem i6 that the 
 angles used in computation are given by the following equations : 
 ^ = /?, + 1(180°-^ - B- C,), ^ = ^, + Mi8o° -A-B^ - CX 
 where ^,, B, , C, are observed vahies. What then is Ja? 
 
 Prob. 18. If 7C' denote the weight of a body and r the radius of 
 the earth, show that for small changes in altitude, Aw/7V'— — 2Ar/r\ 
 whence, if a precision of one part in 500000000 is attainal)Ie in com- 
 paring two nearly equal masses, the effect of a difference in altitude 
 of one centimeter in the scale-pans of a balance will be noticeable.* 
 
 Art. 10. Laws of Error. 
 
 A law of error is a function which expresses the relative 
 frequency of occurrence of errors in term.s of their magnitudes. 
 Thus, using the customary notation, let e denote the magni- 
 
 * This problem arose with the International Bureau of Weights anti Measures, 
 whose \vnri< of intercomparison of the Prototype Kilogrammes attained a pre- 
 cision indicated by a probable error ol 1/500 000 000th part of a kilogramme.
 
 32 
 
 PROBABILITY AND THEORY OF ERRORS. 
 
 tude of any error in a system of possible errors. Then the law 
 of such s)stem may be expressed by an equation of the form 
 
 J/ =0(6). (I) 
 
 Representing e as abscissa and j as ordinate, this equation 
 gives a curve called the curve of frequency, tiie nature of which, 
 as is evident, depends on the form of the function 0. Tiiis 
 equation gives the relativ'e frequency of occurrence of errors in 
 the s)'stem ; so that if e is continuous the probability of the 
 occurrence of any particular error is expressed by ju/e = 0(e)</e; 
 which is infinitesimal, as it plainly should be, since in any con- 
 tinuous system the number of different values of e is infinite. 
 
 Consider the simplest form of 0(6), namely, that in which 
 0(^e) = r, a constant. This form of (p{e) obtains in the case of 
 tlie errors of tabular logarithms, natural trigonometric func- 
 tions, etc. In this case all errors between minus a half-unit 
 and plus a half-unit of the last tabular place are equally likely 
 to occur. Suppose, to cover the class of cases to which that 
 just cited belongs, all errors between the limits —a and -\- a 
 are equally likely to occur. The probability of any individual 
 error will then be cp{e)de =^ cde, and the sum of all such prob- 
 abilities, by equation (2), Art. 4, must be unity. That is, 
 
 +a -fa 
 
 / <p{e)de — c I de — \. (2) 
 
 -a -a 
 
 This gives c = i/2a, or by (i) j/ = 1/2^?. The curve of fre- 
 Q quency in this case is shown in the figure, 
 
 D AB being the axis of e and OQ that of j. 
 It is evident from this diagram that if the 
 errors of the system be considered with 
 respect to magnitude onh', half of them 
 should be greater and half less than a/2. 
 This is easily found to be so in the case of 
 tabular logarithms, etc. 
 
 As a second illustration of (i), suppose j and e connected 
 by the relation^ = c Vd" — e\ where a is the radius of a circle,
 
 TYPICAL ERRORS OF A SYSTEM. Xi 
 
 c a constant, and e may have any value between — a and -j- a. 
 Then the condition 
 
 + a 
 
 cj de Vd' - e' = i 
 
 gives c = 2/{a^Ti). In this, as in the preceding case, 0(-|- e) — 
 0(— e), the meaning of which is that positive and negative 
 errors of the same magnitude are equally likely to occur. li 
 will be noticed, however, that in the latter case small errors 
 have a much higher probability than those near the limit a, 
 Avhile in the former case all errors have the same probability. 
 
 In general, when e is continuous 0(e) must satisfy the condi- 
 tion / (p{e)de = I, the limits being such as to cover the entire 
 
 range of values of e. The cases most commonly met with are 
 those in which 0(e) is an even function, or those in which 
 0(-|- e) = 0(— e). In such cases, if ± a denote the limiting 
 value of e, 
 
 + a a 
 
 fcp{e)de = 2f<p{e)de = I. (3) 
 
 Art. 11. Typical Errors of a System. 
 
 Certain typical errors of a system have received special 
 desienations and are of constant use in the literature of the 
 theory of errors. These special errors are the probable error, 
 the mean error, and the average error. The first is that error 
 of the system of errors which is as likely to be exceeded as 
 not ; the second is the square root of the mean of the squares 
 of all the errors ; and the third is the mean of all the errors 
 regardless of their signs. Confining attention to systems in 
 wliich positive and negative errors of the same magnitude 
 are equally probable, these typical errors are defined mathe- 
 matically as follows. Let 
 
 e^ = the probable error, 
 
 e,„= the mean error, 
 
 Ca =■ the average error.
 
 34 PROBABILITY AND THEORY OF ERRORS. 
 
 Then, observing (2), of Art. 10, 
 
 f(p{6)de =J(p{e)de = f (p{e)de =J'(P{e)de = 
 
 -a -fp +e^ 
 
 + a +a 
 
 3 
 
 y (0 
 
 = / (p(^e)6-de, e^, = 2 / cp{e)ede. 
 
 -a 
 
 The student should seek to avoid the very common misap- 
 prehension of the meaning of the probable error. It is not 
 " the most probable error,'" nor " the most probable value of 
 the actual error"; but it is that error which, disregarding signs, 
 would occupy the middle place if all tlie errors of the system 
 were arranged in order of magnitude. A few illustrations will 
 sufifice to fix the ideas as to the typical errors. Thus, take the 
 simple case wherein 0(e) = ^ = \ /2a, which applies to tabular 
 logarithms, etc. Equations (i) give at once 
 
 1 , ft l/~ I 
 
 ep= ± - a, €,„ = ± - '^ 3 , 6^= ± ~a. 
 
 2 3 2 
 
 For the case of tabular values, a = 0.5 in units of the last 
 tabular place. Hence for such values 
 
 €p= ± 0.25, e„, = ± 0.29, e, = ± 0.25. 
 
 Prob. 19. Find the typical errors for the cases in which the la\v 
 of error is 0(e) = i:Vd' — e\ (pie) = c(±a^£), <p{e)=c cos" in: £/2ay, 
 c being a constant to be determined in each case and e having any 
 value between — a and + a. 
 
 Art. 12. Laws of Resultant Error. 
 
 When several independent sources of error conspire to pro- 
 duce a resultant error, as specified by equation (i) of Art. 9, 
 there is presented the problem of determining the law of the 
 resultant error by means of the laws of the component errors. 
 The algebraic statement of this problem is obtained as follows 
 for the case of continuous errors : 
 
 In the equation (i), Art. 9, write for brevity
 
 LAWS OF KESULTANT ERROR. 35 
 
 and let the laws of error of e, e^, e^, ... be denoted by (p{e), 
 
 ^,(6-,), 0,(^2) • • • Tlien the value of e is given by 
 
 e=^, + ^, +••• (i) 
 
 The probabilities of the occurrence of any particular values 
 of e,, e„ . . . are given by (/),{£, )de^, 0,(ej^/e,, . . . ; and the 
 probability of their concurrence is the probabihty of the cor- 
 responding value of £. But since tiiis value may arise in an 
 infinite number of ways through tiic variations of e,, e„ . . . 
 over their ranges, the probability of e, or 0(e)c/e, will be 
 expressed by the integral of (p^{e^)d£^(p^{£^)d£^ . . . subject to 
 the restriction (i). Tliis latter gives e, = e — e„ — £^ . . ., and 
 </e, = (/£ for the multiple integration with respect to e,, e„ . . . 
 Hence there results 
 
 cp{e)d6 = deJ"(PXe - e, - ^3 — • • •) 0.(^2)^^2 • • • , 
 or 
 
 0(^) = y0:(e -e, - e, - . . .)(p,{e,)d£,J(J^,{£,)d£, ... (2) 
 
 It is readily seen that this formula will increase rapidly in 
 complexity with the number of independent sources of error.* 
 For some of the most important practical applications, how- 
 ever, it suf^ces to limit equation (2) to the case of two inde- 
 pendent sources of error, each of constant probability within 
 assigned limits. Thus, to consider this case, let e^ var\^ over 
 the range — a to -{- a, and e, vary over the range — b to -\- b. 
 Then by equation (2), Art. 10, 
 
 0,(e,) = i/{2a), 0,(6„) = l/{2b). 
 Hence equation (2) becomes 
 
 ^^^) = ifbf'^'^ 
 
 4ab, 
 
 In evaluating this integral e^ miust not surpass ± ^ and 
 e, = e — e„ must not surpass ± (^- Assuming a > b, the limits 
 of the integral for any value of e = e^ -\- e^ b'''\? between 
 — {a -j- b) and — {a — b) are — b and -}- (e -}- c?). This fact is 
 
 * The reader desirous of pursuing this phase of the subject should consult 
 Bessel's Untersuchungen ueber die VVahrscheiiilichkeit der Beobachtungsfehler; 
 Abhandlungen von Bessei (Leipzig, 1876), Vol. II.
 
 3G PROBABILITY AND THEORY OF ERRORS. 
 
 made plain by a numerical example. For instance, suppose 
 ^ = 5 and /^ = 3. Then — [a-]- d) = —8 and — {a ~ d) = — 3. 
 Take e = — 6, a number intermediate to — 8 and — 3. Then 
 the following are the possible integer values of e, and e, which 
 will produce e = — 6: 
 
 e e, e, limits of e^ 
 
 — 6= -5 -I, -i=^{e-\-a), 
 = - 4 - 2, 
 
 = - 3- 3, -3^ -b. 
 
 Similarly, the limits of e^ for values of e lying between 
 — {a — b) and -\- {a — b) are — b and -|- h\ and the limits of 
 e, for values of e between -{- {a — b) and -{-{a-{- b) are + (e — ^) 
 and "1- b. Hence 
 
 e-\-a-\-b 
 
 4ab 
 2b 
 
 for —{a-]-b) < e< — (a—b), 
 
 (3) 
 
 for _ (^ - ^)< e < + (^ - b), 
 
 '-^-^ for -^{a-b)<e<+{a^b). 
 
 Thus it appears that in this case the graph of the resultant 
 
 law of error is represented by the upper base and the two sides 
 
 of a trapezoid, the lower base beingr the 
 IIOII ^ .... 
 
 axis of 6 and the line loining the middle 
 liioiii . . , 
 
 ponits of the bases benig the axis of 0(e). 
 
 (See the first figure in Art. 13.) Tlie prop- 
 erties of (t,), including the determination 
 
 III . . 
 
 of the limits, are also illustrated by the 
 
 nil. . , 
 
 adjacent trapezoid of numerals arranged 
 
 to represent the case wherein a = 0.5 and 
 ^ z= 0.3. The vertical scale, or that for cp(e), does not, how- 
 ever, conform exactly to that for e. 
 
 Ill ion 
 
 II 1 1 ion 
 
 II 1 1 lion 
 
 II iiiiion 
 
 III I I II lOI I
 
 ERRORS OF INTERPOLATED VALUES. 37 
 
 Prob. 20. Prove that the values of 0(e) as given by equation (3) 
 satisfy the condition specified in eciuation (3), Art. 10, 
 
 Prob. 21. Examine equations (3) for the cases wherein a = /"and 
 <^ = o; and interpret for the latter case the hrst and last of (3). 
 
 Prob. 22. Find from (3), and (i) of Art. 11, the probable error of 
 the sum of two tabular logarithms. 
 
 Art. 13. Errors of Interpolated Values. 
 
 Case I. — One of the most instructive cases to which formulas 
 (3) of Art, 12 are applicable is that of interpolated logarithms, 
 trigonometric functions, etc., dependent on first differences. 
 Thus, suppose that %\ and 7', are two tabular logarithms, and 
 that it is required to get a value v lying / tenths of the interval 
 from T', towards v.-^. Evidently 
 
 y ^ y^ _j_ (,;^ _ 2,J / = (l _ t)V^ + tV, ; 
 
 and hence if e, e^, e^ denote the actual errors of v, v^, i\ , re- 
 spectively, 
 
 ^ = (1 -0^'. + ^^.- (0 
 
 It is to be carefully noted here that ^ as given by (i) re- 
 quires the retention in v of at least one decimal place be- 
 yond the last tabular place. For example, let z' = log (24373) 
 from a 5-place table. Then 7\ = 4.38686, 7'., = 4.38703, 
 v^ — 7\ — -|- 0.00017, / = 0.3, and 7> =4.38691.1. Likewise, as 
 found from a 7-place table, ^, = — 0.45, e., = -f- 0.37 in units of 
 the fifth place; and hence by (i) ^= —0.20. That is, the 
 actual error of ?' = 4.38691.1 is = 0.20, and this is verified by 
 reference to a 7-place table. 
 
 The reader is also cautioned against mistaking the species 
 of interpolated values here considered for the species common- 
 ly used by computers, namely, that in which the interpolated 
 value is rounded to the nearest unit of the last tabular place. 
 The latter species is discussed under Case II below. 
 
 Confining attention now to the class of errors specified by 
 
 equation (i), there result in the notation of the preceding 
 
 article 
 
 e, = ( I — t)e^, e„ = te^, and e — e — e^-\-e.,\ 
 
 and since e^ and e^ each vary continuously between the limits
 
 38 
 
 PROBABILITY AND THEORY OF ERRORS. 
 
 ±0.5 of a unit of the last tabular place, a and b in equations 
 (3) of that article have the values 
 
 a — 0.5(1 — /), b — 0.5/. 
 
 Hence the law of error of the interpolated values is ex- 
 pressed as follows : 
 
 0(e) = -^^^C — for values of e betw. — 0.5 and —(0.5—/), 
 (I t)t 
 
 = for values of e betw. —(0.5—/) and -[-(0.5—/), )■ (2) 
 
 0.5 
 
 for values of e betw. -l-(o.5— /) and -(-0.5. 
 
 -(I -/)/ 
 
 The graph of 0(e) for / = 1/3 is shown by the trapezoid 
 AB, BC, CD in the figure on page 40. Evidently the equa- 
 tions (2) are in general represented by a trapezoid, which degen- 
 erates to an isosceles triangle when / = 1/2. 
 
 The probable, mean, and average errors of an interpolated 
 value of the kind in question are readily found from (2), and 
 from equations (i) of Art. 1 1, to be 
 
 = (i/4)(i-0 
 
 = 1/2 - {\/2)^2t{\ — t) 
 = 1/4^ 
 
 _ ( I -(I -2/y )v^ 
 - I 96(1 - /)/i • 
 
 I - (I - 2ty 
 
 for o < / < 1/3, 
 for 1/3 </ < 2/3, 
 for 2/3 </ < 1. 
 
 }- (3) 
 
 e, = 
 
 24(1 -/)/ 
 
 I -(2/- l) = 
 
 for o < / < 1/2, 
 
 for 1/2 < / < I. 
 
 24(1 -/y 
 
 It is thus seen that the probable error of the interpolated 
 value here considered decreases from c.25 to 0.15 of a unit of 
 the last tabular place as / increases from o to 0.5, Hence such 
 values are more precise than tabular values ; and the computer 
 who desires to secure the highest attainable precision with a 
 given table of logarithms should retain -^ne additional figure 
 bej'oiid the last tabular place in interpolated values.
 
 ERRORS OF INIERI'OLATED VALUES 39 
 
 Case II. — ReCLirrinj^ to the ccjuatioii 7' = 7', -f- t{v.^ — t',) for an 
 interpolated value v in terms of two consecutive t.ibular values 
 7', and 7'j, it will be observed that if the quantity t{v^— 7',) is 
 rounded to the nearest unit of the last tabular place, a new error 
 is introduced. For example, if z'^ = log 1633 = 3.21299, and 
 ^2 = log 1^34 = 3-21325 from a 5-place table, v^ — v^ = -f- 26 
 units of the last tabular place ; and if / = 1/3, /(7'.^ — v^) = 8^ ; 
 so that by the method of interpolation in question there results 
 V = 3.21299 +9 = 3-21308. Now the actual errors of 7', and 
 7'j are, as found from a 7-place table, — 0.38 and -|-0 2i in units 
 of the fifth place. Hence the actual error of 7' is by equation 
 (0» I X — 0.38 -f J X + 0.21 — 1 = — 0.52, as is shown di- 
 rectly by a 7-place table. 
 
 It appears, then, that in this case the error-equation cor- 
 responding to (i) is 
 
 e= {i - t)c, + te, + e, , (4) 
 
 wherein ^, and i\ are the same as in (i) and ^'3 is the actual error 
 that comes from rounding t{v^ — 7',) to the nearest unit of the 
 last tabular place. 
 
 The error e^, however, difTers radically in kind from r, and 
 e^. The two latter are continuous, that is, they may each have 
 any value, between the limits — 0.5 and +0-5 \ while ^3 is dis- 
 continuous, being limited to a finite number of values depend- 
 ent on the interpolating factor t. Thus, for / = 1/2 the only 
 possible values of c^ are O -\- 1/2, and — 1/2 ; likewise for / — 
 1/3, the only possible values of ^3 are o, -j- 1/3, and — 1/3. It 
 is also clear that the maximum value of t\ which is constant and 
 equal to 1/2 for (i), is variable for (4) in a manner dependent 
 on /. For example, in (4), 
 
 The maximum of ^ = 1/2 + r/2 = l, for / = 1/2, 
 
 " ^ = 1/2 + 1/3 = 5/6, " /'=i/3. 
 
 « " "^=1/2 + 1/2=1, " / = 1/4, 
 
 'V=: 1/2 4- 2/5 =9/10" / ~ l/5- 
 
 The determination of the law of error for this case presents 
 some novelty, since it is essential to combine the continuous 
 errors (i — t)i\ and te^ with the discontinuous error e^. The
 
 40 
 
 PROBABILITY AND THEORY OF ERRORS, 
 
 simplest mode of attacking the problem seems to be the fol- 
 lowing quasi-geometrical one. In the notation of Arts. 12 and 
 13, put in (4) ^ = e, (i — /)t\ = e,, /r, = e^, and c^ = e^. Then 
 
 e = (e, -f ej + 63. (5) 
 
 The law of error for (e^ -|- e^) is given by equation (2) for any 
 value of A Hence for a given value of t there will be as many 
 expressions of 0(e) as there are different values of e,. The 
 graphs of 0(e) will all be of the same form but will be differently 
 placed with reference to the axis of 0(e). Thus, if /" j= 1/3 the 
 
 values of e.^ are — 1/3, o, and 
 -|- 1/3, and these are equally 
 likely to occur. For e^ = o the 
 graph is given directly by (2), 
 and is the trapezoid A BCD 
 symmetrical with respect to OQ. 
 For £^ = — 1/3 the graph is 
 abQd, of the same form as 
 ABCD but shifted to the left 
 by the amount of e-g = — 1/3. 
 Similarly, the graph for the case 
 of £3 = -|- 1/3 is a'Qb'd\ and is produced by shifting ABCD to 
 the right by an amount equal to -|- 1/3. 
 
 Now, since the three systems of errors for this case are 
 equally likely to occur, they may be combined into one system 
 by simple addition of the corresponding element areas of the 
 several graphs. Inspection of the diagram shows* that the 
 resultant law of error is expressed by 
 
 0(e) = (i/4)(5 + 6e) for - 5/6 < e < _ 1/6, >| 
 
 = 1 for - 1/6 < e < + 1/6, j- (6) 
 
 = (i/4)(5 - 6e) for + I /6 < e < 4- 5/6. J 
 
 This is represented by a trapezoid whose lower base is 10/6, 
 upper base 2/6, and altitude i. 
 
 * Sum the three areas and divide by 3 to make resultant area = i, as 
 required by equation (3), Art. 10.
 
 ERRORS OF INTERPOLATED VALUES. 
 
 41 
 
 As a second illustration, consider equation (5) for the case 
 /= 1/2. In this case e, must be either o or 1/2, the sign of 
 which latter is arbitrary. For e^ = o, equations (2) give 
 0(e) = 2 -|- 46 for _ 1/2 < 6 < O, I 
 
 = 2 — 4e for o < e <-|- 1/2. \ (7) 
 
 This function is represented by the isosceles triangle AQE 
 
 whose altitude OQ is twice the base AE. 
 
 Similarly 0(e) for e, = -|- 1/2 would 
 
 be represented by the tna.ng\e AQE dis- 
 placed to the right a distance 1/2 ; and 
 if the two systems for ^3=0 and 63 = 
 -|- 1/2 be combined into one system, 
 their resultant law of error is evidently 
 0(e)= i-|-2e for— i/2<e<o, \ 
 
 = I for o< e< + i/2, t (8) 
 
 = 2 — 2e for -f- 1/2 < e< I ; ) 
 the graph of which is ABCD. On the 
 other hand, if the errors in this combined system be considered 
 with respect to magnitude only, the law of error is 
 
 0(e) = 2(1 -e) for o < e< I, (9) 
 
 the graph of which is OQD. 
 
 The student should observe that (6), (7), (8), and (9) satisfy 
 
 the condition / (p{e)d£ = i if the integration embraces the 
 
 whole range of e. 
 
 Tiie deteimination of the general form of 0(e) in terms of 
 the interpolating factor / for the present case presents some 
 dif^culties, and there does not appear to be any published solu- 
 tion of this problem.* The results arising from one phase of 
 the problem have been given, however, by the author in the 
 Annals of Mathematics, f and may be here stated without 
 proof. The phase in question is that wherein / is of the form 
 1//1, n being an}' positive integer less than twice the greatest 
 
 * The author explained a general method of solution in a paper read at the 
 summer meeting of the American Mathematical Society, August, 1895. 
 f Vol. II, pp. 54-59.
 
 42 PROBABILITY AND THEORY OF ERRORS. 
 
 tabular difference of the table to which the formulas are ap- 
 plied. For this restricted form of / the possible maximum 
 value of e as given by equation (5) is, in units of the last 
 tabular place, {2n — i)/;/ for n odd and i for h even. 
 
 The possible values of e^ of equation (5) are 
 O, 
 
 I 
 
 2 
 
 71 — I 
 
 2n 
 
 
 
 for 71 odd, 
 
 I 
 
 ± -.. 
 
 71 
 
 2 
 
 71 — 2 
 
 • • • ^ _ » 
 
 211 
 
 ± 
 
 I 
 
 2 
 
 for 71 even. 
 
 o, 
 
 An important fact with regard to the error 1/2 for ;/ even 
 is that its sign is arbitrary, or is not fixed by the computation 
 as is the case with all the other errors. However, the com- 
 puter's rule, which makes the rounded last figure of an inter- 
 polated value even when half a unit is to be disposed of, will, 
 in the long-run, make this error as often plus as minus. 
 
 The laws of error which result are then as follows : 
 
 For n odd. 
 
 0(6) = I for 6 between — 1/2;/ and -\- i/2«, 
 
 71 (271 — I \ 
 
 0(e) = I ± e for ebetw. =Fi/2« and ^{27i—\)/27i. 
 
 ^ 71 — \\ 271 I ' 
 
 For n even. 
 
 n (271 — 2 \ 
 (p(€) = — , ± e for e between o and =F i/«, 
 
 ^ ' 2{7l — \)\ 71 I T^ / ' 
 
 ;/ (271 — I \ , 
 
 = -I— ± e) for e betw. =F ^ "■ and T {71 — \)/7t, 
 
 71 — l^ 271 I ji y 
 
 71 
 
 = —. r(i + e) for e between T {n— i/«) and =F i. 
 
 2{7l — If ^ \ I I 
 
 By means of these formulas and (i) of Art. 1 1 the probable, 
 mean, and average errors for any value of ;/ can be readily 
 found. The following table contains the results of such a com- 
 putation for values of ;/ ranging from i to 10. The maximum 
 actual error for each value of 71 is also added. The verifica- 
 tion of the tabular quantities will afford a useful exercise to the 
 student.
 
 ERRORS OF INTERPOLA'l ED VALUES. 43 
 
 Typical Errors of Intkrpolated Logarithms, etc. 
 
 Interpotaling 
 Factor, 
 
 Probable Error. 
 
 Mean Error. 
 
 Average Error. 
 
 Maximum 
 
 t= i/n 
 
 /^ 
 
 m 
 
 '^a 
 
 Actual Error. 
 
 I 
 
 0.250 
 
 0.289 
 
 0.250 
 
 1/2 
 
 1/2 
 
 .292 
 
 .408 
 
 •333 
 
 I 
 
 1/3 
 
 .256 
 
 •347 
 
 .287 
 
 5/6 
 
 >/4 
 
 .276 
 
 .382 
 
 ■3'3 
 
 I 
 
 •/5 
 
 .268 
 
 .370 
 
 •303 
 
 9/10 
 
 1/6 
 
 277 
 
 •385 
 
 •3'5 
 
 I 
 
 1/7 
 
 .274 
 
 .380 
 
 •3" 
 
 13/14 
 
 1/3 
 
 .279 
 
 •3S9 
 
 .3>8 
 
 I 
 
 1/9 
 
 .278 
 
 .3S6 
 
 • 3'6 
 
 17/18 
 
 i/io 
 
 .281 
 
 •392 
 
 .320 
 
 I 
 
 When the interpolating factor / has the more general forrn 
 J/1//1, wherein w and ft are integers with no common factor, the 
 possible values of e^ are the same as for t = i/n. But equa- 
 tions (3) of Art. 12 are not the same for / = vi/// as for/ = i/;/, 
 and hence for the more general form of /, 0(e) assumes a new 
 type which is somewhat more complex than that discussed 
 above. The limits of this work render it impossible to extend 
 the investigation to these more complex forms of 0(e). It may 
 suf^ce, therefore, to give a single instance of such a function, 
 namely, that for which / = 2/5. For this case 
 
 0(e) = I for e between o and =F i/io, 
 
 = (5/6)(i3/iO ± e) for e between ^ i/io and q: 3/10, 
 = (5/3)(4/5 ± e) for e between T 3/10 ''^'icl T 7/10, 
 = (5/6)(9/iO± e) for e between =F 7/10 and ^p 9/10. 
 The graph of the right-hand half of ^ b 
 this function is shown in the accompany- 
 ing diagram, the whole graph being 
 symmetrical with respect to OA, or the 
 axis of 0(e). 
 
 Attention maybe called to the strik- 
 ing resemblance of this graph to that of 
 the law of error of least squares. 
 
 Prob. 23. Show from equations (3) that e,„ varies from i/r 12 
 
 = 0.29 — , for / = o, to i/v 24 = 0.20 +, for / = 0.5 ; and that e^ 
 varies from 0.25 to 1/6 for the same limits.
 
 44 PROBABILITY AND THEORY OF ERRORS. 
 
 Prob. 24. Show that the probable, mean, and average errors 
 for the case of /= 2/5 cited above (p. 43) are ± 0.261, ± 0.251, 
 and ± 0.290, respectively. 
 
 Art. 14. Statistical Test of Theory. 
 A statistical test of the theory developed in Art. 13 may 
 be readily drawn from any considerable number of actual er- 
 rors of interpolated values dependent on the same interpolating 
 factor. The application of such a test, if carried out fully by 
 the student, will go far also towards fixing clear notions as to 
 the meaning of the critical errors. 
 
 Consider first the case in which an interpolated value falls 
 midway between two consecutive values, and suppose this 
 interpolated value retains two additional figures beyond the 
 last tabular place. Then by equations (2), Art. 13, the law of 
 error of this interpolated value is 
 
 0(e) = 2 -(- 4e for e between — 0.5 and o 
 = 2 — 4e for e between o and -f- 0.5. 
 
 Hence by equation (i) of Art. 1 1, or equation (3) of Art. 12, the 
 probable error in this system of errors is ^ — {^) V2 := o.i^. 
 It follows, therefore, that in any large number of actual errors 
 of this system, half should be less and half greater than 0.15. 
 Similarly, of the whole number of such errors the percentage 
 falling between the values 0.0 and 0.2 should be 
 
 + 0-2 +0.2 
 
 J (p{e)de = 2 J (2 — ^e)de = 0.64 ; 
 
 -0.2 
 
 that is, sixty-four per cent of the errors in question should be 
 less numerically than 0.2. 
 
 To afTord a more detailed comparison in this case, the act- 
 ual errors of five hundred interpolated values from a 5-place 
 table have been computed by means of a 7-place table. The 
 arguments used were the following numbers : 20005, 20035, 
 20065, 20105, 20135, etc., in the same order to 36635. The 
 actual and theoretical percentages of the whole number of 
 errors falling between the limits 0.0 and o. i, o.i and 0.2, etc., 
 are shown in the tabular form following::
 
 STATISTICAL TEST OF THEORY. 45 
 
 Limits of Errors. „ A^'"^' Theoretical 
 
 Percentage. Percentage. 
 
 O.oaiido. I 33.2 36 
 
 O. I and 0.2 30.2 28 
 
 0.2 and 0.3 19.0 20 
 
 0.3 and 0.4 13.2 12 
 
 0.4 and 0.5 4.4 4 
 
 0.0 and O.I 5 51.4 50 
 
 The agreement shown here between the actual and theoretical 
 percentages is quite close, tiie maximum discrepancy being 2.8 
 and the average 1.5 per cent. 
 
 Secondly, consider the case of interpolated mid-values of the 
 species treated under Case II of Art. 13. The law of error for 
 this case is given by the single equation (9) of Art. 13, namel)', 
 0(e) = 2(1 — e), no regard being paid to the signs of the errors. 
 The probable error is then found from 
 
 
 « 
 
 whence 6p = i — -^ V2 = 0.29. Similarly, the percentage of 
 the whole number of errors which may be expected to lie, for 
 example, between 0.0 and 0.2 in this system is 
 
 2 
 
 2 / (i — e)(/6 = 0.36. 
 
 
 
 Using the same five hundred interpolated values cited 
 above, but rounding them to the nearest unit of the last tabu- 
 lar place and computing their actual errors by means of a 7-place 
 table, the following comparison is afforded : 
 
 ... , „ Actual Thcoreiical 
 
 Limits of Errors. Percentage. Percentage. 
 
 0.0 and 0.2 35.8 36 
 
 0.2 and 0.4 27.8 28 
 
 0.4 and 0.6 1 8.6 20 
 
 0.6 and 0.8 1 2.2 12 
 
 0.8 and i.o 5.6 4 
 
 0.0 and 0.29 49.8 50
 
 46 PROBABILITY AND THEORY OF ERRORS. 
 
 The agreement shown here between the actual and theoretical 
 percentages is somewhat closer than in the preceding case, the 
 maximum discrepancy being only 1.6 and the average only 0.6 
 per cent. 
 
 Finally, the following data derived from one thousand act- 
 ual errors may be cited. The errors of one hundred inter- 
 polated values rounded to the nearest unit of the last tabular 
 place were computed * for each of the interpolating factors 
 0.1, 0.2, . , . 0.9. The averages of these several groups of act- 
 ual errors are given along with the corresponding theoretical 
 errors in the parallel columns below: 
 
 Interpolating Actual Theoretical 
 
 Factor. Average Error. Average Error. 
 
 O.I 0.338 0.320 
 
 0.2 o..?88 0.303 
 
 0.3 0.321 0.304 
 
 0.4 0.268 0.290 
 
 0.5 0.324 0.333 
 
 0.6 ... 0.276 0.290 
 
 0.7 0.32 I 0.304 
 
 0.8 0.289 0.303 
 
 0.9 0.347 0.320 
 
 The average discrepancy between the actual and theoret- 
 ical values shown here is 0.017. It is, perhaps, somewhat 
 smaller than should be expected, since the computation of the 
 actual errors to tlnee places of decimals is hardly warranted 
 by the assumption of dependence on first differences only. 
 
 The averatfe of the whole number of actual errors in this 
 case is 0.308, which agrees to the same number of decimals 
 with the average of the theoretical errors, f 
 
 * By Prof. H. A. Howe. See Annals of Mathematics, Vol. Ill, p. 74. 
 The theoretical averages were furnished to Prof. Howe by the author. 
 
 f The reader who is acquainted with the elements of the method of least 
 squares will find it instructive to apply that method to equation (i), Art. 13, 
 and derive the probable error of e. This is frequently done without reserve by
 
 STATISTICAL TEST OF THEORY. 47 
 
 Prob. 25. Apply formulas (3) of Art. 12 to the case of the sum 
 or difference of two tabular logarithms and derive the correspond- 
 ing values of the probable, mean, and average errors. The graph 
 of 4>{e) is in this case an isosceles triangle whose base, or axis of e, 
 is 2, and whose altitude, or axis of 0(e), is i. 
 
 those familiar with least squares. Thus, the probable error of <?i or ^2 being 
 0.25, the probable error of e is found to be 
 
 0.25 Vl — 2/ + 2/-. 
 
 This varies between 0.25 for ^ = o and o.iS for / = 5 ; while the true value of 
 the probable error, as shown by equations (3), Art. 13, varies from 0.25 to 0.15 
 for the same values of t. It is, indeed, remarkable that the method of least 
 squares, which admits infinite values for the actual errors d and e^, should give 
 so close an approximate formula as the above for the probable error of c. 
 
 Similarly, one accustomed to the method of least squares would be inclined 
 to apply it to equation (4), Art. 13, to determine the probable error of e. The 
 natural blunder in this case is to consider ei, ei , and ^3 independent, and e-., like 
 fi and i'i continuous betweer the limits 0.0 and 0.5 ; and to assign a probable 
 error of 0.25 to each. In t'.is manner the value 
 
 0.25 ^2(1 - i + i^) 
 
 is derived. But this is absurd, since it gives 0.25 V^2 instead of 0.25 for t =■ o. 
 The formula fails then to give even approximate results except for values of / 
 near 0.5.
 
 INDEX. 
 
 Average error, 33, 34. 
 
 of interpolated logarithm, 38, 43. 
 of tabular logarithm, 34. 
 
 Babbage: 
 
 Ninth Bridgewater treatise of, 26. 
 Bernoulli, James: 
 
 theorem of, 22. 
 
 work cited, 8. 
 Bert';t;id, work cite,', '>.^. 
 Bessel, work cited, 35. 
 
 Chance, games of, 7. 
 Combinations, 13-16. 
 
 formulas for, 14-16. 
 
 table of, 15. 
 Concurrent events, 19-21. 
 
 De Moivre, work cited, 8. 
 
 De Morgan, work cited, 13, 22. 
 
 Error equation, 31. 
 function, 31. 
 Errors, theory of, 30-47. 
 
 Fermat, 7, 8. 
 
 Games of chance, 7. 
 
 Gamma function, 28. 
 
 Geographical tables (of Smithsonian 
 
 Institution) cited, 10. 
 Graphs of laws of error, 32, 36, 40, 41, 
 
 43- 
 
 Howe, computation of, cited, 46. 
 Huygens, work cited, 8. 
 
 Integral, probability, 23. 
 table of, 24. 
 
 Jevons, work cited, 16. 
 
 Laws of error, 31-33. 
 
 interpolated logarithms, 34-47. 
 
 least squares, 10, 43, 46, 47. 
 
 tabular logarithms, 32. 
 Least squares, 10, 43, 47. 
 Laplace, work cited, 9, 22. 
 Logarithmic tables, 37-43. 
 
 Mean error, ^;i, 34. 
 
 of interpolated logarithms, 38, 43. 
 
 of tabular logarithms, 34. 
 Mere, Chevalier de, 7. 
 Method of least squares, 10, 46, 47. 
 Montmort, work cited 8. 
 
 Observations, errors of, 30, 31. 
 
 Pascal, 7, 8. 
 Permutations, 11-18. 
 
 , formulas for, 11, 12. 
 table of, II. 
 Poisson, work cited, 7, 20, 24. 
 Probable error, ^^, 34. 
 
 of interpolated logarithms, 38, 43. 
 
 of tabular logarithms, 34. 
 Probabilities, 16-30. 
 
 direct, 16-18. 
 
 inverse, 24-27. 
 
 of concurrent events, 19-21. 
 
 of concurrent testimony, 26. 
 
 of future events, 27-30. 
 Probability integral, 23. 
 
 Resultant error, 34. 
 
 Shortrede, tables cited, 13. 
 Statistical test of theory, 44-46. 
 Stirling's theorem, 22, 23. 
 
 Table of combinations, 15. 
 
 of permutations, 11. 
 
 of probability integral, 24. 
 
 of statistical test, 45, 46. 
 
 of typical errors, 43. 
 Tabular values, errors of, 34-38. 
 Theory of errors, 30-47. 
 
 of interpolated values, 37-46. 
 Todhunter, I., work cited, 7, 28. 
 Typical errors, $3, 43. 
 
 Values of combinations, 15. 
 of permutations, 11, 
 of typical errors, 43.
 
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