Probability And Theory Of Errors No 7

05 TV: ro P

Sjjjg OU_1 60667 >m

CQ

OSMANIA UNIVERSITY LIBRARY

Call No. T/ ^ Accession No.

M^t P

Author '"/'. I O t I

l\J &&& tAj OAM K> bj2A>r , S

Title -4

This book should be returned on or before tne date
last marked below. ~ 5 1 \Jvf3- s L2

MATHEMATICAL MONOGRAPHS.

FDITED BY

MANSFIELD MERRIMAN AND ROBERT S. WOODWARD.

No. 7.

PROBABILITY

THEORY OF ERRORS.

BY

ROBERT S. WOODWARD,

PRESIDENT CARNEGIE iNSTiTimdWTTfl WASHINGTON.

NEW YORK:

JOHN WILEY & SONS.
LONDON: CHAPMAN & HALL, LIMITED.

COPYRIGHT, 1896,

v
MANSFIELD MERRIMAN AND ROBERT S. WOODWARD

UNDER THE T I LE

HIGHER MATHEMATI-CS.

First Ediuon, September, 1896.
Second Edition, January, i8g8.
Third Ediuon, AURUSI, 1900.
Fourth Edition, January, 1906.

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 publication 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 publishers 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 elliptic 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 publication may-
tend to promote mathematical study and research over a wider
field than that which the former volume has occupied.

December, 1905.

iii

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. Airy. 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 theory of probability 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 applications to the more precise sciences of as-
tronomy, geodesy, and physics, their range of applicability 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 applications 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 little additional aid; if
they are 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 u

3. COMBINATIONS 13

4. DIRECT PROBABILITIES 16

5. PROBABILITY OF CONCURRENT K VENTS 19

6. BERNOULLI'S THEOREM 22

7. INVERSE PROBABILITIES 24

8. PROBABILITIES OF FUTURE EVENTS 27

9. THEORY OF ERRORS 30

TO. LAWS OF ERROR 31

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 gambler.f The problem in question is
known as the problem of points and may be stated as follows :
two players 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 H istory of the Mathematical Theory of Probability from the time of
Pascal to that of Laplace (Cambridge and London, 1865).

f " Un probleme relatif aux jeux de hasard, proposg aun austere janseniste
par un homme du rnonde, a 6t6 Torigine du calcul des probability. " Poisson,
Recherches sur la Probability des Jugements (Paris, 1837).

g PROBABILITY AND THEORY OF ERRORS.

acute thinkers, however, accomplished 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

*Dc Ratiociniis in Ludo Aleae, 1657.

f Ars Conjectandi, 1713.

J Essai d'Analyse sur les Jeux de Hazards, 1708.

The Doctrine of Chances, 1718.

jTodhunter says of Montmort, for example, "In 1708 he published his
work 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 Thorie 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* may be 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 the 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 1820, and of Vol. VII of the
complete works of Laplace recently republished under the auspices of the
Academic des Sciences by Gauthier-Villars. This Vol. VII bears the date 1886.

f " Cette Introduction," writes Laplace, "est le developpement d'une Legon
sur les Probability, que je donnai en 1795, aux coles Normales, ou je fus ap-
pele com me professeur de Mathematiques avec Lagrange, par un dgcret 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 Smith-
sonion Institution, 1894.

PERMUTATIONS.

11

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 probabilities. This theory is now to be
found in most works on algebra, and it will therefore suffice
here to state the principal formulas and illustrate their mean-
ing by a few numerical examples.

The number of permutations of ;/ things taken r in a group
is expressed by the formula

() r = ( - i)(;/ _ 2) ...(- r + I). (i)

Thus, to illustrate, the number of ways the four letters a, b,

c, dfcan be arranged in groups of two is 4. 3 12, and the groups
are

ab, ba, ac, ca, ad, da, bc y cb, bd, db, cd, dc.

Similarly, the formula gives for
n = 3 and r = 2, (3), = 3.2 =6,

n = 7 " r= 3, (7) 3 = 7- 6 -5 =210,

n = 10 ". r = 6, (io) B = 10.9.8.7.6. 5 = 151200.
The results which follow from equation (i) when ;/ and r
do not exceed 10 each arc embodied in the following table :
VALUES OF PERMUTATIONS.

10

9

8

7

6

5

4

3

2

I

I

10

9

8

7

6

5

4

3

2

I

2

90

72

c6

42

30

20

12

6

2

3

720

504

33^>

210

uo

60

24

6

4

5040

3024

1 680

840

360

120

24

5

30240

15120

6720

2=; 20

720

120

6

i 5 i 200

60480

20160

50-10

720

7

604800

181440

40320

5040

8

1814400

362880

40320

9

3628800

362880

10

3628800

s>

9864100

986409

109600

13699

1956

325

6 4

15

4

I

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, which gives for the number of permutations of n things
in groups of n the same value as for n things in groups of
(n 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 . Calling
this total or sum S p , it will be given by

S f = 2. (2)

To illustrate, suppose ;/ 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 S f = 3 -f- 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 S^ for ;/ = I, 2, . . . 10 are given under the
corresponding columns of the above table. But when ;/ 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 i) is the same as for r = n. Hence it appears

that

n\ , n\

COMBINATIONS. 13

But as n increases, the series by which n\ is here multiplied
approximates rapidly towards the base of natural logarithms;
that is, towards

e = 2.7182818 +, log c = 0.4342945.
Hence for large values of n

Sj, n\c, approximately.* (3)

To get an idea of the degree of approximation of (3), sup-
pose // = 9. Then the computation runs thus (see values in

the above table) :

log

. 91 = 362880 5- 5 597 6 3
c 0.4342945

9!* = 986410 5-9940575
Sfi = 986409 by equation (2).

The error in this case is thus seen to be only one unit, or
about one-millionth of S p .\

Prob. i. 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 Sp 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, b, c, d\\\ 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 PROBABILITY AND THEORY OF ERRORS.

But if the order of arrangement is ignored all of these are
seen to be repetitions of the groups enclosed in parentheses,
namely, (abc), (a fa/), (rftd), (bed). Ilcncc in this case out of
twenty-four permutations there are only four combinations.

A general formula for computing the number of combina-
tions of ;/ things taken in groups of r things is easily derived.
For the number of permutations of ;/ things in groups of r is
by (i) of Art. 2

(/,),. = //( - ])(// - 2 ) ...(//- r + i) ;

and since each group of r tilings gives 1.2.3...?' r! per-
mutations, the number of combinations must be the quotient
of (;/),. by r\. Denote this number by C(n) t . Then the gen-
eral formula is

This formula gives, for example, in the case of the four let-
ters <?, /;, c, d taken in groups of three, as considered above,

n<\ 4 ' 3 ' 2 A
c (4) 3 = , 2>3 =4-

Multiply both numerator and denominator of the light-hand
member of (i) by (;/ ?') ! 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 . . ./ in groups of three or

seven is

10!

The following table gives the values C(ii) r for all values of
;/ and r from i to ro.

The mode of using this table is evident. For example, the
number of 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.

IO

9

8

7

6

5

4

3

2

I

I

IO

9

8

7

6

5

4

3

2

I

2

45

36

28

21

15

10

6

3

I

3

1 20

84

5<>

35

20

IO

4

i

4

2IO

126

70

35

15

5

I

1

252
210

126

84

56

28

21

7

6
I

i

I

I 2O

45

36
9

8
i

i

9

IO

I

10

I

&

1023

5"

255

127

63

3i

15

7

3

It will be observed that the numbers in any column show
a maximum value when ;/ is even and two equal maximum
values when n is odd. That this should be so is easily seen
from (i)', which shows that C(ii] r will be a maximum for any
value of ;/ when r ! (n r) ! is a minimum. For ;/ even this is
a minimum for r = 4-# ; while for ;/ odd it has equal minimum
values for r = (n i) and r (// -|- i). Thus,

maximum of C(n\

n\

for ;/ even,

' n -\- i

;/!
. n

(2)

for n odd.

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 n both inclusive.
Calling this total or sum S c ,

S c = 2C(n) r .

The same sum will also come from (i)' by giving to rail values
from I to (n i), both inclusive, summing the results, and in-
creasing their aggregate by unity, Thus by either process

c i n ( n ~ *) _L n ^ H "" l )( n ~3) , , n ,

c n-\- i 2 -f- 1.2.3 -f- f- -r i.

16 PROBABILITY AND THEORY OF ERRORS.

The second member of this equation is evidently equal to
(! _[_ !) _ i. Hence

S c =2C(n) r = 2-i. (3)

The values of S c for values of ;/ and r from I to 10 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 b 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-{-b), while the probability
that the second will happen is b/(a + b). Such events are said
to be mutually exclusive. Denote their probabilities by/ 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 must give " head " or
" tail" ; a = b = I, and / = q = 1/2. Again, if an urn contain
a = 5 white and b = 8 black balls, the probability of drawing

* Jevons, Principles of Science, New York, 1874, p. 217.

DIRECT PROBABILITIES. 17

a white ball in one trial is / = 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 t c . . . ways respectively, their probabil-
ities /, q, r . . . are given by

a b c

c+. ..'

(2)

For example, if an urn contain a= 4 white, =5 black,
and c = 6 red balls, the probabilities of drawing a white, black,
and red ball at a single trial are ^ = 4/15, =5/15, and
r = 6/1$, respectively.

Formulas (l) 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 symbols of formula (i) have in
this case the values #=210, and a -f- b = 1 3699. 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. 51 . 50... 40

T~f-l f~- = 6 350i3 559 600,

1.^.3... 13

and the probability required is the reciprocal of this number.
The probability against this event is, therefore, very nearly
unity.

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. Evidently 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 PROBABILITY AND THEORY OF ERRORS.

one each of the white, black, and red. There are thus seven
different probabilities to be taken into account. The theory
of combinations shows (see equation (i), Art. 3) that the total
number of

White triads = *'^ = 4 = a

Black triads = * -\ ~ = 10 = *

6

Red triads = 'J ' 4 = 20 = c

9-8.7
White and black triads = ^-r - ( 4+10)=: 70 = d

IO . Q . 8

White and red triads = ~ ( 4+20)= 96 = e

Black and red triads ~ ' * ( 10+20) = 135 =/

White, black, and red triads = 4.5.6 = 120 = g

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, b, r to indicate the colors represented in a triad :
For a triad www /= 4/455,

" " bbb q= 10/455,

" " " rrr r = 20/455,

" " rvivb or wbb s = 70/455,

" " " bbr or brr u = 135/455,

" " wbr v= 120/455.

Prob. 7. When three dice are thrown together, what is the prob-
ability that the throw will be greater than 9 ?

Prob. 8. Write down a literal formula for the probabilities of the
several possible triads considered in the above question of the balls,
supposing the numbers of white, black, and red balls to be /, m< n>
respectively.

PROBABILITY OF CONCURRENT EVENTS. 19

ART. 5. PROBABILITY OF CONCURRENT EVENTS.

If the probabilities of two independent events are/, and
pv respectively, the probability of their concurrence in any
single instance is />/. Thus, suppose there are two urns
U l and / a , the first of which contains a l white and b l black
balls, and the second <z a white and b^ black balls. Then the
probability of drawing a white ball from U l \&p l = a l /(a l + ,)
while that of drawing a white ball from U^ is/ a = tf 9 /(<z 9 + &,)
The total number of different pairs of balls which can be formed
from the entire number of balls is (a l + ^i)(^ a + ^i) Of these
pairs a,a y 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 l a^/(a l + ^i)(# 3 + A.)

white with black = (*A + *A)/0*i + ,)(*, + # a ),

black with black = bjkjfa + *,)(*, -f- *,),

and the sum of these is unity, as required by equations (2) of
Article 4.

In general, if /,, /,, A ... denote the probabilities of several
independent events, and P denote the probability of then
concurrence,

^=AAA-.. (i)

To illustrate this formula, suppose there is required the
probability of getting three aces with three dice thrown simul-
taneously. In this case/j =/ 9 =/ 3 = 1/6 and
/>=(i/6)'= 1/216.

Similarly, if two dice are thrown simultaneously the proba-
bility that the um of the numbers shown will be II is 2/36;
and the probability that this sum 1 1 will appear in two succes-
sive throws of the same pair of dice is 4/36.36.

The probability that the alternatives of a series of events
will concur is evidently given by

Q = g lMt . . . = (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

20 PROBABILITY AND THEORY OF ERRORS-

0j i= a = q % = 5/6, and the probability that they will concur in
this is (2 = 125/216.

Many cases of interest occur for the application 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 which the probability
is p and against which the probability is q. The probability
that the event will occur every time is/*. The probability that
the event will occur (w i) times in succession and then fail ih
P*~ l q. But if the order of occurrence is disregarded this last
combination may arrive in n different ways; so that the prob*
ability that the event will occur (;/ i) times and fail once is
np n ~ l q. Similarly, the probability that the event will happer*
(// 2) times and fail twice is J( l)/'~ V 5 etc * 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 (/ + #) w is tnat * n which the exponent m, say, of q is
the whole number lying between (#+ i)^ I and (n + i)q.
In other words, the most probable result in n trials is the
occurrence of the event (n m) times and its failure m
times. When n is large this means that the most probable of
all possible results is that in which the event occurs n nq
=. n(i q) = np times and fails nq times. Thus, if the 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 wilf fail every time
in n trials, the probability that it will occur at least once in n
trials is I q". Calling this probability r,*

r= i-f = i-(i-/)". (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 Probability des Jugements, pp. 40, 41.

PROBABILITY OF CONCURRENT EVENTS. 21

chances even that the event whose probability is/ will occur
at least once. Thus, in this case, the value of n is given by

log 2
logo-/)'

This shows, for example, if the event be the throwing of double
sixes with two dice, for which p = 1/36, that the chances are
even (r = 1/2) that in 25 throws (;/ = 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, n may be so large as to render
it practically certain that the event will occur at least once.

When n is large

1.2 1.2.3'

= e~** approximately.
Thus an approximate value of r is

r = i - c ~ ">, log e = 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 e" 9 = 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- q) n from p n up to that in ftg> n ~* inclusive. This proba-
bility must be carefully distinguished from the probability that
the event will occur t times only in the n trials, the latter being
expressed by the single term in

Prob. 9. Compare the probability of holding exactly four aces in
five hands of whist with the probability cf folding 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).

22 PROBABILITY AND THEORY OF ERRORS.

ART. 6. BERNOULLI'S THEOREM.

Denote the exponents of/ and q in the maximum term of
(/ + $)" by J* anc * M respectively, and denote this term by T.
Then

n(n - !)( 2)^^+^ = nl_

ml * y ^ '

As shown in Art. 5, p in this formula is the greatest whole
number in (n + i)p, and m the greatest whole number in
( -j- i)^; so that when ;/ is large, JJL and / are sensibly equal
to np and nq respectively.

The direct calculation of T by (r) is impracticable when n
is large. To overcome this difficulty the following expression
is used : *

n I = KV- */2(i + ^ + -^ + ...). (2)

log * = 0.4342495, log 2* = 0.7981799.

This expression approaches n"e~ n ^ 2nn 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 Z"are desired. Making use of
the theorem in (i) it becomes

T=* . T ... (3)

V27tnfq

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 throw 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 (l) gives 0.296.

The probability that the event will occur a number of times

* This expression is due to Laplace, ThSorie Analytique des Probabilitfis.
See also De Morgan's Calculus, pp. 600-604.

BERNOULLI'S THEOREM. 23

comprised between (// <x) and (^ + a) in n trials is evidently
expressed by the sum of the terms in (/> + q) H for which the
exponent of p has the specified range of values. Calling this
probability A 3 , putting

jji = np -f- , and ; = ;/ w,

and using Stirling's theorem (which implies that n is a large
number),*

r / 0\ -<> + >, #\ -(*-*>

jo _ js? . , __^ ^ x I j I __ j / 1 _ _ I

^~2n~npq^ npl \ nql '

very nearly; and the summation is with respect to u from
// = a to u = + a. But expansion shows that the natural
logarithm of the product of the two binomial factors in this
equation is approximately u*/2npq. Hence

and. since n is supposed large, this may be replaced by a definite
integral, putting

dz = i/V2npq, and z* = u*/2npq.
Thus

This equation expresses the theorem of James Bernoulli,
given in his Ars Conjectandi, published in 1713.

The value of the right-hand member of (4) varies, as it
should, between o and I, and approaches the latter limit rap-
idly as z increases. Thus, writing for brevity

* See Bertrand, Calcul des Probability Paris, 1889, for an extended discus-
sion of the questions considered in this Article.

24: PROBABILITY AND THEORY OF ERRORS.

the following table shows the march of the integral :

s

'

z

'

z

'

o.oo

o.ooo

0.75

0.711

1.50

0.966

.25

.276

1. 00

.843

i 75

.987

.50

.520

1.25

.923

2 00

995

To illustrate the use of (4), suppose there is required the
probability 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 -f- !O or a number of times lying
between 990 and 1010. In this case a = 10, n = 6000, p = 1/6,
q = 5/6. Thus, a/V2npq = 10/^2 . 6000 . 1/6 . 5/6 = 0.245.
Hence, by (4) and the table, R = 0.27.

Prob. n. 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, U v containing 3 white and 5 black balls, and {/, contain-
ing 2 white and 4 black balls ; and that the probability in favor
of each urn is required. If U^ is as likely to have been
chosen as 7 9 , the probability that U l was chosen is 1/2. After
such choice the probability of drawing a white ball from U l is
3/8. Before drawing, therefore, the probability of getting a
white ball from U l was 1/2 X 3/8 = 3/16, by Art. 5. Similarly,
before drawing the probability of getting a white ball from [7 9
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
* See Poisson, Probability des Jugemcnts, pp. 81-83.

INVERSE PROBABILITIES. 25

diminished so long as the ratio of white to black 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 that 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 l 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 m causes, C 19 C 9t . . . C m .
Denote their direct probabilities by q^ g^ . . . q m \ their antecedent
probabilities by r lf r^ . . . r m ; and their resultant probabilities
on the supposition of separate existence by A> A> /

That is,

A = $Vi A = ft'i. /*= 4*r m . (i)

Let D be the common denominator of the right-hand mem-
bers in (i), and denote the corresponding numerators of the
several fractions by $., $ tf . . . s m . Then

and it is seen that there are in all (s t -f- s 9 + O equally
possible cases, and that of these s t are favorable to C lt s 9 to
C t , . . . Hence, if />, P>, . . . P m denote the probabilities of
the several causes on the supposition of their coexistence,

P* = *!/(*. + *. + *.) = A/(A + A + . A,)-
Thus in general

/>, = A/3A P* = pJ2p, ...P m = p m /2p. (2)

To illustrate the meaning of these formulas by the above
concrete case of the urns it suffices to observe that
for U 19 q, = 3/8 and r l = 1/2,
for /,, q, = 1/3 and r a = 1/2 ;
whence /, = 3/16, /, = 1/6, A + A = '7/48 J

and / > 1 =9A7t ^=8/'7-

As a second illustration, suppose it is known that a white

26 PROBABILITY AND THEORY OF ERRORS.

ball has been drawn from an urn which originally contained m
balls, some of them being black, if all are not white. What is
the probability that the urn contained exactly n white balls?
The facts are consistent with m different and equally probable
hypotheses (or causes), namely, that there were I white and
(;// l) black balls, 2 white and (;;/ 2) black balls, etc.
Hence in (l), q l = q^ = . . . = I, and

p l = \/m, / = 2/m, .../> = ////, . . . p m = m/m.
Thus ;g/= (i/2)(+i),

and Pu =

This shows, as it evidently should, that n = ;// is the most
probable number of white balls in the urn. The probability
for this number is P m = 2/(m + i), which reduces, as it ought,
to I for m = 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 several witnesses, X^ X . . .
Denote the probabilities that the witnesses tell the truth by
x l % . . . Then, supposing them to testify independently,
the probability that they will concur in the truth concerning
the event is x^^ . . . ; while the probability that they will con-
cur in the only other alternative, falsehood, is (i x^)(\ x^ . . .
The two alternatives are equally possible. Hence by equations
(i) and (2)

... ,...

p _ (!-*,)(!- *.)... (3)

^ ~ *,*, . . . + (i - *,)( i - .o . . .'

P l being the probability for and P 9 that against the event

To illustrate (3), if the chances are 3 to I that X, tells the
truth and 5 to I that X* tells the truth, x l = 3/4, *,, = 5/6, and
P l= z 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
tQ the Ninth Bridgewater Treatise by Charles Babbage (London, 1838).

PROBABILITIES OF FUTURE EVENTS. 87

It is of theoretical interest to observe that if #,, .r 9 , . . . in
(3) are each greater than 1/2, P v approaches unity as the
number of witnesses is indefinitely increased.

Prob. 13. The -groups of numbers of one figure each, two figures
each, three figures each, etc., which it is possible to form from the
nine digits i, 2, ... 9 are printed on cards and placed 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 /\ = 0.999 in equation (3) ?

ART. 8. PROBABILITIES OF FUTURE EVENTS.
Equations (2) of Art. 7 may be written in the following

1J = 1J = _~ _ d\

P. A "*P, ^/ ()

If /,, / a , .../, are found by observation, P lt P 9 , . . . P m will ex-
press the probabilities of the corresponding causes or their
effects. When, as in the case of most physical facts, the num-
ber of causes and events is indefinitely great, the value of any
p or P in (i) becomes indefinitely small, and the value of 2p
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
(;;/ + ) cases the event in question has occurred m times and
failed n time's. The probability of such a concurrence is, by
Art. 5, cx m (i .r)*, where c is a constant. Since x 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 m (i x) n * .

Then evidently the probability that x will fall within any as-
signed possible limits a and b is expressed by the fraction

PROBABILITY AND THEORY OP ERRORS.

so that the probability of any particular x is given by

y> ( , -

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 + -*) 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

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 Q,

- x)*dx

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 *

~~ r\slmlnl(m + n + r +

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 s times in succession,

* It is a remarkable fact that formula (5) is true without restriction as to
values of m t n, r, s. The formula may be established by elementary considera-
tions, as was done by Prevost and Lhuilier, 1795. See Todhunter's History oi
he Theory of Probability, pp. 453-457-

PROBABILITIES OF FUTURE EVENTS, %&

the factor (r + s)l/rlsl in (3), (4), (5) must be replaced by
unity.

To illustrate these formulas, suppose first that the event
has happened m times and failed no times. What is the prob-
ability that it will occur at the next trial? In this case (4)
gives

Q = J'x m ^dx I j'x m dx = (01 + i)/(m + 2).

When m 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

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 = 7, n = 3, r = 2, s = 3,

m + T = 9, n + s = 6, r + ^=5, m + # -f- I = 1 1,

w + w + f + -y + I ^ *6.
Thus by (5)

n _ 5! 9 !6!n!
- ~

Suppose there are two mutually exclusive events, the first
of which has happened m times and the second n times in
m -[- trials. What is the probability that the chance of the
occurrence of the 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.

'(i - xTdx

I

/'*"(i

(6)

Thus, if m = i and n = o, P = 3/4 ; or the odds are three to
one that the event is more likely to happen than not. Simi-
larly, if the event has occurred m times in succession,

P = I - (1/2)-+',
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 are 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 actual 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 will show how different sources of error enter
and affect the aggregate error ; and the statement of such an
expression is of fundamental importance in the theory of errors.

To fix the ideas, suppose a quantity Q to be a function of
several independent quantities x, y, z . . . ; that is,

0=/(*,^ *...)>

and let it be required to determine the error in Q due to errors
in x, y, 8 . . . Denote such errors by AQ % Ax, Ay, As . . .
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
of the function in terms of the component actual errors, since
the actual value of AQ is known when the actual errors of
x, y, z . . . are known. It should be carefully noted that the
quantities x, y, z . . . 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-equation in accordance with (i)
for the function Q xyz + .# a iog (y/z).

Prob. 1 6. In a plane triangle a/b sin A/sin B. Find the error
in a due to errors in b, A, and B.

Prob. 17. Suppose in place of the data of problem 16 that the
angles used in computation are given by the following equations :
A = A, + HiSo -^- B," Q, B = B, + i(i8o - A, -B, - C t ),
where A t , B, , C t are observed values. What then is Aat

Prob. 18. If w denote the weight of a body and r the radius of
the earth, show that for small changes in altitude, Aw/w 2Ar/r\
whence, if a'precision of one part in 500 ooo ooo is attainable 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 terms of their magnitudes.
Thus, using the customary notation, let e denote the magni-

* This problem arose with the International Bureau of Weights and Measures,
whose work of intercomparison of the Prototype Kilogrammes attained a pre-
cision indicated by a probable error ot 1/500 ooo oooth part of a kilogramme*.

PROBABILITY AND THEORY OF ERRORS.

tude of any error in a system of possible errors. Then the law
of such system may be expressed by an equation of the form

y = *(e). (0

Representing e as abscissa and y as ordinate, this equation
gives a curve called the curve of frequency, the nature of which,
as is evident, depends on the form of the function 0. This
equation gives the relative frequency of occurrence of errors in
the system ; so that if e is continuous the probability of the
occurrence of any particular error is expressed by yds = 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(e), namely, that in which
0(e) = c, a constant. This form of 0(e) obtains in the case of
the 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 (f>(e)de = cde, and the sum of all such prob-
abilities, by equation (2), Art. 4, must be unity. That is,

J<t>(e)de = c

(2)

This gives c = i/2a, or by (i) y = i/2a. The curve of fre-
quency in this case is shown in the figure,
D AB being the axis of e and OQ that of y.
It is evident from this diagram that if the
errors of the system be considered with
respect to magnitude only, half of them
should be greater and half less than a/2.
This is easily found to be so in the case of

o B

tabular logarithms, etc.

As a second illustration of (i), suppose y and e connected
by the relation^ = c Va* - e 9 , where a is the radius of a circle,

TYPICAL ERRORS OF A SYSTEM. 33

c a constant, and e may have any value between a and + <*
Then the condition

a

cj'de Vtf^ 4 =

gives c = 2/(a*7t). In this, as in the preceding case, 0(-f- e) =
<p( e), the meaning of which is that positive and negative
errors of the same magnitude are equally likely to occur. It
will be noticed, however, that in the latter case small errors
have a much higher probability than those near the limit a,
while in the former case all errors have the same probability.

In general, when e is continuous 0(e) must satisfy the condi-

y* *

<t>(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(-f- e) = 0( e). In such cases, if a denote the limiting
value of e,

ART. 11. TYPICAL ERRORS OF A SYSTEM.

Certain typical errors of a system have received special
designations 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
which 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,

e a = the average error.

34 PROBABILITY AND THEORY OF ERRORS.

Then, observing (2), of Art, 10,

/0(eWe = C<j>(e)de = f(p(e)d = l'<t>(e)de =
*/ */ t/

CO

, = zj*(t>(e)ede.

The student should seek to avoid the very common misap-
prehension of the meaning of the probable error. It is not
" the rriost probable error/' nor " the most probable value of
the actual error" ; but it is that error which, disregarding signs,
wpuld occupy the middle place if all the errors of the system
were arranged in order of magnitude. A few illustrations will
suffice to fix the ideas as to the typical errors. Thus, take the
simple case wherein 0(e) = c = 1/20, which applies to tabular
logarithms, etc. Equations (i) give at once

>=, e m =lV3, e.= La.

23 2

For the case of tabular values, a = 0.5 in units of the last
tabular place. Hence for such values

e, = 0.25, e m = 0.29, e a = 0.25.

f . Prob., 19. Find the typical errors for the cases in which the law
of error is 0(e) = cVa* e a , 0(e) = c(a^e\ <f>(e)=c cos 9 (?re/2);
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
resviltant 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 RESULTANT ERROR. 35

and let the laws of error of e, e l9 e a , . . . be denoted by 0(e),
j&,( e i)> 9 (e 8 ) . . . Then the value of e is given by

=,+.+... (i)

The probabilities of the occurrence of any particular values
of e,, e a , . . . are given by 0,(e 1 )dfe 1 , t (e,K e . . . ; and the
probability of their concurrence is the probability of the cor-
responding value of e. But since this value may arise in an
infinite number of ways through the variations of e lf e a , . . .
over their ranges, the probability of e, or 0(e)dfe, will be
expressed by the integral of ffr^e^de^^e^de^ . . . subject to
the restriction (l). This latter gives e l = e e a e, . . ., and
dfej = dfe for the multiple integration with respect to e a , e f , . .
Hence there results

or

(O*. (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 suffices 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 l vary over
the range a to -\-a, and e 2 vary over the range b to -f- b.
Then by equation (2), Art. 10,

^fe) = I/(2*), 9 (6 8 ) = l/(2b).

Hence equation (2) becomes

In evaluating this integral e a must not surpass b and
e, = e e a must not surpass a. Assuming a > b y the limits
of the integral for any value of e = e, + e, lying between
(a -|n b) and (a b) are b and -(- (e + #) This fact is

* The reader desirous of pursuing this phase of the subject should consult
Bessel's Untersuchungen ueber die Wahrscheinlichkeit der Beobachtungsfehler;
Abhandlungen von Bessel (Leipzig, 1876), Vol. II.

36

PROBABILITY AND THEORY OF ERRORS.

made plain by a numerical example. For instance, suppose
a = S and =3. Then (a-\-b) = 8 and (a b) = 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 a limits of e a

- 6 = - 5 - i, - i = + (e + a\

= - 4 - 2,

= -3-3, - 3 = - *

Similarly, the limits of e a for values of e lying between
(# b) and -f- (# V) are and -f- b\ and the limits of
e 9 for values of e between + (a V) and + (a + b) are -f- (e a)
and + . Hence

for -

< e< - <

- (3)

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 being the

axis of e and the line joining the middle
. , , , ,.,.,, x
points of the bases being the axis of 0(e).
. ^ \ /

(See the first figure in Art. 13.) The prop-

V . , / v - , ,. 11 . .

erties of (3), including the determination

f , ,. . , Ml

of the limits, are also illustrated by the

,.... -j r i j

adiacent trapezoid of numerals arranged

J , , & ,

to represent the case wherein a = 0.5 and
b = 0.3. The vertical scale, or that for 0(e), does not, how-
ever, conform exactly to that for e.

1 1 ion I
Ill 101 in

Ill! 101 III I

IIIIIIOIIIIII

IIIIIIIOIIIIIII
IIIIIIIIOIIIIIIII

ERRORS OF INTERPOLATED VALUES* 37

Prob. 20. Prove that the values of 0(e) as given by equation (3)
satisfy the condition specified in equation (3), Art. 10.

Prob. 21. Examine equations (3) for the cases wherein a = and
b = o; and interpret for the latter case the first and last of (3).

Prob. 22. Find from (3), and (i) of Art. u, 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 fir^t differences.
Thus, suppose that v v and v 9 are two tabular logarithms, and
that it is required to get a value v lying / tenths of the interval
from v l towards v v Evidently

v = v, + (v % - vj/= (i /X + to a ;

and hence if ^, e l9 e^ denote the actual errors of v, v^v^ , re-
spectively,

* = (!-/)*, + /',. (I)

It is to be carefully noted here that e 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 v = log (24373)
from a 5-place table. Then 7;, = 4.38686, 7> a = 4.38703,
v a v l = +0.00017, / = 0.3, and v =4.38691.1. Likewise, as
found from a 7-place table, e l == 0.45, e. t = +0.37 in units of
the fifth place; and hence by (i) e= 0.20. That is, the
actual error of v = 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

6i== (i f)e it e a = /^ a , and e = e = e, + e* I

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 = o.$t.

Hence the law of error of the interpolated values is ex-
pressed as follows :

o c -4- e
, _- jJ2_J for values of e betw. 0.5 and (0.5 /),

= for values of e betw. (0.5 /) and +(o.5~/),

= for values of e betw. +(0.5 /) and +0.5.

(2)

The graph of 0(e) for / = 1/3 is shown by the trapezoid
A B y 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. n, to be

e, = (1/4X1 - /) for o < / < 1/3, ^

= 1/2 (j/2)V2t(i /) for 1/3 </< 2/3,

= i/4/ for 2/3 <t < i.

I

^fiTTjTj Y (3)

for

--!--- *"/><,.

It is thus seen that the probable error of the interpolated
value here considered decreases from 0.25 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
beyond the last tabular place in interpolated values.

ERRORS OF INTERPOLATED VALUES 39

Case II. Recurring to the equation v = v v -f- t(i\ z/,) for an
interpolated value v in terms of two consecutive tabular values
v l and z> a , it will be observed that if the quantity /(z/ a z/,) is
rounded to the nearest unit of the last tabular place, a new error
is introduced. For example, if v l = log 1633 = 3.21299, and
^a = lg *634 = 3.21325 from a 5-place table, z/ 3 # , = -{- 26
units of the last tabular place ; and if / = 1/3, /(z' a z/ a ) = 8| ;
so that by the method of interpolation in question there results
v 3.21299 + 9 = 3.21308. Now the actual errors of i\ and
v z are, as found from a /-place table, 0.38 and +o 21 in units
of the fifth place. Hence the actual error of v is by equation
(0 I X ~ a 38 + i X + 0.21 = 0.52, as is shown di-
rectly by a 7-place table.

It appears, then, that in this case the error-equatipn cor-
responding to (i) is

, = (i-/X + /, a +, s , ., (4)

wherein e l and e y are the same as in (i) and ^ s is the actual error
that comes from rounding /(?' ?',) to the nearest unit of the
last tabular place.

The error e^, however, differs radically in kind from e l and
e y . The two latter are continuous, that is, they may each have
any value, between the limits 0.5 and +0.5 ; while e s is dis-
continuous, being limited to a finite number of values depend-
ent on the interpolating factor /. Thus, for t = 1/2 the only
possible values of c 9 are o + r / 2 > anc ^ */ 2 likewise for / =
T/3, the only possible values of e a are o, + J /3 anc ^ ~~~ J /3- ^
is also clear that the maximum value of e, 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 e = 1/2 + 1/2 = i, for t = 1/2,

"e= 1/2 + 1/3 = 5/6, " / = 1/3,

" " e = 1/2 + 1/2 = i, " / = 1/4,

" " e = 1/2 + 2/5 = 9/10 " / =: 1/5.

The determination of the law of error for this case presents
some novelty, since it is essential to combine the continuous
errors (i t)e l and te a with the discontinuous error e 3 . 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 = e, (l /)*, = e,, te^ = e a , and e t =. e s . Then
e=(6 1 +e a ) + e 8 . (5)

The law of error for (e x -j- e a ) is given by equation (2) for any
value of /. Hence for a given value of / 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 / = 1/3 the

values of e 8 are 1/3, o, and
-{- 1/3, and these are equally
likely to occur. For e 8 = o the
graph is given directly by (2),
and is the trapezoid ABCD
symmetrical with respect to OQ.
For e s = 1/3 the graph is
abQd, of the same form as
ABCD but shifted to the left
by the amount of e a = 1/3.
Similarly, the graph for the case
of e 8 = -f i/3 is a'Qb'd', and is produced by shifting ABCD to
the right by an amount equal to -f- 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) = (1/4X5 + 6s) for - 5/6 < e < - 1/6, >|

= l for - 1/6 < e < + 1/6, I (6)

= (1/4X5 - 6e) for + 1/6 < e < + 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

e

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 a = o, equations (2) give
0(e) = 2 + 46 for - 1/2 < e < o,
= 2 - 4e for o < e <+ 1/2.
This function is represented by the isosceles triangle AQE
whose altitude OQ is twice the base AE.
Similarly 0(e) for e f == -f- 1/2 would

be represented by the triangle^f^ dis-
placed to the right a distance 1/2 ; and
if the two systems for e s = o and e, =
-f- 1/2 be combined into one system,
their resultant law of error is evidently
0(e)= i+2e for i/2<e< o,

= i foro< e<+i/2, V (8)

== 2 2e for + 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 C(f)(e)de = i if the integration embraces the
whole range of e.

The determination of the general form of 0(e) in terms of
the interpolating factor t for the present case presents some
difficulties, 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,-)- and may be here stated without
proof. The phase in question is that wherein / is of the form
i/n, n being any 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-

4$ 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, (2# i)/n for n odd and i for n even.

The possible values of e 3 of equation (5) are

o, ~, -, . . . -^- for n odd,

I 2 n 2 I

- ** "" -^r> - 2 for even -

An important fact with regard to the error 1/2 for n 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, wfyich 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.
<f>(e) = i for e between \/2n and + i/2,

0(e) = ( e) for ebetw. ^F i/2n and

ft *"" I 2fl I

For n even.

0(e) = - ; A ej for e between o and ^ i/,

ft 1 2H I \

_- ( -j. J for betw. =F i/n and ^F (n i)/,

n i ^ 2^ *

= -p r(i e) for e between T ( i/) and qp I.

2( I;

By means of these formulas and (i) of Art. 1 1 the probable,
mean, and average errors for any value of n can be readily
found. The following table contains the results of such a com-
putation for values of n ranging from i to 10. The maximum
actual error for each value of n is also added. The verifica-
tion of the tabular quantities will afford a useful exercise to the
student.

ERRORS OF INTERPOLATED VALUES.
TYPICAL ERRORS OF INTERPOLATED LOGARITHMS, ETC.

Interpolating
Factor.

/Sl/tt

Probable Error.
t

Mean Error.
m

Average Error.
*

Maximum
Actual Error.

I

0.250

0.28Q

0.250

1/2

1/2

.292

.408

333

I

V3

.256

347

.287

5/6

1/4

.276

.382

.313

i

1/5

.268

370

.303

9/10

1/6

277

385

.315

i

1/7

.274

.380

3"

I3/H

1/3

.279

.389

.318

i

1/9

.278

.386

.316

17/18

I/IO

.281

.392

.320

i

When the interpolating factor t has the more general form
m/n, wherein m and n are integers with no common factor, the
possible values of e, are the same as for / = i/n. But equa-
tions (3) of Art. 12 are not the same for t = m/n 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
suffice, therefore, to give a single instance of such a function,
namely, that for which / = 2/5. For this case

0(e) =l for e between o and ^f i/ IO >

= (5/6)03/io e) for e between qp i/ 10 ancl T 3/*o,
= (S/3)(4/S 6 ) for e between qp 3/10 and T 7/10,
= (S/^)(9/ 10 e) for e between =F 7/IO and qp 9/IO.

The graph of the right-hand half of A .
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 m varies from
= 0.29 , for / = o, to 1/^24 = 0.20 +-, for / = 0.5 ; and that
varies from 0.25 to 1/6 for tfye 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 ol
error of this interpolated value is

0(e) = 2 + 46 for e between 0.5 and o
= 2 46 for e between o and + 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 J () V~2 =. 0.15.
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 o.o and 0.2 should be

+V +o,

J <t>(e)<te = 2j (2 - 46)^6 = 0.64 ;

.0.3

that is, sixty-four per cent of the errors in question should be
less numerically than 0.2.

* To afford 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, 2OO35i
20065, 20105, 20135, etc - m ^e same order to 36635. The
actual and theoretical percentages of the whole number of
errors falling between the limits o.o and o.i, o.i and 0.2, etc.,
are shown in the tabular form following :

STATISTICAL TEST OF THEORY. 45

T s^jto ^f Vrmr* Actual Theoretical

Limits of Errors. Percentage. Percentage.

o.o and 0.1 33.2 36

o.iando.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

o.o and o.i 5 51.4 50

The agreement shown here between the actual and theoretical
percentages is quite close, the 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, namely,
<p(e) = 2(1 e), no regard being paid to the signs of the errors.
The probable error is then found from

whence e p = I ^2 == 0.29. Similarly, the percentage of
the whole number of errors which may be expected to lie, for
example, between o.o and 0.2 in this system is

O a

2 / (i e)de = 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 /-place
table, the following comparison is afforded :

T -t c *< T?rrc Actual Theoretical

Limits of Errors. Percentage. Percentage.

o.o and 0.2 35.8 36

0.2 and 0.4 27.8 28

0.4 and 0.6 18.6 20

0.6 and 0.8 12.2 12

O.8 and i.o $.6 4

0.0 and 0.29 \ 49.8 50

46 PROBABILITY AND tkEORY 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
O.I, 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.

0.1 0.338 0,320

0.2 0.288 0.303

0.3 0.321 0.304

O.4 O.268 O.29O

0.5 0.324 0.333

O.6... 0.276 0.290

0.7 0.321 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. I fc i s perhaps, somewhat
smaller than should be expected, since the computation of the
actual errors to three places of decimals is hardly warranted
by the assumption of dependence on first differences only.

The average 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. r lowe. 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 <p(e) is in this case an isosceles triangle whose base, or axis of 6,
is 2, and whose altitude, or axis of 0(e), is i.

those familiar with least squares. Thus, the probable error of e t or e being
0.25, the probable error of e is found to be

0.25 Vi 2/-J-2/ 2 .

This varies between 0.25 for t = o and 0.18 for t = \ ; 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 /. It is, indeed, remarkable that the method of least
squares, which admits infinite values for the actual errors <?j and <? 9 , should give
so close an approximate formula as the above for the probable error of e.

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 *. The
natural blunder in this case is to consider <?i, <?a , and e independent, and e 9 like
fi and <?3 continuous betweer the limits o.o and 0.5 ; and to assign a probable
error of 0.25 to each. In t'Js manner the value

0.25^2(1 -

is derived. But this is absurd, since it gives 0.25 4/2 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'-a:id, work cite;', 03.
Bessel, work cited, 35.

Chaice, 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, 33, 34.

of interpolated logarithms, 38, 43.

of tabular logarithms, 34.
Me*re\ 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, n, 12.
table of, ii.

Poisson, work cited, 7, 20, 24.
Probable error, 33, 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, 2730.
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, n.

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, 33, 43.

Values of combinations, 15.
of permutations, n.
of typical errors, 4$.