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fondo: PRINTED FOR LONGMAN, BROWN, GREEN &
LONGMAN S, PATERNOSTE
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PREFACE.
In order to explain the particular object of this Trea- tise, it will be necessary to give a brief account of the science on which it treats.
At the end of the seventeenth century, the theory of probabilities was contained in a few isolated problems, which had been solved by Pascal*, Huyghens, James Bernoulli, and others. They consisted of questions re- lating to the chances of different kinds of play, beyond which it was then impossible to proceed: for the dif- ficulty of a question of chances depending almost en- tirely upon the number of combinations which may arise, the actual and exact calculation of a result be- comes exceedingly laborious when the possible cases are numerous. A handful of dice, or even a single pack of cards, may have its combinations exhausted by a mode- rate degree of industry: but when a question involves the chances of a thousand dice, or a thousand throws with one die, though its correct principle of solution would have been as clear to a mathematician of the six- teenth century as if only half a dozen throws had been considered ; yet the largeness of the numbers, and the
* Un probléme relatif aux jeux de hasard, proposé @ un austére janse- niste par un homme du monde, a été l’origine du calcul des aegeag a Ge oisson.
A
CMQGROYD
vi PREFACE.
consequent length and tediousness of the necessary operations, would have formed as effectual a barrier to the attainment of a result, as difficulty of principle, or want of clear perception.
There was also another circumstance which stood in the way of the first investigators, namely, the not hav- ing considered, or, at least, not having discovered, the method of reasoning from the happening of an event to the probability of one or another cause. ‘The questions treated in the third chapter of this work could not therefore be attempted by them. Given an hypothesis presenting the necessity of one or another out of a certain, and not very large, number of consequences, they could determine the chance that any given one or other of those. consequences should arrive ; but given an event as having happened, and which might have been the consequence of either of several different causes, or explicable by either of several different hypotheses, they could not infer the probability with which the happening of the event should cause the different hypo- theses to be viewed. But, just as in natural philosophy the selection of an hypothesis by means of observed facts is always preliminary to any attempt at deductive discovery ; so in the application of the notion of proba- bility to the actual affairs of life, the process of reasoning from observed events to their most probable antecedents must go before the direct use of any such antecedent, cause, hypothesis, or whatever it may be correctly termed. These two obstacles, therefore, the mathema- tical difficulty, and the want of an inverse method, pre- vented the science from extending its views beyond problems of that simple nature which games of chance present. In the mean time, it was judged by its fruits;
PREFACE Vil
and that opinion of its character and tendency which is not yet quite exploded, was fixed in the general mind. Montmort, James Bernoulli, and perhaps others, had made some slight attempts to overcome the mathema- tical difficulty ; but De Moivre, one of the most pro- found analysts of his day, was the first who made decided progress in the removal of the necessity for tedious operations. It was then very much the fashion, and particularly in England, to publish results and con-: ceal methods ; by which we are left without the know- iedge of the steps which led De Moivre to several of his most brilliant results. These however exist, and when we look at the intricate analysis by which Laplace ob- tained the same, we feel that we have lost some im- portant links * in the chain of the history of discovery. De Moivre, nevertheless, did not discover the inverse method. This was first used by the Rev. T. Bayes, in Phil. Trans. liii. 370.; and the author, though now almost forgotten, deserves the most honourable remem- brance from all who treat the history of this science. Laplace, armed with the mathematical aid given by De Moivre, Stirling, Euler, and others, and being in possession of the inverse principle already mentioned, succeeded both in the application of this theory to more useful species of questions, and in so far reducing the dif- ficulties of calculation that very complicated problems may be put, as to method of solution, within the reach of an ordinary arithmetician. His contribution to the science was a general method (the analytical beauty and power of which would alone be sufficient to give him a high rank among mathematicians) for the solution of
wp The same may be said of several propositions given by Newton.
(|
Vill PREFACE. | }
all questions in the theory of chances which would otherwise require large numbers of operations. The instrument employed is a table (marked Table I. in the Appendix to this work), upon the construction of which the ultimate solution of every problem may be made to depend.
To understand the demonstration of the method of Laplace would require considerable mathematical know-~ ledge ; but the manner of using his results may be de- scribed to a person who possesses no more than a common acquaintance with decimal fractions. To reduce this method to rules, by which such an arithmetician may have the use of it, has been one of my primary objects in writing this treatise. JI am not aware that such an attempt has yet been made: if, therefore, the fourth, and part of the fifth chapters of this work, should be found difficult, let it be remembered that the attainment of such results has hitherto been impossible, except to those who have spent a large proportion of their lives in mathe- matical studies. I shall not, in this place, make any remark upon the utility of such knowledge. Those who already admit that the theory of probabilities is a desir- able study, must of course allow that persons who cannot pay much attention to mathematics, are benefited by the possession of rules which will enable them to obtain at least the results of complicated problems ; and which will, therefore, permit them to extend their inquiries further than a few simple cases connected with gambling. By those who do not make any such concession, it will readily be seen, that the point in dispute may be argued in a more appropriate place than with reference to the question whether others, who hold a different opinion,
PREFACE. ix
should, or should not, be supplied with a certain arith- metical method.
The first six chapters of this work (the fourth, and part of the fifth exclusive) may be considered as a treatise on the principles of the science, illustrated by questions which do not require much numerical com- putation. To this must be added the first appendix, on the ultimate results of play. Omitting the first pages of the latter, the discussion on the noted game of rouge et noir will, with the problems in page 108. &c., serve to show the real tendency of such diversion. I am informed that this game is not played in England at any of the clubs which are supposed to allow of gambling: but it was permitted in the Parisian salons until the very recent suppression of those establishments ; and the ac- count given of it will show what has taken place in our own day. The game of hazard is more used in this country; but I have been prevented from giving it the
same consideration by the want of aclear account of the
manner in which it is played. Nothing can be more unintelligible than the description given by the cele- brated Hoyle.
The fourth chapter has been already alluded to: it contains the method of using the tables at the end of the work in the solution of complicated problems. The seventh chapter, and the fourth appendix, contain the application of the preceding principles to instruments of observation in general.
The remainder of the work is devoted to the most common application of this theory, the consideration of life contingencies and pecuniary interests depending upon them, together with the main principles of the management of an insurance office. As this portion was
x PREFACE.
not written for the sake of the offices. but of those who deal with them, I have confined myself to such points as I considered most requisite to be generally known. Common as life insurance has now become, the present amount of capital so invested is trifling compared with what will be the case when its principles are better un~ derstood ; provided always that the offices continue to act with prudence until that time arrives. At present, while the public has little except results to judge by, the failure of an office would cause a panic, and perhaps re- tard for half a century the growth of one of the most useful consequences of human association : but the time will come when knowledge of the subject will be so diffused, that even such an event as that supposed, if it could then happen, would not produce the same result.
There are, however, one or two things to which I should call the attention of those whose profession it is to calculate life contingencies : —
1. The notation for the expression of such contin- gencies (pp. 197—204.). This notation was suggested by that of Mr. Milne, from which it differs in what I believe to be acloser representation of the analogies which connect different species of contingencies. Thus, an annuity to last a number of years certain does not differ from a life annuity in any circumstance which requires a difference of notation ; nor an insurance from an an- nuity certain of one year deferred till a life drops. Since writing the pages above referred to, I have learned that I was not the first who considered an insurance in that light. Some years ago the government granted annuities for terms certain, ‘to ccmmence at the death of an individual ; but refused to insure lives: the consequence was, that, by a very obvious evasion, insur-
PREFACE, xi
ances were effected by buying annuities for one year certain, to commence at the death of a person named. This had the effect of putting an end to such annuities.
2, The form of the rule for computing the value of fines, and its introduction into the method of calculating the present value of a perpetual advowson (pp. 231. 236. and Appendix the Second). It will be found that the rule of every writer on the subject is palpably wrong in principle, with the exception of that of Mr. Milne.
3. The rule for the valuation of uniformly increasing or decreasing annuities, given in the fifth appendix. A simple application of the differential calculus is made a striking instance of the position, that the labour of a person of competent knowledge is seldom lost. The annuities given by Mr. Morgan and Mr. Milne, are for every rate of interest, from three to eight per cent.; and perhaps those gentlemen may have had some doubts as to the necessity of inserting the two last rates. It now appears, however, that, in consequence of the extent to which their tables are carried, the values of increasing or decreasing annuities, can be calculated with great accuracy for three and four per cent., and with sufficient nearness for five per cent. ; and with very little trouble, compared with that which it must have cost Mr. Morgan to calculate the table referred to in page xxviii. of the Appendix.
The rules, in page xxix. of the Appendix, contain a point which, as no demonstration is given, may cause some difficulty. In turning an annuity or insurance which cannot be extinguished during the life of the party into one which can, a direction to add is given which will at first sight, perhaps, be supposed to be a mistake, and that subtract should be written instead. But
Xli PREFACE.
it must be remembered that an annuity of, say £3 a year, diminishing by £1 every year, is equivalent, by the first part of the rule, to an annuity of which the suc- cessive payments are as follows:
£3, £2, £1, £0, £(—1), £(—2), £(—3), &e.
That is, the first part of the rule, when the annuity is extinguished during the tabular life of the party, gives the value of his interest upon the supposition that he is to begin to pay as soon as he ceases to receive. If then, this is not to be the case, the value of his interest must be increased accordingly.
4, The method of the balance of annuities, or the determination of complicated annuities by the addition and substraction of simple ones. This has been done before ; but it has not, to my knowledge, been carried to the extent of making all the questions which commonly occur deducible from the fundamental tables, without the aid of any new series. It is desirable that the beginner should be accustomed to deduction by reasoning, without having recourse to the mechanism of algebra, which, as a quaint editor of Euclid observed, “is the paradise of the mind, where it may enjoy the fruits of all its former labours, without the fatigue of thinking.” Of no part of algebra is this more true, than of the method by which complicated annuities are deduced from simple ones, by the resolution of the series which’ represent them into the simpler series of which they are composed. The education of an actuary does not neces- sarily imply the study of geometry ; and such processes, for instance, as those by which are found the values of a contingent insurance or a temporary insurance (pp. 222. 226.), will serve, as far as they go, to ac-
i Y a « a 3 b 4 ; : ie a 4 :
i ae
PREFACE. Xili
custom him to make those efforts of mind, and to bear that tension of thought, the necessity for which is the distinction between a problem of geometry, and one of ordinary algebra.
The considerations contained in this volume have, in my opinion, a species of value which is not directly de- rived from the use which may be made of them as an aid to the solution of problems, whether pecuniary or not. Those who prize the higher occupations of intel- lect see with regret the tendency of our present social system, both in England and America, with regard to opinion upon the end and use of knowledge, and the purpose of education. Of the thousands who, in each year, take their station in the different parts of busy life, by far the greater number have never known real mental exertion ; and, in spite of the variety of subjects which are crowding upon each other in the daily business of our elementary schools, a low standard of utility is gain-
ing ground with the increase of the quantity of instruc-
tion, which deteriorates its quality. All information be- gins to be tested by its professional value; and the know- ledge which is to open the mind of fourteen years old is decided upon by its fitness to manure the money-tree. Such being the case, it is well when any subject can be found which, while it bears at once upon questions of business, admits, at the same time, the application of strict reasoning ; and by its close relation to knowledge of a more wide and liberal character, invites the student to pursue from curiosity a path not very remote from that which he entered from duty or necessity. Such a subject is the theory of life annuities, which, while it will attract many from its commercial utility, can hardly fail to be the gate through which some will find their
XVi PREFACE,
a person whose ambition it is to walk in the brightest boots to the cheapest insurance office, he has my pity: for, grant that he is ever able to settle where to send his servant, and it remains as difficult a question to what quarter he shall turn his own steps. ‘The matter would be of no great consequence if persons desiring to insure could be told at once to throw aside every prospectus which contains a puff: unfortunately this cannot be done, as there are offices which may be in many circumstances the most eligible, and which adopt this method of ad- vertising their claims. If these pompous announce- ments be intended to profess that every subscriber shall receive more than he pays, their falsehood is as obvious as their meaning; if not, their meaning is altogether concealed.
Public ignorance of the principles of insurance is the thing to which these advertisements appeal: when it shall come to be clearly understood that in every office some must pay more than they receive, in order that others may receive more than they pay, such attempts to persuade the public of a certainty of universal profit will entirely cease. To forward this result, I have en-
deavoured, as much as possible, to free the chapters of |
this work which relate to insurance offices from mathe- matical details, and to make them accessible to all edu- cated persons. Whether they act by producing convic- tion, or opposition, a step is equally gained: nothing but indifference can prevent the public from becoming well acquainted with all that is essential for it to know on a subject, of which, though some of the details may be complicated, the first principles are singularly plain.
August 3. 1838.
eg eS en
ADEE SCG Sees Be a
CONTENTS.
CHAPTER IL
On the Notion of Probability and its Measurement; on the Province of Mathematics with regard to it, and Reply to Objections - Page 1
CHAPTER II. On Direct Probabilities - - . ax = - = 30
CHAPTER III. On Inverse Probabilities - S ~ fe e
Cr 09
CHAPTER IV.
Use of the Tables at the end of this Work - » ~ - 69 CHAPTER V.
On the Risks of Loss or Gain - oer o s - ¥8 CHAPTER VL
On common Notions with regard to Probability - . - IR
CHAPTER VIL On Errors of Observation, and Risks of Mistake - - - 18
CHAPTER VIIL
| On the Application of Probabilities to Life Contingencies - - 158 CHAPTER IX.
: On Annuities and other Money Contingencies - ~ - 181 CHAPTER X,
ro coy b>
On the Value of Reversions and Insurances « = ie
XVill OONTENTS.
CHAPTER XI. On the Nature of the Contract of Insurance, and on the Risks of Insurance Offices in general . Page 237
CHAPTER XIi. On the Adjustment vf tne Interests of the different Members in an Insurance Office - . - e - 267
CHAPTER XIII. Miscellaneous Subjects connected with Insurance, &c. a - 294
APPENDIX.
APPENDIX THE FIRST.
On the ultimate Chances of Gain or Loss at Play, with a particular Application to the Game of Rouge et Noir - . ~ eas |
APPENDIX THE SECOND.
On the Rule for determining the Value of successive Lives, and of Copyhold Estates - .. - - - Xv
APPENDIX THE THIRD. On the Rule for determining the Probabilities of Survivorshir - xxii
APPENDIX THE FOURTH. On the average Result of a Number of Observations - XXiV
APPENDIX THE FIFTH.
On the Method of calculating uniformly decreasing or increasing Annuities - - - - - - XXvVi
APPENDIX THE SIXTH. On a Question connected with the Valuation of the Assets of an In- surance Office - . a “ - - XXX1
Table I. ” ° m » S<Xxtv Table II. . ie . ~ XXXvVili .
ON
PROBABILITIES.
CHAPTER I.
ON THE NOTION OF PROBABILITY AND ITS MEASURE- MENT 5 ON THE PROVINCE OF MATHEMATICS WITH REGARD TO IT, AND REPLY TO OBJECTIONS.
~“Wuen the speculators of a former day were busily employed in constructing celestial tables for the use of prophets, or investigating the qualities of bodies for the manufacture of gold, no one could guess that they were accelerating the formation of sciences which should themselves be among the most essential foundations of navigation and commerce, and, through them, of civilis- ation and government, peace and security, arts and liter-. ature. That good plants of such a species require the warmth of mysticism and superstition in their early growth is not a rule of absolute generality, for there are eases in which cupidity and vacancy of mind will do as well. Cards and dice were the early aliment of the branch of knowledge before us; but its utility is now generally recognised in all the more delicate branches of experimental science, in which it is consulted as the guide of our erroneous senses, and the corrector of our fallacious impressions. And more than this, it is the source from whence we draw the means of equalising the B
of ee . , A - , é “ s « <
Bis ESSAY ON PRVBABILITIES. aiccidlents of life, and contains the principles on which jt is found practicable to induce many to join together, and consent that all shall bear the average lot in life of the whole. But the ill educated offspring of a vicious parent is frequently fated to bear the stigma of his de- scent, long after his own conduct has created the good opinion of those who know him. The science which I endeavour, and I believe almost for the first time, to ren- der practically accessible in its higher and more useful parts to readers whose knowledge of mathematics ex- tends no farther than common arithmetic, is still often considered as foreign to the pursuits, and dangerous in the conduct, of life. It is said to be necessary only to gam- blers, and calculated to excite a passion for their worthless and degrading pursuit. This refers to its practical and moral consequences : with regard to its title to confidence, it is often supposed to rest upon pure conventions of an uncertain order, and to depend for the connection of results with principles upon the higher branches of ma- thematics ; things understood by very few, and frequently distrusted, if not by those who have reached them, by those who have passed some way up the avenue which leads to them. All these impressions must necessarily be removed before the theory of probabilities can occupy its proper place ; and it is, therefore, my preliminary task to meet the arguments which arise out of them: There is an indefinite dislike in many minds to all know- ledge which they cannot reach ; it may tend to remove this if I show that results, at least, are very easily at- tained, and methods practised: but the notion that asserted knowledge is not knowledge must be met by preliminary reasoning, and imperfect as it must neces- sarily be, considered as a view of the subject, it may yet afford the means of dwelling on the first principles to a greater extent than is usually done in formal treatises on recognised subjects.
Human knowledge is, for the most part, obtained under the condition that results shall be, at least, of that degree of uncertainty which arises from the possibility of
SS ore
}
PE eR aA
INTRODUCTORY EXPLANATIONS, 3
their being false. However improbable it may be, for in- stance, that the barbarians did not overturn the Roman empire, we do not recognise the same sort of sensible cer- tainty in our moral certainty of the fact which we have in our knowledge that fire burns, or that two straight lines do not enclose space. And we perceive a difference in the quality of our knowledge, when any alteration takes place in our circumstances with respect to exterior objects. That fire does burn is more certain than the account of the fall of Rome: that fire yet to be lighted wild burn may or may not be more certain than the historical fact, according to the temperament and knowledge of the in- dividual. And thus we begin to recognise differences even between our (so called) certainties ; and the com- parative phrases of more and less certain are admissible and intelligible. It is usual to begin the subject by saying that our certainties are only very high degrees of probability. This is not practically true at the outset ;
yet so far as deductions can be made numerically.
with respect to our impressions of assent or dissent, it will be shown to be correct so to consider the subject.
We have a process to go through before we can arrive
at such a conclusion, as follows: — When a child is born, there is a certain degree of force which we allow to the assertion that he will die aged 50. To it we answer that it may be, but that that particular age is unlikely compared with all the rest, though, at first sight, as likely as any other. If the assertion be made of two children, that one or other will die aged 50, we readily admit that our “it may be, but it is not likely,”’ is no longer the same assertion as it was before. it is of the same sort, but not of the same strength: the assertion is more probable, and wherever we have the notion of more and ess, we feel the possibility of an answer to the question, ‘‘ how much more or less? ” and which we should produce if we knew how. First impressions would induce us to suppose it twice as probable that the assertion may be made of one or other of two children, as of one alone; and so on. Let this false measure (for B 2
4, ESSAY ON PROBABILITIES.
such it is) remain ; we are not here considering what is the proper measure, but whether we can conceive the possibility of a measure or not. Let the preceding me- thod of measurement be admitted ; and let us ask how we stand with regard to the same assertion, predicated of one or other of a million of children born together. The answer is, we feel quite certain, that many of them will die at the age of 50, Supposing humanity to en- dure 50 years, we feel as confident of the truth of the assertion, as we do that Rome was taken by Alaric, or that fire will burn. Without entering into the very different sources through which conviction comes to us, we put four propositions together : —
The Romanem- | Two straight | Fire will | Of 1,000,000 of
pire was over- | lines cannot burn. | children born,some turned bynorth- | enclose a will die aged fifty, ern barbarians. space. if the race of man
last fifty years.
and, we ask, if you were to receive a certain advantage upon naming a truth from among these four assertions, what would guide your choice? There is certainly a little difference in the impressions of assent with which we regard the four; but whether. it be of any real strength, we may test in this way: — Supposing the benefit in question to be 1000/., would you not let another person choose for you, almost at his pleasure, and certainly for a shilling ?
On this we remark, firstly, that by it we feel sensible of our assent and dissent to propositions derived in very different ways, being a sort of impression which is of the same kind in all. To make this clearer, observe the following: — A merchant has freighted a ship, which he expects (is not certain) will arrive at her port. Now suppose a lottery, in which it is quite certain that every ticket is marked with a letter, and that all the letters enter in equal numbers. If I ask him, which is most probable, that his ship will come into port, or that he will draw no letter if he draw, he will apswer, unques- tionably, the first, for the second will certainly not hap-
es ste i
INTRODUCTORY EXPLANATIONS. §&
pen. If I ask, again, which is most probable, that his ship will arrive, or that he will, if he draw, draw either @ Oe, OF Cy 08. or 2, or y, or 2, he will answer, the second, for it is quite certain. Now suppose I write the following series of assertions : —
He will draw no letter (a drawing supposed). He will draw a.
He will draw either a or 0.
He will draw either a, or b, or c.
ie will draw either a or b OF seccsseee OF ry. He will draw either a or 5 or .,...006. OF Y OF %
and making him observe that there are, of their kind; propositions of all degrees of probability, from that which cannot be, to that which must be, I ask him to put the assertion that his ship will arrive, in its proper place among them. ‘This he will perhaps not be able to do, not because he feels that there is no proper place, but because he does not know how to estimate the force of his impressions in ordinary cases. If the voyage were from London Bridge to Gravesend, he would (no steamers being supposed) place it between the last and last but one: if it were a trial of the north-west passage, he would place it much nearer the beginning ; but he would find difficulty in assigning, within a place or two, where it should be. All this time he is attempting to compare the magnitude of two very different kinds (as to the sources whence they come) of assent or dissent ; and he shows by the attempt that he believes them to be of the same sort. He would never try to place the weight of his ship in its proper position in a table of times of high water.
We also see, secondly, that the impression called cer-
tainty is of the character of a very high degree of
probability. Out of 1,000,000 of children born, it is
certain some will die aged 50. But by gradual pro-
gression, our unassisted judgment makes us _ believe
that we may correctly say that it is 1,000,000 times as B 3
an
6 ESSAY ON PROBABILITIES.
probable the assertion will be true of one or other out of 1,000,000 as of one alone. The method of measuring is wrong, but that is here immaterial; suffice it that, come how it may, the multiplication of the degree of assent implied in “ there is a remote chance of it” is found to give that which is conveyed in “ we are quite sure of it.” We have thus a sort of freezing and boiling point of our scale of assent and dissent, namely, absolute certainty against on the one hand, absolute certainty for on the other hand, with every description of intermediate state.
Thirdly, we have proposed two ascending scales of assertions, in both of which first impressions would make us suppose the probability of the second is double that of the first, that of the third treble, and so on, as follows : —
A child born will die aged fifty. | a must be drawn. Of two children born, one or | a or 6 must be drawn. other will die aged fifty. Of three children born, one or | a, or 6, or c must be drawn. other will die aged fifty. &e. &e. &c. &e. &e. &C.
Now it will hereafter be positively proved that our notion is correct in the second case, but incorrect in the first; or at least that it cannot be correct in both. Even then, if we should fail in assigning positive mea- surements, we may succeed in drawing useful distinctions. When we imagine two things to have a point of re- semblance which they have not, it is worth while to in- vestigate methods of correction, even though we cannot assign how much the two properties differ which we supposed were alike.
The quantities which we propose to compare are the forces of the different impressions produced by different circumstances. The phraseology of mechanics is here extended: by force, we merely mean cause of action, considered with reference to its magnitude, so that it is more or less according as it produces greater or smaller effect. It is one of the most essential points of the
INTRODUCTORY EXPLANATIONS. |
subject to draw the distinction we now explain. Pro- bability is the feeling of the mind, not the inherent property of a set of circumstances. It is frequently referred to external objects, as if it accompanied them independently of ourselves, in the same manner as we imagine colour, form, &c. to abide by them. ‘Thus we hold it just to say, that a white ball may be shut up in a box, and whether we allow light to shine on it or not, it is still a white ball. And if we were to translate the common notion, we should also say that in a lottery of balls shut up in a box, each ball has its probability of being drawn inseparably connected with it, just as much as form, size, or colour. But this is evidently not the case: two spectators, who stand by the drawer, may be very differently affected with the notion of likelihood in respect to any ball being drawn. Say that the question is, whether a red or a green ball shall be drawn, and suppose that A feels certain that all the balls are red, B, that all are green, while C knows nothing whatever about the matter. We have here, then, in reference to the drawing of a red ball, absolute certainty for or against, with absolute indifference, in three different persons, coming under different previous impressions. And thus we see that the real probabilities may be dif- ferent to different persons. The abomination called intolerance, in most cases in which it is accompanied by sincerity, arises from inability to see this distinction. A believes one opinion, B another, C has no opinion at all. One of them, say A, proceeds either to burn B or C, or to hang them, or imprison them, or incapacitate them from public employments, or, at the least, to libel them in the newspapers, according to what the feelings of the age will allow ; and the pretext is, that B and C are morally inexcusable for not believing what is true. Now substituting* for what is true that which A be- lieves to be true, he either cannot or will not see that it
“* The refusal of this substitution is what soldiers call the key of A’s position: he himself sees the absurdity of his own arguments the moment it is made; and he is therefore obliged to contend for a sort of absolute truth external to himself, which B or C, he declares, might attain if they pleased.
B 4
8 ESSAY ON PROBABILITIES.
depends upon the constitution of the minds of B and C what shall be the result of discussion upon them. Let it be granted that the intellectual constitution of A, B, and C is precisely the same at a given moment, and there is ground for declaring that any difference of opinion upon the same arguments must be one of moral character. Granting, then, that it were quite certain A is right, he might be justified in using methods with B and C which are reformative of moral character ; that is to say, granting that state punishments are reform- ative of immoral habits, as well as repressive of im- moral acts, he would be justified in direct persecution. But to any one who is able to see with the eyes of his body that the same weight will stretch different strings differently, and with those of his mind that the same arguments will affect different minds differently — by difference not of moral but of intellectual construction — will also see that the only legitimate process of alteration is that of the latter character, not of the former ; namely, argument * and discussion. In the mean time, we bring it forward as not the least of the advantages of this study, that it has a tendency constantly to keep before the mind considerations necessarily corrective of one of the most fearful taints of our intellect.
Let us now consider what is the measure of proba- bility. Any one thing is said to measure another when the former grows with the growth of the latter, and diminishes with its diminution. For instance, in the tube of a thermometer, the height of the mercury above freezing point (a line) measures the content of a cy- linder ; not that a line is a solid, but twice as much length beiongs to twice as much content, and so on. Again, the content of the cylinder measures the quantity of expansion in a given quantity of mercury (and in this case not only measures, but is). Thirdly, the
* It is frequently asserted, that opinions dangerous to the existence of public order must not be promulgated. This is a question distinct from the one in the text, so far as it is political. If we grant no morals except expe.
diency, (which, it appears to us, is necessary for the affirmation of the pre. ceding,) the answer is, simply, that persecution is ineffective. .
INTRODUCTORY EXPLANATIONS. 9
quantity of expansion measures the quantity of heat
which produces it.
The exactness of mathematical reasoning depends upon that of our knowledge of the circumstances em- ployed. No theorem about triangles, for instance, is true of any approach to a triangle such as we make on paper ; but only more and more nearly true, the more nearly we make our lines lengths without breadths, and straight. Similarly, we cannot apply any theory of pro- babilities to the circumstances of life, with any greater degree of exactness than the data will allow. But as in geometry we invent exactness by supposing the utmost limits of our conceptions attainable in practice, so in the present case we begin by reasoning on circumstances de- fined by ourselves, and require adherence to certain axioms, as they are called, meaning propositions of the highest order of evidence.
Axiom 1. Let it be granted that the impression of probability is one which admits perceptibly of the gradations of more and less, according to the circum- stances under which an event is.to happen.
Axiom 2. Let it be granted that when one out of a certain number of events must happen, and these events are entirely independent of one another, the probability of one or other of a certain number of events happening must be made up of the probabi- lities of the several events happening. For instance, in the lottery of letters, in which there are 26 inde- pendent possible events, the probability of drawing either a, b, c, or d is made up of the probabilities of drawing a, of drawing 6, of drawing c, and of drawing d, put together.
The latter axiom may excite some discussion ; but we must observe that it is the uniform practice of mankind to act upon it, which is a sufficient justification ; for what are we doing but endeavouring to represent that which actually exists? With regard to the value of each chance, suppose that one of the letters is a prize of 26/., and that the 26 letters have been bought, If I buy
10 ESSAY ON PROBABILITIES.
up all the vested interests at less than 1/. a piece, I am certain to gain; if at more, I am certain to lose. 1/, a piece is what I ought to give for each, if I buy all: it is the universal practice to consider that 1/. a piece is still the value, if I buy a part. To say this is in fact to say that the force of the impression called certainty should, in this case, be considered as made up of 26 equal parts, each of which is to be considered as the representative of the impression of probability which a right-minded man would derive from the possession of one ticket.
On this I have to remark, 1. That so soon as any notion receives the exactness of mathematical language, though it be thereby not altered, objections are taken to it. The reason is, that we frequently not only use ex- pressions which can be rendered quite exact, but also fairly act upon them as if they were exact, but not be- cause we consider them exact. Why does the lottery ticket of the preceding instance bear the character of being exactly worth 1/.? Not as any consequence of the accuracy of the preceding process, supposing it ac- curate, but because we do not know why we should exceed rather than fall short of it. It appears to me that many of our conclusions are derived from this principle, which is called in mathematics the want of sufficient reason. <A ball is equally struck in two dif- ferent directions, the table being uniform throughout. In what direction will it move? In the direction which is exactly between those of the blows. Why? No posi- tive reason is assignable (experiment being excluded) ; but from the complete similarity of all circumstances on one side and the other of the bisecting direction, it is impossible to frame an argument for the ball going more towards the direction of one blow, which cannot imme- diately be made equally forcible in favour of the other. The conclusion remains, then, balanced between an in- finity of possible arguments, of which we can only see that each has its counterpoise. Now whether we adopt the above conclusion as to probability for its exactness.
eee
INTRODUCTORY EXPLANATIONS. iI
or for its want of demonstrable inexactness one way more than the other, it is still a principle of human action, and as such is adopted. Many writers on pro- bability speak of it as being a maxim which, if it were not adopted, ought to be. Certainly, such an assertion has some strong arguments in its favour ; but with me they would not outweigh the importance I should attach to exact deduction from the conceptions which actually ° prevail.
Let the prospect of drawing any given letter be of a degree of force represented by 1, all the several prospects being equal. Then 2 is the chance of draw- ing one or other out of any given pair; and so on up to 26, which is here the representative of certainty. But if the lottery had 50 letters, the prospect of draw- ing a given letter would no longer be represented by 1; or if so, the certainty of drawing one out of 50 in the second would be represented by 50, while the certainty of drawing one out of 26 in the first is repre- sented by 26. Now certainty, absolute certainty, should have the same representation whatever contingencies it may be supposed to be compounded of. If a man be sure of 100/., it matters nothing whether his certainty arise from the announcement of a prize in a lottery of 1000 tickets, or of a legacy to which 20 other people were looking forward. ‘To use a common phrase, a man can but be certain ; and therefore it would be desirable to use the same symbol for certainty in all cases. Let this symbol be unity or 1; then in the first lottery the chance of any given letter is represented by =',, and in the second by =!;. Similarly the chance of 1 out of 10 given letters in the first lottery is 4°, and in the se- cond, 4°.
Now I pause upon this result, which, in fact, con- tains all the theory I shall be obliged to use; grant this, and you can be constrained, by demonstration, to admit all the rest as simple logical consequences. A writer on this subject, therefore, must take care not to let an opponent of its principles choose his own ground of
12 ESSAY ON PROBABILITIES.
attack, so as to wait until he can take advantage of the length of a deduction, or of the mathematical character of the steps. Do you admit, 1. That a certainty, if you have it, of drawing a 10/. prize in a lottery, is precisely the same thing whether there be 100 or 1000 tickets ? and 2. That if there be 3 white balls and 17 black in a lottery, of which either white ball is to be a prize, you ’ are compelled to regard your chances of success and failure with impressions of which it is reasonable to suppose the force to be as 3 to 17; or to say, “ the degree in which I fear failure is, to my degree of hope of success, in the proportion of 17 to 3.” If you say this, it matters nothing whether you say it because you feel the correctness of the proposition, or because you feel a want of data to deny it in one way more than in the opposite. _ Provided only that you do not deny it, your occupation of opponent is gone; for all that suc- ceeds is merely a mathematical use of this mathematical definition. In the words of the ritual, Speak now, or ever after hold your tongue.
But it may be asked, with regard to the mathematical part of this subject, What is the province of the science of calculation? Are we, because we reject the higher mathematics, entirely without evidence; or can we ob- tain any thing like conviction of the truth of our me- thods? Now it happens unluckily for objectors, that the duty of mathematics in this science is very much more simple in character than the same in astronomy, mechanics, optics, music, or any other part of mathe- matical physics. For in the whole of these sciences, we have principles, as well as results, deduced by long trains of mathematical reasoning ; whereas, in the science before us, we ask nothing of mathematics but the abbreviation of long numerical operations. For in- stance : — “ If bodies move round another body, circu- larly, and so that each body, in its own circle, describes equal lengths in equal times, and if, moreover, the squares of the times of revolution are in the same proportion as the cubes of the distances, then it follows that the cause of motion can be nothing but an attractive
é ‘ . ¢ 4 4 ; ; ;
Le ef ee a ee A i SES ain EMO Oe ee eS te en ee es
en
INTRODUCTORY EXPLANATIONS, 13
force directed towards the central body, which, for dif- ferent distances, changes inversely as the square of the distances.” This is well knoWn to be a fundamental part of the system of astronomy which has enabled one century to do more towards correct prediction of the state of the heavens than the twenty centuries which preceded it; and yet the apparatus of mathematics which is required to establish this result, which is of the nature of a principle, is enormous. But in the present subject we shall establish all our principles without the aid of any more mathematics than is contained in arith- metic ; and when we draw upon the science, it shall be for nothing but abbreviation of long processes. The principle upon which mathematical abbreviation fre- quently proceeds is this: that where the calculation of a few results materially aids the production of a great many more, it is advisable to calculate a multitude of results, to arrange them in convenient tables of reference, and to publish them ; so that by means of one person taking a little more trouble than would otherwise fall to his share, all others may be saved labour altogether. Mathematical tables are frequently nothing but the re- sult of labour performed once for all ; but it also some- times happens, that the principle on which the labour is performed can be exemplified by a familiar case of it. We shall take that of logarithms as an instance.
Every table of logarithms is an extensive table of com- pound interest. Not to embarrass ourselves with frac- tions, let us take a table of cent. per cent. compound interest. We have then the forlawsne: amounts of 1/. in 1, 2, 8, &c. years :—
Yrs. Am. | Yrs..| Am. i Yrs. Am. Yrs. Am. O 1 7 128 # 14 16564 § 21 2097152 1 g 8 256 415 82768 } 22 4194304 2 4 9 512 | 16 65536 | 23 8388608 3 8 } 10 | 1024 4 17 | 181072 § 24 | 16777216 4/16 ;11 | 2048 | 18 | 262144 } 25 | 33554432 5 | 562 $12 | 4096 § 19 | 524288 | 26 | 67108864 | 6 | 64 £18! 8192 § 20 |1048576 | 27 1134217728
14 ESSAY ON PROBABILITIES.
The property of this table is, that if we wish to multiply together any two numbers called amounts, we have only to add togetMer the number of years they belong to, and look opposite the sum in the table of years. Thus, 11 and 12, added together, give 23; 2048 and 4096, multiplied together, give 8,388,608. The reason is as follows: if 1/.in 11 years yield 20481, and if this 2048/. be put out for 12 years more, then, since 1/. in 12 years yields 4096/., 2048 times as much will yield 2048 x 4096/.; or the amount in 11+12 years is the product of the amounts in 11 and 12 years. The only reason why the preceding table is not in the common sense of the words a table of logarithms, is, that its construction leaves out most of the numbers. We can deal with 2048 and 4096, but there is nothing between them. The remedy is, to construct a table of compound interest, at such an excessively small interest, that a year shall never add so much as a pound through- out. Certain considerations, by which the table may be _ shortened, but with which we have here nothing to do, make it convenient to suppose such a rate of interest, that 1/. shall increase to 10/. in not less than 100,000 years, at compound interest. Or we may suppose interest pay- able 100,000 times a year, and say, let the whole yearly interest be 1000 per cent. per annum. ‘Taking the first supposition, we have a part ot a table of logarithms as follows :-—
Am.
1000 1001 1002 1003
Vrs.
Am,
Yrs.
Am
| Yrs.
Am.
Yrs.
300043 300087 300130 300173
5232 5233 5234 5235
371867 371875 371883
9997 9998
371892
9999 10000
399987 399991 399996 400000
10001 10002 10003 10004
400004 400008 400015 400017
&e. | &c. | &e.| &c.
This is the light in which a common reader may view a table of logarithms. Let 1 increase to 10 at compound interest in 100,000 equal moments, then 1 will become 5234 in 371,883 such moments; and soon. We can thus manage to put down every number, within certain
INTROPRUCTORY EXPLANATIONS, £5
limits, as an amount; and thus, within those limits, we reduce all questions of multiplication and division to addition and subtraction, by reference to the tables.
We thus perceive a simple principle applied with much labour, but such as is performed once for all. The notion above elucidated was the first on which logarithms were constructed ; in time came more easy methods. We now take another abbreviation which is perpetually occurring in our subject. It is the multi- plication of all the successive numbers from 1 up to some high number ; thatis, the continuation of the pro- cess following. Let [10], for instance, represent the product of ali the numbers, from 1 up to 10, both in- | clusive, or let
[10] stand forl x2x3x4x5x6x7x8x9x 10=3628800
[l]is 1; [2] is2; [8] is 6; [4] is 245 [5] is 1203 [6] is 720; [7] is 5040; [8] is 40,320; and soon. This labour becomes absolutely unbearable when the numbers become larger; thus, [$0] contains 33 places of figures, and [1000] contains 2568 figures. But, nevertheless, we cannot deal with problems in which there are 1000 possible cases without knowing, either nearly or exactly, — the value of [1000]. It will, however, be sufficient to know this value very nearly ; within, say, a thou- sandth part of the whole ; that is, as nearly as when, the answer of a problem being 1000, we find something be- tween 999 and 1001. We now put before the reader who can use logarithms a rule for this approximation, with an example; intending thereby to show the reader who does not comprehend the process how mathematics enter this subject in the abbreviation of tedious comput- ations,
Ruute.— To find very nearly the value of [a given number], from the logarithm of that number, subtract *4342945, and multiply the difference by the given number, for a first step. Again, to the logarithm of the given number add *7981799, and take half the sum, for a second step. Add together the results of the first and
16 ESSAY ON PROBABILITIES.
second steps, and the sum is nearly the logarithm of the product of all numbers up to the given number inclusive. For still greater exactness, add to the final result its ali- quot part, whose divisor is 12 times the given number. Exampie.— What is [30] or 1X2 X3X...... x 29
x 30?
log. 30 1°4771213 1°4771213 *4342945 °7981799
Subtract 1:°0428268 2)2-2753012 Add. 30 wae
en 1°1376506 Second step.
Multiply 31-284804 First step. 1°137651 Result of second step.
Ce ee
32°422455 log. of result.
The result has, therefore, 33 places of figures, of which the first six are (nearly) 264,518 ; or, if this be increased by its 300th (12 times 30) part, or about 735, the result is 265,253, followed by 27 ciphers ; or
the approximate result is —
265253000000000000000000000000000
The true result is — 965252859812191058636308480000000
and the error is not so much of the whole, as one part out of 500,000.
In this way, we are able to do with more than sufficient nearness, and in a few minutes, what it would take days to arrive at by the common method, and with much greater risk of error.
If we wish to find the product of all the numbers, say from 31 to 100, both inclusive, we find [100] and [30] approximately, and divide the first by the second. We shall represent this by [31,100]: thus,
[7,15] stands for 7x8x9x10x11x12x13x%14x15
But, though we can thus simply put the logarithmic computer in possession of a great acquisition of power we can get through much the greater part of our task without such a process, by means of a table of which
INTRODUCTORY EXPLANATION. 17
it is not possible to explain either the principle, or the reason of its utility, to any but a mathematician. We can only explain its mere construction, as follows: —
B
a Renae anne } " a eo
A N \ wie.
Let A B be one (inch, for example) ; and take an inde- finitely extended line A X, perpendicular to A B: from A towards X let a curve be conceived to be described, so that every ordinate NP shall be connected with its abscissa AN, by the following law. Measure AN in inches and parts of inches; and multiply the result by itself; and the product by °4342945. Find the num- ber to which this product is the common logarithm, and divide 1°1284 by the result. The quotient is the fraction of an inch in NP, and in the table We find, not N P, but the area AN P B expressed as a fraction of a square inch. The curve itself is what is called an asymptote to A X, continually approaching, but never reaching, AX: and the whole area, AX being continued for ever, is one square inch. To this table I shall have continual occasion to refer: into it, in fact, is condensed almost the whole use I shall have to make of the higher mathematics.
I have thus drawn the distinction between the prin- ciples of the subject, as derived from very obvious results of self-knowledge, and the principles of mathe- matics, applied merely to the abbreviation of the tedious operations which large numbers require. I now pro- ceed to the several assertions which have been made upon the nature and tendency of the subject.
I, That it is not true. The whole weight of this assertion, and of all arguments in its favour, falls entirely upon the method of measurement in page 11., and ultimately upon the second axiom, in page 9. Again, as we are most unquestionably justified in say- ing that it is more probable we shall draw one of the
c
18 ESSAY ON PROBABILITIES.
two, a or b, than that we shall draw a, the argument must be directed against the method of measurement, not against the possibility of a measure: for wherever more or less are applicable terms, twice, thrice, &c. must be also conceived to be possible, whether we can ascertain how to find them or not. But no other method of measurement has ever been proposed, nor, in truth, have the assertors been aware that they could be brought to such close quarters, but have generally ob- jected to the theory as a whole, without any particular knowledge of its parts. It will be time enough to refute their notion, when they begin to be so particular that refutation becomes possible.
II. That it is not practical. By this it is either meant, (for practical is one of the words employed in shifting an argument, which are sometimes so con- venient), that it has not been reduced to practical form, og else that it is not capable of being so reduced ; or perhaps that it is not useful. The working results hitherto obtained may be divided into: —1. The method of obtaining probabilities. —2. The method of estimating the probability of more or less departure from the results indicated by the main branch of the theory as most probable. The first has been frequently made practical ; the second not hitherto, except to mathema- ticians. That the whole can be made practical, I hope to establish by the contents of this work. To the asser- tion that it is not useful, we oppose: —1. The unanimous opinion of astronomers, (meaning thereby persons capa- ble of applying the subject to astronomy ) that the exactness of our present knowledge is very much owing to the application of it, and their uniformly continuing to use it in the deduction of results from the necessary discordances of observations. —2. The extent to which it has been applied in the very choicest view of the word practical, (which frequently means money- making) in concerns which now employ many millions sterling. — 3. The light in which it is regarded by a very large majority of those who have studied it, as a
INTRODUCTORY EXPLANATION. 19
corrector of false impressions, and indicator of just and necessary, though not always perceptible, distinctions. — 4. The beauty of the study itself, considered merely as a speculation, and as a method of exercising certain powefs of mind, which might otherwise lie useless. — 5. The necessity of informing the public as to the real nature of the occupation called gambling, and of the class of men who live by it; the latter being persons who are using knowledge of these principles success- fully, to the daily loss and ruin of those who are not aware of what constitutes unequal play. If such argu- ments be not sufiicient to counterbalance a simple asser- tion, to the extent of making it worth while to decide the question by an examination of the subject itself, we may safely dispute the utility of any branch of know- ledge.
III. That it has a tendency to promote gambling. Those who make this objection generally use the common signification of the term gambling ; and the motives for this pursuit are, in their view, either the pleasure of suspense, acting as a stimulus to a mind weary of its own vacancy, or the desire of gain. On the first notion, the assertion is self-destructive; it amounts to saying that knowledge which diminishes suspense, by giving a better view of the circumstances, has a tendency to promote gambling, by affording the pleasure arising from suspense. So far as the theory of probabilities bears upon gaming in general, its tendency is to convert games of chance into something more resembling games of skill. Now games of skill are seldom made the ve- hicles of very high play. So far, then, the tendency of our study is to substitute the satisfaction of mental exercise for the pernicious enjoyment of an immoral stimulus. With regard to the desire of gain, we may safely admit that those who are already actuated by this motive in an undue degree, will sometimes be led to -gamble by knowing how to do so properly ; and just in the same manner some of them will be led to make forgery the means of increasing their store, from knowing
c 2
20 ESSAY ON PROBABILITIES.
how to write. But the fear that those who seek a livelihood by what is commonly called gambling, which always means cards, dice, or horse-racing, &c., would be much increased in number, if at all, by such a pursuit as the mathematical appreciation of proba- bilities, seems to me grounded upon a want of know- ledge of human nature. Putting out of view the tendency of all serious thought to lead the mind toa perception of its own resources, and to furnish methods of employing time; and not even considering that the demand for this baneful excitement is controlled by the opinion of society, and lessened by the amount of edu- cation: there still remain the means of showing that the balance is in favour of a study of the theory of probabilities, even as a preventive of this very gambling which it is said to provoke. Nemo repenté fuit tur- pissimus: and, it will be one of our objects to show, that the person who lives by gaming, deserves the strongest form of the adjective. No one ever said to himself, I have not played hitherto, but I will begin henceforward to make it my trade. A young man who is ruined by play in the first instance, or who, at least. has begun by courting as an amusement what he ends by requiring as an occupation, is the subject of which a gambler is made. Now, suppose that all those who have been ruined by play had been trained to under- stand the true nature of their pursuit. Let it be granted that some of them are so fond of acquisition, that it is only necessary to point out a plausible method to insure their following it: yet we must grant, on the other side, that there will be some who can be per- suaded that when they play against a bank or a gamester, they are almost certain of playing on very unequal terms, which is never what they contemplate and intend. The only question is, which of the two numbers will be the greatest; 1. Of those who be- come gamesters prepense, or, 2. Of those who either take a total or a partial warning; the latter in a degree sufficient to insure a fair chance for them-
pee ie
INTRODUCTORY EXPLANATION. 21
selves, The thoughtlessness of youth will be urged against my opinion, that the latter number would be very much the greatest. I reply, that, comparatively speak- ing, and with respect to maturer years, young men are thoughtless ; but, absolutely speaking, they are not so with respect to dangers of which they know the risks, The ill success of others does not deter them, be- cause they attribute it to fortune ; and, because they have superstitions hanging about them with respect to luck which are tolerably prevalent in all classes. They think they are trying their luck, as the phrase is; but if they could be convinced that it is not their luck which they are trying, but only a fraction of it, their opponent having the rest in his pocket, they would show themselves in this, as in other matters, averse to risks in which it is more than an even chance against them. They come to the consideration of the subject fraught with wrong notions, which have been carefully instilled as preventives. The character of a gambler is represented as dishonest, in the com- mon sense of the word. That is to say, the term gambler is confounded with that of sharper, meaning a person who would mark a card, or load a die. They find the falsehood of this notion in their commerce with the world: gamblers show themselves in the face of day who really appear to be, and are, men of ho- nour in the common sense of the word, and who would scorn any under-handed proceeding, under ordinary temptation at least. What then becomes of the pre- vious warning? It is proved to be false in an essential part; and is therefore lost altogether. Add to this that the principle of the occupation is misrepresented : admonition is given against trying fortune, instead of proof that fortune is not tried. A proposition is ad- vanced which is an absurdity: equal play is supposed, and yet it is maintained that the luck will generally be against the inexperienced. Skill is considered as only adding to the chances against the unskilful, instead of creating a certainty, and arguments drawn from a single . c 3
92 ESSAY ON PROBABILITIES.
game, which are really good, are applied to collections of many games, with regard to which they are not ap- plicable. I will leave it to any one to say, whether the considerations pointed out in the succeeding pages have the tendency to promote the pursuit of fair gaming as a means of profit.
With regard to gambling as a stimulus, it must be observed, that the passion has every where subsided with the increase of education and occupation. If the his- torians who write for schoolboys could spare a little space among their interminable accounts of kings, treaties, battles, to insert some account of the manners of the several ages of Europe, it would be matter of surprise that the universal rage for games of chance, should have left any time for the (so called) great actions which fill the books. The wars of the middle ages would be looked upon as belonging only to one particular class of the stimuli by which the universal vacancy was sought to be filled up. From the old Germans, who played away, to one another, their wives, their children, and lastly themselves, down to the time * of the French re- volution, the continent of Europe (and during a part of the time, Great Britain, though in a less degree,) — gives, comparatively to ourselves, the notion of succes- sive races devoted to gambling throughout the upper class, the only one upon whose occupations we get fre- quent details.
IV. That the basis of it is an irreligious principle. There is a word in ‘our language with which I shall not confuse this subject, both on account of the dis- honourable use which is frequently made of it, as an imputation thrown by one sect upon another, and of the variety of significations attached to it. I shall use the word anti-deism to signify the opinion that there does not exist a Creator, who made and sustains the universe. The charge is, that a theory of probabilities (called chances) is necessarily anti-deistical, because it refers
* Quand, avant la révolution francaise, les états de certaines provinces - aient assemblés, on y jouait un jeu terrible, et tel que Vendroit ot il se
enait, dans la c- devant province de Brétagne, s’appellait Venfer. — Dict. he Jews (Encyc. Meth.) 1792.
INTRODUCTORY EXPLANATION. 983
events to chance. Various modifications of this asser- tion present themselves, but they may all be referred either to that just made, or to a tendency argument of the same character. All the sciences have had to encounter this aspersion, each in its turn ; but it is to be remarked, that philosophy and philosophers have always been charged with the worst thing going. The believers in sorcery never failed to attribute an intimate connection with infernal spirits to all who investigated nature in any form: the believers in anti-deism follow in their steps. There is in the proposition above mentioned, a shifting of the meaning of terms: it has been customary to designate anti-deism as the opinion that the world was made by chance, meaning, without any law or purpose existing ; but the word chance*, in the acceptation of probability, refers to events of which the law or purpose is not visible. Thus a great part of the application of this subject has been destroyed by successive discoveries, When the observatory at Greenwich was founded, the chance errors of observation were large in the fixed stars. Nothing could be said but that there was a deviation which appeared of one sort in one observation, and of another in another, without visible law or order. Brad- ley’s discoveries removed much of this, that is, pointed out law where law was not seen to exist before. Im- provement of instruments and methods of observation has still more distinguished the error into parts with a visible, and parts with an invisible, cause. As an answer to the species of argument employed, nothing more is necessary: those who can, may consider this science as not bearing on religion, either in one way or the other, so far as anything in the preceding argu- ment is concerned, or in the explanation which is no more than necessary for an answer. But there is a view of the subject, and that one most indispensable, which
* Generally speaking, the abstract singular term chance has the anti- de.stic meaning, while the plural chances is used for the several possibilities of an event happening. Thus Hume says : — ‘‘ Though there be no such thing as chance in the world..... there is certainly a probability which arises from a superiority of chances.”
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24, ESSAY ON PROBABILITIES.
better deserves to be made a fundamental principle, than an incidental answer to a futile objection. The past contains our grounds of expectation for the future: Why? Because we cannot help supposing that there were causes which produced the past, and which con- tinue to act. If there be any one to whom this is not a truth, he cannot proceed with us one step. Suppose that 100 drawings out of an urn all give white balls, the presumption is very strong that the 101st will give a white ball also. But if there neither be, nor ever were, some reason why the balls so drawn should be white rather than black, that is, if the event be pure chance, then the 100 drawings afford no presumption whatever — that the 101st will be either white or black. So far then as we have yet gone we have the following positive and negative conclusion :
The theory of probabilities absolutely requires, in its fun-
The theory of probabilities, so far as considerations of ab-
damental principles, the rejec- tion of the notion that pure chance can produce any two events alike; that is, it pre- sumes causation and order of some kind or other, that is, providence of some kind or
solute necessity are concerned, neither denies nor asserts, in whole or in part, any thing whatsoever respecting the mo- ral or intellectual character of the providence which it re- quires to be granted.
other.
From the preceding we may be certain that no con- clusion in any way leading to natural religion, however faint, is tacitly assumed in the premises. If there- fore such a conclusion should follow legitimately, it stands upon a basis of absolute security. This is not often the case in arguments drawn from nature in gene- ral, on account of the mixture of considerations with which the mind is affected by them. When we speak of the vastness, the regularity, and the permanency of the solar system in general, the very immensity of the argument would prevent the mind from being aware whe- ther there was or was not either an appeal to constitutional feeling, distinct from reason, or even an assumption of the question in the manner of deducing it. The cele-
INTRODUCTORY EXPLANATION. 25
brated work of Paley may be considered as a treatment of the following syllogism : — “ If there be contrivance, there was design; but there is contrivance, there- fore there was design,” the minor of which is proved by appeal to observation. But the author refers the opponent to the beauty and ingenuity of the methods in which the contrivance is brought about: to the general effect on our notions of what is done, compared with what we could do. It may very often be discovered what the real tenor of an argument is, by observing what would refute it: now imagine an individual possessed with the notion that he could execute * better contrivances, and Paley’s argument must (to him ) be imagined to be ineffec- tive.t Itappears to me that the result of the treatise in question is this: ‘‘ If there be a contriver, he must be one of infinite power and intellect.” But the argument of contrivance against chance cannot, from the complication and non-numerical character of the instances, be illustrated. by any reference to what might have been if chance had prevailed. Taking, for example, the chamois, as the result of a contrivance for the support of animal life on frozen mountains, we have no method of comparing the chamois of design with any notion that we can form, and call the chamois of chance. But where a consider- ation is pure number, we then have other ideas, of the homogeneity of which with that in question, we feel assured: and we can absolutely try the question with chance in precisely the same manner as we try it in the common affairs of life. Let us assume, as we must, that a number produced by chance alone (in the anti- deistical sense of the word,) might as well have been any other as what it is. And further, let us require before we grant intelligence and contrivance, not merely the presence of an adaptation which would have been unlikely from chance alone, but two such phenomena,
* Such as is said actually to have struck Alfonso of Portugal, when the Ptolemaic theory of the heavens was explained to him.
+ The fault of most treatises on Natural Theology is to draw the reader’s attention from the mere design, to the complication and ingenuity of the design. The Bridgwater Treatises have a consistent title, and it is worthy of remark, that this was the doing of the testator himself,
26 ESSAY ON PROBABILITIES.
perfectly distinct from each other considered as phe- nomena, each of which might have existed without the other, and both tending to the same object, which would have been defeated by the absence of either. Let it also be granted, to fix our ideas, that we admit as proved, a proposition which has a hundred million to one in its favour.
This being premised, and laying it down as our object to show that the necessary results of the theory of pro- babilities lead to the conclusion that the existence of contrivance is made at least as certain, by means of it, as any other result which can come from it, we proceed to state a consequence :— The action of the planets upon each other, and that of the sun upon all (the most cer- tain law of the universe), would not produce a perma- nent * system unless certain other conditions were fulfilled which do not necessarily follow from the law of attrac- tion. The latter might have existed without the former, or the former without the latter, for any thing that we know to the contrary.t Two of these conditions are, that the orbital motions must all be in the same direc- tion, and also, that the inclinations of the planes of these orbits must not be considerable. Granting a planetary sys-. tem which is what ours is, in every respect except either of these two, and it is mathematically shown that such a system must go to ruin: its planets could not preserve their distances from the sun. Neither of these phe- nomena can be shown to depend necessarily on the other, or on any law which regulates the system in general. For any thing we know to the contrary, then, they are distinct and independent circumstances of the organisation of the whole. Now let us see what are the phenomena in question : —
* Permanent, not liable so to change as to destroy the organisation of the parts. Ifthe earth could ever approach so near to the sun that all the water should be vaporised, the permanency of the system would be de- stroyed, so far as our planet is concerned.
¢ The only way in which we can guess any two things to be independent. It must be remembered, as a result of the theory, that, ot things perfectly unknown, the probability of their coming to act, when known, against an
argument, is counterbalanced by the equal probability of the future dis. covery being on the other side.
INTRODUCTORY EXPLANATION. oF
I. All the eleven planets yet discovered move in one direction round the sun.
I], Taking one of them (the earth) as a standard, the sum of all the angles made by the planes of the orbits of the remaining ten with the plane of the earth’s orbit, is less than a right angle, whereas it might by possibility have been ten right angles.
Now it will hereafter be shewn that causes are likely or unlikely, just in the same proportion that it is likely or unlikely that observed events should follow from them. The most probable cause is that from which the observed event could most easily have arisen. Taking it then as certain that the preceding phenomena would have fol- lowed from design, if such had existed, seeing that they are absolutely necessary, ceteris manentibus, to the main- tenance of a system which that design, if it exist, actu- ally has organised, we proceed to inquire what prospect there would have been of such a concurrence of circum- stances, if a state of pure chance had been the only ante- cedent. With regard to the sameness of the directions of motion, either of which might have been from west to east, or from east to west, the case is precisely similar to the following :—- There is a lottery containing black and white balls, from each drawing of which it is as likely a black ball shall arise as a white one; what is the chance of drawing eleven balls, all white. Answer, 2047 to 1 against it. With regard to the other question, our position is this. — There is a lottery containing an infinite number of counters marked with all possible dif- ferent angles less than a right angle, in such manner that any angle is as likely to be drawn as another; so that in 10 drawings the sum of the angles drawn may be any thing under 10 right angles. What is the chance of 10 drawings giving collectively less than 1 right angle? Answer, 10,000,000 to 1 against it. Now what is the chance of both of these events coming together? An- swer, more than 20,000,000,000 tu 1 against it. It is consequently of the same degree of probability that there has been something at work which is not chance, in the
28 ESSAY ON PROBABILITIES.
formation of the solar system. And the preceding does not involve a line of argument addressed to our percep- tions of beauty or utility, but one which is applied every day, numerically or not, to the common business of life.
Now whether what precedes amounts to the means of producing rational conviction, it is not necessary for me to stop and inquire. The question is, how do the results of this theory affect those moral and intellectual considerations which it has been stated to have a tendency to overthrow? It matters nothing to my present purpose how much of the preceding a reader will admit ; for the point, considered with reference to the objection before us, is this : -— Does the preceding deduction weaken the probability of the existence of an intelligent creative power? for if not, the objection is overthrown, and whatever strength is conceded to belong to the reply is so much addition to other arguments in favour of the same conclusion. Let us suppose a reader so much biassed against the higher parts of mathematics that he does not feel any confidence in the united work of all ages and countries amounting to more than a millionth of certainty. There still re- mains 20,000 to 1 in favour of the conclusion above stated, after weakening the preceding by introducing a probability that all the exact sciences may be wrong, such as his state of mind requires. With respect to the bearings of the theory, we may now add the follow- ing to the statement in page 24.
Applying the first principles of the theory of proba- bilities, by means of mathematics, to the phenomena of the universe, it is a necessary conclusion that the ex- istence of something which combines together different and independent arrangements to produce an end which could not, ceteris manentibus, be produced without them, must be added to the notion of a Providence, in- telligent or not, which is required in the first prin- ciples.
With regard to the existence of a revelation from the
INTRODUCTORY EXPLANATION. 29
Supreme Being, this theory leaves the question exactly where it found it; and the same of all questions of his- torical evidence. If we were to assume fictitious data, we night, as in all other sciences of inference, produce a consequence which should be as true as the premises, standing or falling with them. The science itself is the deduction of the probability in a complicated case from the probability in a known and simple case. But where is the known and simple case in the historical question? In valuing testimony, no theory of the method in which conflicting evidence should be com- bined will help us to the original value of the several parts of it, any more than an investigation of the method of solving an equation will help us to a know- ledge of the particular equations which apply in any given case. ,
The two great theoretical questions before us are :— I. What is the measure of probability? II. What is the way of using it? — The necessary preliminary to application is, the result of the measurement in a case to which the method of measuring can be applied, and has been applied. The mistakes which have arisen from confounding these considerations are numerous. For in- stance, tell me how many times per cent. a given man will be wrong in his judgment, and I can tell you exactly, positively, and mathematically, how much more likely a unanimous jury-(not starved) is to have arrived at a true decision, than another in which the voices are 8 to 4. But that does not put me one step nearer to ascertaining what is the per centage of erroneous conclusions in the judgments of a single individual. The miscon- ceptions just alluded to are equally prevalent with regard to all the sciences; a person who studies astronomy is frequently asked what the moon is made of.
Much of the objection, religious or not, made against probability in general, is connected with the notion already mentioned, (page 7.) that it is a fundamental quality of events, external to ourselves, which is under consideration: on which the rational feeling must be,
30 ESSAY ON PROBABILITIES.
that there is no such thing. The term probability is as difficult to explain as gravitation: and the method of proceeding is the same with regard to the properties of both. We cannot tell what they are, in simpler terms, but we know them by, or rather trace and define them by, their manifestations. In both, we first see a com- pound result, depending upon the patient as well as the agent. In the case of the mental phenomena, we can- not decompose the effect produced, still less ascend a step, and find any of the laws which regulate the hu- man disposition to doubt or expect. I shall conclude by again reminding the reader, that the impression produced by circumstances upon his own mind is the thing in question ; and that nothing can be more liable to cause confusion than a lurking notion that the results of theory are anything more, before the event arrives, than a re- presentation of the relative force of his own impressions, as they should be if unassisted reason could follow the le- gitimate consequences of some simple and universally admitted principles.
I proceed in the next chapter to develope the leading rules of the science.
CHAP. if.
ON DIRECT PROBABILITIES.
We now proceed on the supposition that the probability of an event is measured by the fraction which the number of favourable cases is of all that can happen. Thus, if there be 20 white balls and 27 black (20+ 27 or 47 in all), the probability of drawing a white ball is measured by $7, and that of drawing a black ball by 214. We shall say that these probabilities ave %2 and 24, Nothing is more common than to substitute a measure
ON DIRECT PROBABILITIES. 31
for the thing itself: thus we talk of a temperature of 60°, when, in fact, each 1° only means a certain length on the tube of the thermometer. It will illustrate this to take a case in which we do not confound the thing and its measure; in the barometer, for instance, we never say that the air’s weight is 30 inches, but that the height of the barometer is 30 inches.
The technical words, probability and improbability, must now be considered as meaning the same thing in different degrees. If there be only one white ball out of a thousand, we usually say that to draw the white ball is possible, but not probable. We now speak of it as having asmall probability, namely —;;!;5 ; we might say it has a great improbability, namely ~°°99., but this phrase is not customary. The moment we obtain either numerical measures, or distinctions which are not verbal, the distinctions which are verbal frequently become su- perfluous and inconvenient. Thus in the art of book- keeping, profit and loss never appear as separate words, but only as part of a complex term profit-and-loss, meaning, one or the other, according to the side of the account on which the item is found. ‘To a mercantile reader, we should say that probability means probability- and-improbability the first or second, according as its measure is greater or less than}. When the number of favourable and unfavourable cases is the ee say 50 of each, the probability of the event is °°; or 4; and in this case we say in common life that there isa “balance of probabilities, or that the event has an even chance.
By the word chance with the article (a chance) we mean one single way in which an event may happen, as when we say that every white ball adds a chance to the prospect of drawing a white ball. In the first instance above, the chances of white and black are as 20 to 27. It is also usual to say that the odds are here 27 to 20 in favour of black against white,
Questions on probability are twofold in character : 1. Where we know the previous circumstances and re- quire the probability of an event. 2. Where we know
32 ESSAY ON PROBABILITIES.
the event which has happened, and require the proba- bility which results therefrom to any particular set of circumstances under which it might have happened. The first I call direct, and the second inverse, questions.
We must begin with direct questions, though the in- verse precede the direct in practice. For, as we know the whole range of possible cases in hardly one instance, we cannot proceed with points which have reference to matters of life until we know what presumptions arise with respect to the whole, from observation of a part.
In direct questions of probabilities, the event may either be simple, that is, depending on one indivisible event ; or compound, consisting of several events which may happen together. Thus, suppose four events, either of which may happen, and call them A,B,C, D. Knowing the circumstances of each, I may ask the fol- lowing questions, every one of which states an event simple or compound.
1. What is the chance of A. 2, What is the chance that one shall happen and only one. 3. What is the chance that one or more will happen. 4. What is the chance that one at least will happen, and one at least will not. 5. What is.the chance that a given pair, and no others, will arrive. 6. What is the chance that a given pair at least will arrive. 7. What is the chance that some pair or other will arrive, but only a pair. 8. What is the chance that a pair at least, will happen, &c. &c. All these are most evidently distinct questions when they are clearly proposed; but it is almost as evident that they are very liable to be confounded.
The mathematical definitions and theorems which will be necessary for our purpose, are the following : —
1. A permutation means a number of cases selected out of all possible cases, in some particular order ; so that different arrangements of the same things make dif- ferent permutations. Thus if all the possible cases be A,B, C, and D, we have the following
Permutations of one out of four, A, B, C, and D.
33
Permutations of two out of four, AB, BA, AC, CA, AD, DA; BC; CB, BD, DB, CD, DC.
Permutations of three out of-four, ABC, ABD,BAC,BAD, ACB, ACD, CAB, CAD, ADB, ADC, DAB, DAC, BCA, BCD, CBA, CBD, -BDA, BDC, DBA, DBC, CDA, CDB, DCA, DCB.
Permutations of four out of four (different arrangements of four), ABCD, ABDC, BACD, BADC, ACBD, ACDB, CABD, CADB, ADBC, ADCB, DABC, DACB, BCAD, BCDA, CBAD, CBDA, BDAC, BDCA, DBAC, DBCA, CDAB, CDBA, DCAB, DCBA.
To find the number of permutations of one number (in) out of another (n), begin with the whole number (n) and write down as many numbers, reckoning down- wards. as there are units in the number (m) which are to be in each permutation: then multiply all together. For instance: How many permutations are there of 4 out of 12, The answer is.
be M22 AR IG 2 See
ON DIRECT PROBABILITIES.
11880
The following table will furnish examples or serve for reference.
| 10 i 9 8 7 6 5 /4.|8 |2 [1 | 1} 10 | 9 8 7 6 15 |a|s left | 2) 90 | 72 56 | 42 | 30 | 20 |1ale {2
3) 720 504 336 | 210 | 120 60 joe | | 4| 5040 | s024 | 1680 | 840 | 860) 120 24
5| 30240 | 15120 | 6720 | 2520 | 720] 120 6| 151200 | 60480 | 20160] 5040 | 720 7| 604800 | 181440 | 40390 5040 | s| 1814400] 362880 | 40320 | 9| 3628800] 362880 |
10] 3628800 | |
4 ESSAY ON PROBABILITIES.
Thus the number of permutations of 6 out of 9 is opposite to 6 under 9, or 60480.
2. By a combination is meant the same thing as a permutation, except only that arrangement is no part of the idea. Thus, of all the permutations of three out of four in A,B,C, D, the following, ABC, BAC, ACB, CAB, BCA, andCBA,are all the same combination. ‘Thus we have a permutation : a selection in a certain order.
a combination : a selection without reference to order.
To find the number of combinations, divide the number of permutations by the product of all the numbers up to the number in each combination. Thus, how many combinations can be made of 4 out of 12? The answer is found by
dividing 12 x1llxlOx9 by lx 2x 3x4
The process may always be shortened by striking common factors from the dividend and divisor. Thus, if we wish to know the number of different hands which can be held at whist (combinations of 13 out of 52) we must divide 52. 51. 50. 49. 48. 47. 46. 45. 44. 43. 42. 41. 40.
by 1, 62-8. 4. °5. -6e 7. 8. 9. 10. 13.122
The shortest way is to decompose every number in both into its factors, when it will appear that all the factors of the divisor are found among those of the di- vidend ; as follows :— 1B. 2.2. 17.3..5.562.467. 2 28.2.2 47. 2:23.: 5.9.8. 2438.
43. 2.7.3. 41. 2.2.2.5. 1, 2. 3. 2.2. 5. 3.2. 7, 2.2.2. 3.3. 5.2. 11. 2.2.3. 184.5
} which gives 495.
and the factors which must be multiplied to give the result are
V7. 7. 47, 23. §& 43. 2.7. 41. 2.2.2.5. or 17. 47. 23. 43. 41. 35. 560 giving 635013559600.
When the number in each combination is more than half of the whole number, the rule may be shortened.
ON DIRECT PROBABILITIES. 35
Thus, if ! ask how many combinations of 21 can be taken out of 25, I do in effect ask how many combinations of 4 may be taken. For there are just as many ways of taking 21 as there are of leaving 4.
The following table corresponds to the one preced- ing: —
10 9 | 8
1 10 ie,
@1 45 36 | $051.0). 13K L1G kth ae diad
3 | 190 84 | 56.1 $5 | PO IO et
41210 {126 | 70 51 is) s+ 4
9 10 1
10 1
Thus the combinations of 5 out of 9 are 126 in number.
3. When the same event may be repeated in a per- mutation, the number of permutations is the product of as many numbers, all equal to the whole number of possible events, as there are of events in each permutation. Thus, of four events, A, B,C, D, the number of per- mutations of two, with repetition, is4 x 4 or 16 ; with- out repetition 4 x 3 or 12. The additional 4 in the first case, are AA, BB, CC, and DD. The number of permutations of three, with repetition, is4 x 4 x 4, or 64; without repetition, 4 x 3 x 2 or 24, Of the addi- tional 40, 10 are where A onlyis repeated ; namely, AA B, AAC,AAD, ABA, ACA,ADA,BAA,CAA,
D2
36 ESSAY ON PROBABILITIES.
DAA, that is nine where A is repeated twice, and AA A, one, where A is repeated three times. Ten more are those in which B is repeated twice, &c.
4. The number of combinations in which there are repetitions must be determined without rule, in every particular case. Suppose I wish to know how many com- binations, including repetitions, can be made of four out of seven. Let the seven be A, B, C, D, E, F, G.
I. The number, without repetition, is 35.
II. Those in which A is repeated twice, and no other, are evidently as many as the number of pairs in B, C, D, E, F, and G, that is, 15. Thus we have AA BC, AACD, &c. There are as many in which B is re- peated twice, &c., so that 7 x 15 or 105, is the num- ber in which one only is repeated twice.
III. The number in which A is repeated three times is evidently 6, and the number in which one or other is repeated three times only, is 6 x 7 or 42,
IV. The number in which one is repeated four times, is 7.
V. The number in which two are repeated twice, is as many as the number of pairs in 7, or 21.
Consequently, the whole number is made up of 35 105, 42, '7, and 21, or it is 210.
We have not so much to do with combinations allow- ing repetition, as with permutations of the same kind.
To avoid the perpetual occurrence of long phrases © for simple ideas, I shall use the following abbrevia- tions: — By P f 4,20 } is meant the number of per- mutations of 4 out of 20, without repetition: by PP § 4,20} the same with repetition. By C {4,20} is meant the number of combinations of 4 out of 20, without repetition: by CC {4,20} the same with repetition. Again, by [4,20], as in page 15., is meant the product of all numbers from 4 to 20, both inclu- sive; and, by 7!°, asin algebra, is meant ten sevens multiplied together. Thus, a reference to the preceding rules will show the following : — —
ON DIRECT PROBABILITIES. 37 P {5,12} ig [12,8], pP{ 5,12} is 19 cf 4 25} is (25, 22] divided by [1, 4]
5. Tf there be an event composed of several others in succession, of which the first may happen in, say 10 different ways, the second in 14, and the third (let there be but three) in 6, then the compound event must be one out of 10 x 14 x 6, which is the number of dif- ferent ways in which it may happen. When all the component events may happen i in the same number of ways, this reduces itself, both in principle and rule, to the case of permutation with repetition.
I now proceed to some problems requiring nothing but a knowledge of the measure of probability and the preceding rules. Any event of known circumstances may be familiarly represented by a lottery of balls of different colours. Thus, suppose ten Russian ships, twelve French, and fourteen English, are expected in port, or may have arrived. Let one be as likely to ar- rive as another, and suppose it known that two have arrived, but not of which nation they are. Let a cer- tain advantage accrue to A, if they should happen to be Russian and French, of which he is desirous of selling his chance immediately. The question is precisely the _ following : — Let there be a lottery of green, white, and red balls, 10, 12, and 14 in number. Let two have been drawn, and suppose a certain advantage to arise to A, if they be green and white. What should be given to A, certain, in consideration of his chance P
Here we must first consider the number of ways in which two can be drawn. All the balls are 36 in num. ber, and the whole number of pairs is 36 x 35 + 2 or 630. Of these a green ball (1 out of 10) may be paired with a white ball (1 out of 12) in 120 different ways, ssp gg the probability of his gaining the advantage is 749 or 54. The probability against it is therefore | +1; or out of every 21 possible events, 4 are in favour of, and 17 against, the advantage. It is there- D $
38 ESSAY ON PROBABILITIES.
17 to 4, or 44 to 1, against the advantage. If the con- tingent gain were 21/. the value of the chance would be 4]., as will hereafter be more fully shown.
Of all the pairs which can be drawn, there are 10 x 12=120 green and white. 10x 9+2=45 both green. 12 x 14=168 white and red. 12 x 11+2=66 both white. 14x 10=140 red and green. 14x 13+2=91 both red. The sum of these is 630, as it should be. The reader should explain to himself the reason of the difference of process. The least probable ,case is both green, the most probable white and red Nothing is more common than the idea that the event most likely to happen, which is compounded of two or more events, is a repetition of the event which is individually the most probable. This is true of repetitions: red being most probable, it is more probable that red should be repeat- ed than that white should be repeated, or that green should be repeated ; but that two white ones should be drawn is not so probable as that a white and red should be drawn. The drawing, whatever it ‘may be, is con- sidered independently of succession — and this makes an important difference. If the balls were drawn succes- sively, both red is more probable than white followed by red, or than red followed by white, but not more probable than the chance of one or other, which is s the preceding case.
I here also take occasion to notice the common error, that because an event is more probable than any other, it is the one to be looked for. The question ought to be, Is that event more probable than some one or other out of all the other events which may happen P If ten persons engage in a competition, with an equal chance of success, and if two of them, A and B, enter into partnership, it is now more probable that the firm of A and B will win than that C will win, or that D will win, &c. But the chances against the firm are still 8 to2 or 4 to 1. If a hundred halfpence be tossed up into the air, the result which is more probable than any other, is 50 heads and 50 tails. But com- mon sense will tell us that the chances of this result are
ON DIRECT PROBABILITIES. 3
©
very small, owt of the whole, and calculation would con- firm it.
Before laying down any more specific rules, I shall take an instance of a somewhat complicated deduction, in order to show that we do not really require any other principle than is contained in the measure of probability. Suppose A and B to play at whist against Cand D. A begins, holding ace, king, and queen of trumps, and these trumps only: what right has he to ex- pect that, by playing them out in succession, he will force all the trumps of the other party? That is to say, ten trumps being distributed among B, C, and D, and any single card having the same chance of belonging to either of the three, what is the probability that neither C nor D holds more than three trumps ?
Firstly, as to the number of tenures, which are per- fectly distinct. To find the number of ways in which any number of distinct objects can be divided among any number of persons, use the following RuLE: —
Multiply together numbers equal to the number of
persons as often as there are things to be divided among them. Thus, to find in how many different ways ten distinct cards can be divided among three persons, find
3x3x3, &c. (ten threes) or 3!° which is 59049.
The question now is, how many of these 59049 ways favour the supposition that neither C nor D holds more than three out of ten. Both together they cannot hold more than six: if, then, we pick out any given siz of the ten trumps, that set may be divided among C and D in 2° or 64 ways. But of these, there are two ways in which 0 and 6 may be held, twelve ways for 1 and 5, and thirty ways for 2 and 4, none of which must be included. It must be observed, that in the last sentence, the six distinct ways into which 6 may be divided into parcels of 1 and 5 must be doubled, because * each gives two of our cases, ac-
* It is important to observe that no duplication must take place on this account, if the two numbers be the same; for instance, in dividing 6 into pareels of 3,
Dn 4
40 ESSAY ON PROBABILITIES.
cording as C holds 1 and D holds 5, or C holds 5 and D holds 1: and so of the other cases. Consequently, out of a given set of six, there are 64 all but 44, that is, 20 ways, in which, when they happen, A could force all the adversary’s trumps. But every set of six yields 20 such cases: hence the ways in which C and D can together hold six trumps, are 20 times C f 6,10} or 20 X 210, or 4200. Similarly, a given set of 5 trumps may be held by C and D in 2° or 32 ways, of which 2 and 10 must be excluded. Hence 20 x C f 5,10 or 20 x 252, or 5040 sets are the number favourable to the event in this case. Four trumps can be held in 2* or 16 ways, of which 2 must be excluded, or 14 xC j 4,10 } that is, 2940, is the number of favour- able cases. On the supposition that C and D together held only 3, or 2, or 1 trump, no exclusions are neces- sary, and the number of cases are 2° x C f 3,10 ¢ or 960, and 22 x C $2,10$ or 180, and 2 x C 1,10 or 20; and there is one case in which C and D hold no trumps. All the favourable cases are, therefore, in number
4200 + 5040 + 2940 + 960 + 180+ 20+1 or 13341.
The chance of A being able to force all the adversary’s trumps is +3344, or nearly 3! to 1 against it.
Given a fraction less than unity, and which has high numbers in its terms, required a set of fractions which shall be very nearly equal to it, and each of which shall be nearer than any other fraction of the same order of simplicity. Required, also, a near estimate of the error committed in each case.
Ruz. First perform the process for finding the greatest common measure of the numerator and deno- minator.
ON DIRECT PROBABILITIRS. 4 1 1834i)59049(4—4 53364
2 5685)13341(2x44+1=9 11870 Qx2+1=5 1971)5685(2x 9-+4+4=22 3942 5x14+2=7 1743)1971(1x22+9=31 1743 7Xx7+5=54 228) 1743(7 x31 +22=239 1596 54x 1+4+7=61 147)228(1 x 239-431 =270 147 81)147(1 81 66 &e.
Opposite to the first quotient write 1, and proceed te form columns out of the several quotients, in the fol- lowing manner ; —
1 =1st Numerator. Ist Quotient = Ist Denomina-
tor.
9nd Qu.=2d Num. 2d Qu. x Ist Qu. + 1 = 2d
Den.
2d Num. x 3d Qu. + Ist Num. | 3d Qu. x 2d Den. + 1st Den. =3d Num. =3d Den.
3d Num. x 4th Qu. + 2d Num*| 4th Qu. x 3d Den. + 2d Den. = 4th Num. =4th Den.
4th Num, x 5thQu. + 3d Num. | 5th Qu. x 4th Den. + 3d Den. = 5th Num. = 5th Den.
Any numerator multiplied by | Any denominator multiplied the next quotient, and pro-| by the next quotient, and duct increased by preceding product increased by pre- numerator, gives succeeding ceding denominator, gives numerator. succeeding denominator.
Thus, +, 2, 35, do vss, 3%, &e. &c., isa set of fractions which approach nearer and nearer to 4+33-}. The first is always too great, the second too small, the third too great, the fourth too small: every odd one too great, every even one too small.
Test of correctness. Take any two successive numer- ators and denominators, multiplied crosswise ; they give products which differ by unity.
42 ESSAY ON PROBABILITIES.
54 61 54 x 270=14580 239 270 239 x 61=14579 1
Estimation of the error. The error of 3} is less than sis (4 x 9 = 36); that of § is less than +}5 (9 x 22 = 198); that of ,9; is less than ;4,5 (22 x 31 = 682), &c.
Th ‘pomed ctbsieeblactd is less than : rror of ; ; s les oy ae its Denominator That Den. x the next.
I now take the following problem: A die is thrown time after time; in how many times have we an even chance of throwing an ace. The common error attached to this problem is, that since there are six faces, it is most likely all will have come up in six throws.
In the first throw there are six events, five of which are unfavourable. In the first two throws, considered as giving one event, there are 6 x 6 or 36 possible cases, for every possible case of the first throw may com- bine with any case of the second. But of these 36 throws, any one of the five unfavourables of the first throw may combine with any one of the second throw, and there are therefore 5 x 5, or 25 unfavourable com- pound events. Hence the following table : —
One throw gives 6 cases ; 5 unfavourable
Two throws give 36 cases ; 25 unfavourable
RIOD dnsvenseseae BLG secacs 3 125 ..rccrcccsenieee KUNG against. )
Bc ocaswounnce A906 scscss $1 OBS. coun. 0 ccekiunin (odds turned in favour. )
oo Ee eee TUT@ icthie 3 3125 &ce.
Answer: There is not quite an even chance of doing it in three throws, but more than an even chance of doing it in four.
Ruxte. When the odds against success in one trial are n to 1, then -'; of n. (or the nearest whole num- ber to it) is about the number of trials in which there is an even chance of one success: more correctly 19° of n. ‘Thus, if it be 144 to 1 against a single attempt,
t ON DIRECT PROBABILITIES. 43
there is about an even chance of one success in 100 throws.
A table of the number of trials (cdds being n. to 1) in which there are various odds of succeeding once : —
69 | 13 to 1 | 264 | 3O0tol | 343 § 1000 to1 | 691 110 | 14to1 | 271 § 40 tol | 371 | 2000 to1 | 760 139 | 15 to1 | 277} 50to1 | 393} 3000 tol | 8v1 161 | 16 to 1 | 283} 60 tol | 411f 4000 to 1 | 829 17 to1 | 289 | 70to1 | 426{ 5000 tol | 852 195 | 18 to 1 | 294 | 80to1 | 439 | 6000 to 1 | 876 208 | 19to 1 | 300 | 90to1| 451 | 7000 tol | 885 220 | 20 to 1 | 304 {100 to 1 | 462 | 8000 to 1 | 899 230 | 21 to 1 | 309 [110 tol | 471 | 9000 to 1 | 911
22 to1 | 314 120 to 1 | 48) | 10,000 to 1 | 921
248 | 23 to 1 | 318 | 130 to 1 | 488 12 to 1 | 256 | 24tol | 322 $140 tol | 495
The method of using this table is as follows: —- Sup- pose the odds to be 20 to 1 against success in a single trial: required in how many throws it is 100 to 1 there shall be one success (or more). Look in the table opposite to “100 to 1,” and we see 462 ; multiply 20 by 462 and divide by 100 (which is always to be done) this gives 924, or 92 is about the number of throws required.
I now proceed to the method of compounding the probabilities of single events, so as to find those of com- pound events: that is, the way of methodising the re- sults of actual inspection.
Let there be two events, one of which may happen in 7 ways, and may not happen in 5; and the other of which may happen in 4 ways, and may not happen in 9: whence the probabilities of the two events (A and B) are =4, and =4;. Now, the compound event may hap- pen in 12 x 13 different ways, for any case of the first (12 in all) may come up with any case of the second (13 in all), By similar reason, the compound case in which both A and B happen may come up in 7 x 4 different ways ; hence the probability of A and B both happening, is
1x4 7 4 omen cone Cee jax ig Whichis the product of 3 and 13
(© CONT CrP 69 DD Ch Oh Oh et ct ct ct er ct coooseosgse Pht ek ek ed ek et
—
~I
io)
ood © aot oo —oy
=
Rute. When events are perfectly independent, so
AA ESSAY ON PROBABILITIES.
that the happening of one has nothing to do with that of the other, the probability that both will happen is the product of the probabilities that each will happen.
This rule applies to any number of independent events.
It is indifferent whether the events are to happen to- gether, or one after the other: thus the chances of a compound drawing out of two lotteries, one drawing out of each, are the same whether two persons draw simultaneously, or one person first draws out of one, and then out of the other.
Examp.e. Let there be three lotteries, as follows : —
6 white v neat { 8 white 5 black 2 black 10 black
What is the chance of drawing from the three, white, black, and white? The probabilities of ee events are, °;, 3, and -8,, the product of which is 54°, or about 17 to 1 against the compound event.
The probability that A will happen, and B will not, is the product of the chance for A, and that against B. The probability that one will happen, and one only, is the sum of the probabilities —1. that A will happen and B will not; 2. That A will not happen and B will happen. The probability that one or both will hap- pen is the remainder when the probability that neither will happen is subtracted from unity. The following are evident results of the measure of probability : —
1, When either P, @, or R must happen, the sum of their probabilities must be unity, which is always the representative of certainty. Thus, if a lottery contain 7 white, 5 black, and 3 red, either bene black, or red must be drawn, and =4, -3., and -3,, together make 1, and the same if the events be more or less than three in number.
2. When of two events, each excludes the other, the probability that one or other of them will happen is the sum of their probabilities. For to say that each excludes the other, is to say that they are connected eveuts, or different possible cases of the same set. Thus,
‘
ON DIRECT PROBABILITIES. AS
in the preceding lottery, if I draw white I cannot draw black, and vice versd, there being one drawing only supposed. Hence the probability of drawing black or white is = += or +2. But suppose there had been two lotteries, as follows :—
(7 white, 8 red) (5 black, 10 red),
and I draw in one without knowing in which I am drawing. The individual probabilities of Pagan in the two, if the lottery be known, are ;‘, and -°- as before ; but the question, what is the chance of padre white or black, and not red, is a different one from the pre- ceding. If I draw without knowing in which lottery 1 am drawing, my position is the same as if one lottery had been thrown into the other, and I had drawn from both in one, giving (7 white, 5 black, 18 red).
This union, which will readily be admitted to make no alteration in the probabilities, is not admissible unless the total number of balls in both be the same. For, in . the mixture, the 8 red balls of the one and the 10 of the other, are considered as severally of equal proba~ bility. But this they are not, unless the number of possible cases in the two were the same. Suppose, for instance, I had the following lotteries : —
(10 white, 10 red) (4 black, 5 red) ;
the probability of each ball of the first is =1,, but of each of the second 4. Before I can draw a white ball,
two events must happen. 1. I must happen to select the first lottery. 2.1 must happen to draw a isi ball rather than a red. The cist of these are 4 and 4° or +5 consequently, 4 x + or 4 is my chance of a white ball, But if I mix the two lotteries, that chance will be 48, which is more than 1. The reason is, that I have mixed together balls which had unequal chances of being drawn, and treated them as if they were to have equal chances. But it is evident, from the measure of probability, that I do not alter the chances for an event
46 ESSAY ON PROBABILITIFS.
or the general chances which any set of circumstances affords, if I multiply the chances for an event any num- ber of times, provided I multiply the chances against it as often. Reduce the preceding lotteries te common num- vers of balls, by putting 20 balls for one into the second, and 9 for one into the first. We then have —
(90 white, 90 red) (80 black, 100 red). Now mix the two, and we have — (90 white, 80 black, 190 red) ;
and the chance of a white is ,%.°,, or 4, that is, the pro- blem is not altered by the mixture. And the same may be shown in any other case.
Let us now suppose that the chance of A is 2 and that of B 4. The chances against A and B are there- fore } and 2.
| The possible cases are The probability of which is That A and B shall both happen 2x# or }f ‘That A shall happen, and not B 2x# Or sf
_ That B shall happen, and not A 4x? or 3; That neither shall happen. tx#2 or 3
One of these cases must happen, and the sum of the chances is 31 or 1, each compound event being ex- clusive of the others. Again, we find
{ Both or neither is 12 One or other, but only one § One or other, or both if
a One or neither The probability A or both }4 A or neither $;
B or both 45 B or neither | A or neither, or both }$ UB or neither, or both f The latter set of events does not consist of those which are mutually exclusive. Any thing which happens falls under six of them, that is, six of them must happen. Thus, A, and not B, actually arriving, would secure any gain which depended upon either of the following : — One or other, but only one A, or both
One or other, or both A, or neither One or neither A, or neither, or both,
ON DIRECT PROBABILITIES. 47
If we add popademah all the probabilities of the preceding, we find 4%, or 6. This is the indication that every possible event enters in six different ways into the contingencies whose probabilities united amount to six.
Exampie. Twelve halfpence, A, A,...Aj,o, are thrown up, required the probability of all the cases which can happen, and which we shall symbolise thus: (H3T.) means that there are three heads and nine tails. The chance of H or T in any one piece is 1; consequently, the chance of the several pieces A, A, &c., yielding each any Pee letter is ixt xi X wn (twelve factors), or ;545- Now, the ‘4096 pos- sible cases are thus classified : —
cole
H, T,,. and H,, T, happen in 1 case each. 1 ia 1] 11 ceesecceece 12 cases each.
H, a4 eeoeos is oe as eeseeceseeose cfe, 12 or 65 eeeeeoeeeo
H, Ty eeece ° H, T e8Bcervescss CG 35 12 or BQO Riese ade
H, Ts eeeese H, I, seeereveeses G 4, 12 or AOS ccasnares
*H, Ty,’ «0... o Hy Vg cdssecesesee ©} 55 12°F Or 799 ee H, T, happens in 924 cases.
It appears, then, that the most likely individual re- sult is He T,, against which, however, it is about 34 to 1. But if we ask for the probability either of this or of a single variation on one side or the other, that is, of the following event —
Either H; T,, or H, T,, or H, T; —
we find 792 x 2+ 924 or 2508"(more than half of 4096) cases in which the event arrives. That is, the chances are in favour of one or other of these arriving. As the number of events increases, a given degree of nearness to the most probable event becomes more and more likely. To illustrate this: suppose 24 halfpence thrown up, the notation remaining as before. The total number of cases is now 22+ or 16777216: calculating the number
4S ESSAY ON PROBABILITIES.
of combinations of 1,2,3, &c. out of 24, we have the following summary :—
H, T.4 and H,, T, happen in 1 case each. Big Digg oonens wy Ag ‘inenabipee ves 24 cases each, (Faas ere $e bg sevvesesoess ZIG scccoosreses Phe! hes sbsces je Pline orp creer 2024. .cccsoseces H, Ty « es we rere ry 10626 J, .ccccceos Bg “Tyg sees: Fig Tis cscseccecses 4DSO® ce cctve die’ eee wee I! ear ne 134596 .....s050 se Bly Ty ovene Hyg Ty soeerees wad: 7PACIO‘ wecsioccones Bog hag coer Ehig Tg scponpenenee HORT L onspavaieoes ee cig: EN eats 1807504 ..eceeee ae cS ag Ree 5 las UB gi ag fo 1961256 isch ececeus Bt ee Pe 2496144 35.
ia ss happens i in 2704156 cases.
The odds are now about 6 to 1 against the even di- vision of the pieces into heads and tails. But let us consider the same degree of departure from the most probable case as we took before. An alteration in one piece out of twelve answers to that of 2 pieces out of 24. Now the number of cases in which either Hy T,,,0rH,,T,;,,orH,.T,.,01H,,T,,,orH,,T,, arrives, and the odds of one or other of them, is com- puted as follows :— i H,, T,,0r Hy, Tyo 3922512 16777216 whole No. of cases H,, T,3 0r H,, T,;, 4992288 11618956 No. favourable
Hoot. s1g61s6 4 ——-___‘ 5158260 No. unfavourable 11618956 or it is now more than 2 to 1 in favour of the heads lying between 10 and 14, both inclusive.
I now put together the principles on which we have hitherto gone, adding two more, the first of which (Principle III.) is obviously a consequence of the pre- ceding, and the second of which will be presently ex- plained.
Principle I. When all the ways in which an event may happen are equally probable, the chance of its happening is the number of ways in which it may happen, divided by all the number of ways in which it raay happen and fail.
~~
ON DIRECT PROBABILITIES. 49
Principle II. The probability of any number of in- dependent events all happening together, is the product of their several probabilities.
Principle IJI. he probability of two events arriving together being known ; and also, that of one of them: the probability of the other is found by dividing the first mentioned probability by the second.
Principle IV. When an event may happen in several ways, whether equally probable or not, the probability of the event is the sum of the probabilities of its hap- pening in the several different ways.
The best way of illustrating the last principle is by beginning with one of the numerous errors into which we may fall, either in proceeding towards it, or applying it. Let there be an event which may happen in two different ways, for each of which there is an even chance. Then, according to the principle, it is certain (4 + }=1) that the event must happen. In this there is nothing inconsistent. Let there be a lottery containing ten white balls, and let them be sub-divided into two sets of five by amark. Then there are two ways of drawing a white ball, for each of which there is an even chance ; namely, I may choose a ball of one mark, or of the other. But it is evidently certain that in this case a white ball must be drawn. Now, suppose an event can happen in three different ways, for each of which there is an even chance. Then the event is more than certain (4 + 4 + 4= 14); which is absurd. But the absurdity is in the supposition : an event can only have an even chance of happening in one particular way, when that way involves half of the total number of individual cases of the event ; and it is impossible that three different ways of arriving can each contain half of the whole number of possible cases. Consequently, when we have made a calculation of the probabilities which different ways of arriving give to an event, there has certainly been an error if the sum of the probabilities exceed unity.
But when we throw three half-pence into the air, are there not three different ways of throwing head, for each
E
50 ESSAY ON PROBABILITIES.
of which there is an even chance? If by H we here mean a single H, the three events by which we propose to-attain it are not H, H, H, but H TT, THT, and TTH, the probability of each of which is 4 and, by the application of the principle, } + 4+ 4, or 4, is the probability of the event,—one single head. If by: throwing a head we mean one head or more, the ways under which it may be brought about are— one head’ only (involving the cases H T T, THT, T T H)—two heads only (involving HHT, H TH, and TH H)— and three heads (involving + HH.) The probabilities of these are 3, 3, and 4, or 4 is the chance required. Another error to which we are liable, is the wrong estimation of the probability of the different cases. For instance, a person is to go on until he throws H, and is to win if he do it in less than five throws. There are then five possible cases ; namely, H, TH, T TH, T T T Hi, T TTT. In four of these cases he wins ; in the fifth he loses. His chance of winning then appears to be +. But this supposes the five cases to be equally likely, whien is i true. tase several probabilities are +, 4, t, zz and =1,: not 1, +, &c., as supposed. Consequently the sum of ee probabilities which the several winning cases actually have is 12, or it is 15 to 1 that he wins. If we put together all the cases, which four throws pre-~ sent, thus,— HHHH 5. HTHH 9 THHH ts. TTA . HHHT 6: HLHT . 10. THHT 14. TTHT
HHTH 7. HTTH 11 THEE 15. TTTH HHTT 8 HTTT 12. THTT ig. TITTs
all those which begin with H are 8 in number, those which begin with T H are 4, with T T H, 2, and with TTTH, one. Hence 8+ 4 +2 +41, or 15, is the number of winning cases. But the argument against us, is this: most of the preceding cases are impossible, for the condition is, that the play shall stop as soon as H occurs: so that, in fact, the only possible cases are, Het, FTTH DT TH,-.T T TD, sees te to; we must then represent the several events as follows :—
Aer
ON DIRECT PROBABILITIES, 51
Ist event ; sigan 5 sh hen and gives either H or T. Probability of H Be
2nd event; does not ue happen, but is con- tingent upon ‘the first throw being T ; and gives either H or T. Probability + that the throw is made, + that if it be made it gives H; probability ao ee throw is made, and that, being made, it gives H, 4 x = 14,
5rd event; contingent upon the two first throws giving T and T, of which ihe chance is 1. Probability of winning at this throw, + x 4=1.
4th event ; cietaeck: ‘upon the three first throws giving T, T, T. Probability of winning at this throw, gxXo= is
It is obvious enough, when stated, that every cone tingency must enter into the consideration of a question ; so that if some of the circumstances depend upon the manner in which preceding contingencies arrive, this circumstance itself influences the method of proceeding. If we wish to avoid the necessity of considering a con- tingency the trial of which is itself contingent, that is, if we wish to make a contingency certain, we must in- troduce all the new events which such change of con- tingency into certainty brings with it. The whole problem is exactly the same as if we made the four throws certain, and made the gain dependent upon one head or more being thrown: but we revert again to the former state of the question, if we agree to mark the first H as the winner. In this point of view, the distinction between the two is evidently immaterial.
Exampie. There are seven lotteries, as follows (W means white, B black) :—
I Wy:1 ae
II. (3 W, 2P) Lia Ww rf 2 W, 5B) VI.
IIE. (1 W, 3 B) Law 1 B) VIL; and the conditions of drawing are the following. I. is drawn, and then II. or III., according as I. gives W or B. If II. be drawn, then IV. or V. is to be drawn, according as II. gives W or B. But if III. be drawn, then VI. or E 2
“
I. (2 W,3 ®)|
52 ESSAY ON PROBABILITIES.
VII. is to be drawn, according as III. gives W or B. What are the probabilities of the several possible drawings P
If from I. we draw W (of which the chance is 2), we proceed to II. If we still draw W (chance, 3), we proceed to IV. And here the chance of W is 1. Hence,
the chance of WWW is 2 x?xi=3
Computing all the other chances in the same way, we
get the following :—
1WWW?2 2 4=2 5. BWW? ] t= 2 WWBA 8 j= 8 6 BWB 2? } =} 3. WBW 2 2 2=<28 7% BBW 2 %4=% 4.WBB 2? 2 4=4 8 BBB 3 % = ih
The sum of all these is equal to unity, as it should be, since one or other of these cases must happen. And by reducing all to the common denominator 5. 5. 4. 6, or 600, we have the following chances : —
1. WWW 22 | s. WBW £4, | 5. BWW 4, | 7. BBW 3218 2. WWB 2 | 4.WBB {| 6.BWB %| 8. BBB $4,
That all shall be white, 7 to 1 against (nearly),
Two white and one black, $ to 1 against (very nearly)
Two black and one white, an even chance (nearly)
All black, 10 to 1 against (nearly).
From what has been said in this chapter, no great difficulty will be found in ordinary questions. The cir- cumstances are supposed to be fully known, and the probabilities will be found, of the strength which it follows they must have, to those who admit the axioms on which the measure of probability is founded,
ON INVERSE PROBABILITIFS. 53
CHAPTER III.
ON INVERSE PROBABILITIES,
In the preceding chapter, we have calculated the chances of an event, knowing the circumstances under which it is to happen or fail. We are now to place ourselves in an inverted position: we know the event, and ask what is the probability which results from the event in favour of any set of circumstances under which the same might have happened. This problem is fre- quently enunciated as follows: — An event has hap- pened, such as might have arisen from different causes : what is the probability that any one specified cause did produce the event, to the exclusion of the other causes ? By a cause, is to be understood simply a state of things antecedent to the happening of an event, without the introduction of any notion of agency, physical or moral. ‘
In order that we may secure a problem of sufficient simplicity, we must limit the number of possible ante- cedent states. Let us suppose that there is an urn, of which we know that it contains balls, three in number, and either white or black, all cases being equally pro- bable: that is, before any drawing takes place, all we ean say is, that we are going to draw out of one of the following, having no reason for supposing one in pre- ference to anather > —
A BC DEF
LCs eo). 1 (aee) TH Goo «)}.. BV: 000)
A drawing takes place, and a white ball is produced,
consequently I. is immediately excluded; for from it
the observed event could not have been produced, This
much is certain; but we are also tempted to say that E 3
54 ESSAY ON PROBABILITIES.
II. is rendered unlikely, because, from such an ante. cedent state of things, a black ball would have been more likely than a white one. On the same prin- ciple III. is more likely than II., and IV. the most likely of all. We have then to decide the relative pro- babilities of IJ., III., and IV.
Before the drawing took place, the probability of each set of circumstances was 4; and, the lottery being given, the probability of any one ball in it was 1. Thus the chance of III. being the lottery, and the second white ball being drawn from it, was} x 1, or jx The same of other balls: so that, in fact, our primitive position was that of having to draw from 12 balls, 6 white and 6 black, all equally probable. But the observed event changed that position ; a white ball was drawn: was it a given ball (namely, the white ball in II.), or was it one of two given balls (those in III.), or was it one of three (those in IV.)? There are siv cases in question, namely, A, B, C, D, E, F, and one of them happened,— we do not know which. We have used all the knowledge we have (namely, that a white ball was drawn,) in excluding the black balls.
Hence the chance that A was drawn, or that
II. was the lottery - is 7: That either B or C was Gran, or that Il.
was the lottery - = is 3% That either D, E, or Fo was drawn, or that
IV. was the lottery - - - is 3
In the preceding instance, owing to the number of balls being the same in every lottery, the antecedent proba- bility of each ball was the same. Previous to deducing a rule, I take an instance in which this is not the case. Prosiem. A white ball has been drawn, and from one or other of the two following urns: (2 white, 5 black) (3 white, 1 black).
What are the probabilities in favour of each urn?
The case is not now that of a lottery of 5 white and 6 black balls; for the chance of our going to the first urn (which is$), and thence drawing a given white
ON INVERSE PROBABILITIES. 55
ball (chance +), is 4 x +r =!;; while our chance of going to the second urn (which is }), and thence draw- ing a given white ball (chance }), is + x fori. But since we do not alter the chance of producing a white ball from either urn, if we double, or treble, &c. the number of white balls, provided we at the same time double, or treble, &c. the number of black balls, let us put four times as many balls into the first, and seven times as many into the second, as there are already. Thus we have :
(8 white, 20 black) (21 white, 7 black).
There are now 28 balls ineach: every individual ball has the antecedent probability x -=1,; and since our knowledge of the event (a white ball was drawn) ex- cludes the black balls, the question is simply this: — Out of 29 possible, and equally probable cases, was the event which did happen one out of a certain 8, or one out ofthe remaining 21? ‘The chances of these are 8, and 24; consequently it is 21 to 8 that the second lottery was that which was drawn, and not the first.
On looking at the resulting chance for the first urn, namely, =, or the (21 + 8)th part of 8, we see that 8 and 21 are in proportion to the two chances for a white ball being drawn, when we know that we are drawing from the first urn, or from the second. For these chances are 2 and 3, which, reduced to a common denominator, are 8; and $4, which are in the propor- tion of 8 to 21. The same reasoning may be applied to any other cases, and the result is as follows : —
Principle V.— When an event has happened, and the state of things under which it happened must have been one out of the set A, B, C, D, &c., take the different states for granted, one after the other, and ascertain the probability that, such state existing, the event which did happen would have happened. Divide the probability thus deduced from A by the sum of the probabilities deduced from all, and the result is the
E 4
56 ESSAY ON PROBABILITIES.
probability that A was the state which produced the event: and similarly for the rest. [Or, reduce the re- sults of the first part of the rule to a common deno- minator, and use the numerators only in the second © part of the rule.]
Exampxte I. There is a lottery which is one or other of the two following:
(3 white, 7 black) (all white). |
A ball is drawn, and restored; this takes place five times, and the result is always a white ball. What are the chances for each lottery?
Upon the supposition that the first lottery was that in question, the chance of the observed event is the product of 3°55 o> vo» To» and 35, Of tooo00: When the second is the lottery, the observed event is certain, and its probability is 1 or 190°8°0, Consequently, the probability for the second lottery is 499009, or the second has the odds 100000 to 243, or more than 411 to 1 in its favour.
Examp.ie II. Two witnesses, on each of whom it is 3 to 1 that he speaks truth, unite in affirming that an event did happen, which of itself is equally likely to have happened or,not to have happened. What is the probability that the event did happen?
The fact observed is the agreement of the two wit- nesses in asserting the event: the two possible ante- cedents (equally likely) are,—1. The event did happen. 2. It didnot happen. If it did happen, the probability that both witnesses should state its happening is that of their both telling the truth, which is? x #, or =%. If it did not happen, then the probability that both witnesses should assert its happening is that of their both speaking falsely, which is + x 4, or51,. Conse- quently, the probability that the event did happen is the (9 + 1)th part of 9, or =% ; that is, it is 9 to 1 in favour of the event having happened.
Exampxe II]. There are two urns, having certainly © 3 and 2 white balls; and in one or other, but which
ON INVERSE PROBABILITIES. 57
is not known, is a black ball. A ball is drawn and replaced ; and this process is repeated, but whether out of the same urn as before is not known. Both draw- ings give a white ball: what is the probability of the several cases from which this result might have hap- pened ?
Since the black ball is as likely to be in one as in the other, the antecedent state of things is (so far as a single drawing is concerned,) the same as if there were four urns, as follows:
I. (3 white) II. (3 white, 1 black) III. (2 white) IV. (2 white, 1 black).
There are 16 possible cases, PP [2, 4,] numbered in the first columns following, described in the second, and having the probability which each would give to the observed event (both drawings white) registered in the third, together with the numerator, when all the fractions are reduced to a common denominator 144.
PiLs 1144-8 GF ed 3, 108 Bick ied, Wa 61k eet Sige le 60 Aen Pb? ca fy da ei a blo 4'T IV. | 4, 96 [8] ILIV: | &, 72 S(t (bei fy ae 10 | ILI. If. | 3;108 | 14] Iv. 11. | 45, 72 41. | TMI. IT. | 1,144 |. 15° | IV. IT & 96
12/ IIL Iv. |2, 96 | 16|IvV.Iv.| 4,64
That is to say, if (case 8), II. and IV. were the urns of the first and second drawings, the chance of the observed event is 4 or ,4°%. But, it must be remem- bered, that we do not suppose the black ball may have been removed from one urn into the other before the second drawing takes place. Most of the preceding cases are, therefore, to be rejected; in fact, I. can combine with nothing but I. or IV., and II. with nothing but II. or III. Reject, therefore, cases 2, 3, 5, 8, 9, 12, 14, 15, and the sum of the numerators in the rest is 841. Hence the probability (for instance, )
:
58 ESSAY ON PROBABILITIES.
1. That the black ball is with the three white ones. 2. That the first drawing is from the lottery which has the black ball, and the second from the other, is (case 7) 198, To find the total probability that the black ball is with the three white ones, we must add the proba- bilities of all the cases (as to ss rab ap hae can take wee under this arrangement, namely #4,, 398, 298, 444, giving #41. Consequently, from the observed event, it is slightly more probable that the black ball is with the three white ones, than with the two.
The principle which we have illustrated, though a mathematical consequence of those which precede, is nevertheless received in common life upon its own evidence. When an event happens, we immediately look to that cause or antecedent which such event most often follows. When it rains, we suspect the barometer must have fallen; because, when the barometer falls, it usually rains.
Our next step is to inquire, what is the probability which an event gives to its several possible antece- dents, upon the supposition that they are not all equally likely beforehand ; as in the following instance.
Prosiem. A white ball is drawn, and from one or other of the following urns:
(3 white, 4 black) (2 white, 7 black):
but before the drawing was made, it was three to one that the drawer should go to the first urn, and not to the second. What is the chance that it was the first urn from which the drawing was made P
We may immediately reduce the preceding to the case where all the antecedent circumstances are equally probable, by introducing urns enough of the first kind to make it 3 to 1 that the drawing is made from one or other of them. Let us suppose the urns to be as follows : (3 white, 4 black) (3 white, 4 black) (3 white, 4 black)
(2 white, 7 black):
these urns being equally probable, the hypothesis of
ON INVERSE PROBABILITIES. 59
the problem exists. If we number the urns A,, A,, A, B, the chances which they severally give to the observed event are #, +, 4, and %, the numerators of which, reduced to a common denehiion diel, ae: 27,27, 27, and 14, Consequently, the probability that A , was chosen, is 34; and the same for A, and A,. There- fore, the chance that one a other of the three, A,, A,, and A., was chosen, is 84; which is the probability of the ball having been drawn from the urn (3 white, 4 black,) in the first statement of the problem.
The rule to which the preceding reasoning conducts us is as follows: When the different states under which an event may have happened are not equally likely to have existed, then having found the probability which each state would give to the observed event, multiply each by the probability of the state itself before using the rule in page 55. The following is another example.
An event has happened, the possible preceding states of which are represented by A, B, and C. The chances of the existence of these different states (independently of all knowledge of the observed event) are, say, 4, %, and : the probabilities that the observed event would have happened are -’,, =4;, and =, if A, B, or C were certainly existing. Form the three products Probability that the event rie} he.
Probability of A x { happen if A were known to exist
there are *
4 5 3 4 2 Hae +X pp $ X pp and 5 x +73
- the numerators of which (the denominators being com- mon) are 20, 12, and 4. Then the probability that A was the state under which the event happened, is 20 divided by 20 + 12 + 4, or 2 3 ; those of B and C are z¢ and 4
Let us now suppose, that “havihig only a first event by which to judge of the preceding state of things, we ask what is the probability of a second event yet to come. For instance, an urn contains two balls, but whether white or black is not known; the first draw-
60 ESSAY ON PROBABILITIES;
ing gives a white ball, and the ball is replaced. What is the chance that a second drawing shall give a black ball P
The preceding states under which the first event may have happened, are —
(2 white) (1 white, 1 black);
and 1 and 4 are the chances of a white ball, if one or other state were absolutely known to exist. Hence, by the last principle, { and + are the chances which the observed event gives to the two states; that is, it is two to one that both balls were white. Now, the black ball can only appear at the second drawing, upon the supposition of the second state existing; and this sup- position being made, the chance of a black ball at the second drawing is 4. Hence, page 43., the second event depending upon two contingencies, of which the chances are + and 4, its chance is 4, or it is five to one against the second drawing being black. But let us now ask what is the chance of a white ball at the second drawing? LEither of the preceding states admit of such an event, and, in fact, the event proposed — a white ball at the second drawing — means One or other of these i (2 white) and white drawn.
two combinations. ( (1 white, 1 black) and white drawn, © In the first combination, the first contingency (the chance of which is 1) ensures the second: so that + x 1 is the chance of a white ball being drawn, and of (2 white) being the lottery from which it is drawn. In the second combination, the chances of the two con- tingencies are } and 1, whence } is the chance of a white ball being drawn, and being drawn from (1 white, 1 black). But the event proposed i if either of these cases occur; therefore, 2 + 4, or 2, is the chance of a white ball ‘at the second drawing, as might have been inferred from the probability already obtained for a black ball. By such reasoning as the preceding, the following principle is established :
Principle VI. Having given an observed event A,
ON INVERSE PROBABILITIES. 61
to find the probability which it affords to the suppo- sition that a coming event shall be B, find the proba- bility which A gives to every possible preceding state ; multiply each probability thus obtained by the chance which B would have from that state, and add the results together.
Propiem. There is a lottery of 10 balls, each one white or black, but which is not known: drawings are made, after each of which the ball is replaced. 'The first five drawings are white; what chance is there that the next two drawings shall be white ?
Let S (20) denote the sum of all numbers up to 20 ; S (202) the sum of the squares of all numbers up to 202 or 400; and soon. The possible preceding states are —
(1W,9B) (2W,8B.)....(10 W,0B); and the probabilities of W five times running from each, are
To ioedesto+ds Tot rot rot rote &. up to 49.40.49 .10 . 49, or 1, the event being certain, if the last state existed, The numerators of these products (the common denominator being 10°,) are 15, 25, cree 105: whence, page 55., the proba- bilities of the several states are —
15 25 105
Sic’ Sie” .. .810 By the same reasoning, the probabilities of the proposed events (two more white balls,) are — SS Se ene 102
102102 102 ; the different preceding states being successively sup-
posed to exist; whence the actual chance which the observed event gives to the proposed is —
15 12 25 2 5 2 a agora x —_— —_—_—— x 2 + eceese + econ x te" S105 102 § 105 102 $105 1023 ree S 107 which is
102 § 105,
62 ESSAY ON PROBABILITIES.
By precisely the same reasoning, if there had been 1000 balls in the lottery, and if 157 had been drawn white, the probability that 27 more drawings would have given white balls, would have been
S 1000184 (157 + 27 = 184) 100027. S$ 1000157
The difficulty of calculating S 1000!54 is insuperable : but a mathematical theorem which we shall proceed to explain, makes it very easy to find a near approxima- tion to the preceding result, and the nearer the greater the numbers in question. :
Take the sums of the powers of the different num- bers, as follows :
Firstpowers 14+ 2+ S+ 44+ 5 GH sseoee Squares 1+ 4+ 9+ 164 25+ 36+ cence, Cubes 1+ 8+ 27+ 64+ 125+ 216 +...
Fourth powers 14+ 164 81+ 256+ 62541296 + sess Fifth powers 1432 +243 4+102443125 4+ 7776 + cess.
a mm] mp] ogee be] re
and examine the sum of any number of terms in any line, as compared with the term immediately below the last in the sum; thus : —
14+24+3+4+4 1+164+81+4+256 +625
16 3125
Form fractions with such sums as numerators, and their compared terms as denominators, and observe how much each fraction, so formed, differs from the fraction written in the last column, as follows: —
14+2 3 see 142+3 6 1 1 a ee geese ores Fae! he pe
4 4.2 4 9 Se ae
eee ead ees 18 eh aot on; 46-16, 2: 8 25. 25 2 10
hanes S (any number) 1 1
square of that number 2” twice that number
Hence it follows, that when the number is large, the preceding fraction is very nearly one half, or 1 + 2 + 3 + .. up to a large number, is very nearly one half the square of that number.
ON INVERSE PROBABILITIES. | 63
; 044 34°02 14449 1 £ Again — ——— Se Se Ht = = e394 27 3 27 S 42 $0 °3. 418 S 52 | ice 6horae 64S BE i2sor58 3. 75
B10? (S850: Bb 485
ak ce te ; and so on. 103 1000 3. 8000
In this way it appears that the sum of all the squares of numbers is nearly one third of the cube of the last number, and that the greater the number of squares taken, the greater the proximity in question. This proposition is general, namely, that the sum of the nth powers of numbers is nearly the (n + 1)th part of the (n + 1)th power of the last of the numbers : thus, the sum of all the 13th powers 1/85 + 218 4. up to 100015, is very nearly the 14th part of 100014. This proposition, never absolutely true, may be made as near the truth as we please, by taking the number of terms sufficiently great; and the error made by the substitution, is nearly such a fraction of the whole as has one more than the index of the power for its nu- merator, and twice the number stopped at for its denominator. Thus, if the tenth powers of all num- bers were summed up to 10,0001°, the substitute for this sum given by the theorem, namely, =, of 10,000!!, would be too small by about
oT eee 2x10,000 20,000
of the whole.
Prosiem. A lottery contains 10,060 balls, each of which may be white or black. A ball is drawn and then replaced, and 100 such drawings give nothing but white balls: what is the chance that the five next drawings shall all be white ?
S 10,000100+5 10,0005 S 10,000100
This chance, by what precedes, is
But § 10,000100+5 mie J x 10,000106 very nearly . 106
64 ESSAY ON PROBABILITIES.
S 10,000100__! 4 10,000!91,,,...006 101
Consequently, the chance as TOE of 10,0001" or abe nearly. sor of 10, 000106 106 If the number of balls had been a million, instead of 10,000, the preceding odds, namely, 101 to 5, would still more nearly have represented the chance that after 100 drawings, all white, the next five should be white also. If the number of balls had been abso- lutely unlimited, the preceding odds will correctly ex- press that same chance. But a lottery with an unlimited number of balls, each of which may be either white or black, is a lottery which may be anything whatever. For instance: [3 W, 7 B] is equivalent to an unlimited lottery, in which for every 7 black balls there are 3 white ones. Again, an unlimited lottery in which any number of balls may be black, and the rest white, is one in which the chance of drawing a white ball may be any whatever, and is absolutely unknown. The draw- ing of a white ball from such a lottery may be likened to the occurrence of an event, about the preceding chances of which we are in total ignorance. The pre- ceding process furnishes us with the following theorem. When an event which may, for any thing we can see to the contrary beforehand, happen in either of two different ways, happens one way m times in succession, it ism + 1 to n that it shall happen » times more in the same way, if it happen n times more at all. Thus, suppose a person on the bank of a river, not knowing in what country he is, and not having the smallest reason to know whether the vessels which come up the river carry flags or not: the first ten ships which come up all carry flags (m = 10); thenitis 10 4+ 1 to 3, or 11 to 3 that the next three ships shall carry them, and 10 + 1 to 1, or 11 to 1 that the next ship shall carry a flag. And it is always m + 1 to 1, that an event which has occurred in one out of two possible Ways m times in succession, shall happen the same way on the next occasion.
ON INVERSE PROBABILITIES. 65
The preceding affords some view of the way in which chances are obtained, in cases where the ante- cedent probability of the events stated may be any whatever. The following are conclusions upon the same subject, obtained by a more complicated reasoning of the same kind.
If an event, each repetition of which may be either A or B, have happened m+ 7 times, and if A have occurred m times, and B n times; then it is m+1 to m+1 that the next event shall produce A, and not B. And in the same case the chance that out of p+4q events to come, p shall produce A, and q shall pro- duce B, is (see pages 15 and 16, for explanation of [ ]).
[p+q] x[m+1,m+p] x [nm+]1,n+q] |
[p] x [q] x [m+n+2,m4+n+p+q4+1]
[m+1,m+p] aed [z+1,2+9] [m+n+2,m+n+pt+1} [m+n+2,m+n+q+1] are the chances that in p new events, all shall give A. and that in q new events all shall give B.
Exampire. In a lottery containing an unlimited number of balls, in which the proportion of black and white is absolutely unknown, six drawings give four white and two black; what are the chances that four drawings more shall give all white, or one only black, or two only black, &c.
Let us first take the case of three white and one black : here ot = 4, n= %, 9.4 3, ¢ = 1,
[p+q] = 1.2.3.4, [m+1,m+p] = 5.6.7, In+l,n+g] =3[p] = 1.2.3. [g] =1, [m+n+2,m+n+p+q+1] = 8. 9.10.11. 1.2.3.4. % 5.6.7. x 3. 7 1.2.3. x 1.x 8.9.10.11. 99 That two shall be white and two black (m = 4, n = 2, p = 2, q = 2), the chance is 1 2.3.4, x 5.6.x 3.4. 9 3 1.2. %1.2.%8.9.10.11. 11 That one shall be white and three black (p = J, q = 3), the chance is Pe
and
Chance required is
F
66 ESSAY ON PROBABILITIES.
1.2.5,4%5%8.4.5 9.5 1x 1-2.3x8.9.10.11 33
| That all shall be white, or all black (p = 4,¢ = 4, second and third formule), the chances are
5.6.7.8 , or irs 3.4.5.6 pe 1
§.910.11 33 89.10.11 22
and the verification of the whole is
7 3 em |
a3 *11* 33" 33* a2 —
which must be, since one or other of the cases con- sidered must happen.
When it is known beforehand that either A or B must happen, and out of m + nm times A has hap- pened m times, and B n times, then (page 65.) it is m+1 to n+1 that A will happen the next time. But suppose we have no reason, except what we gather from the observed event, to know that A or B must happen ; that is, suppose C or D, or E, &c. might have happened: then the next event may be either A or B, or a new species, of which it can be found that the respective probabilities are proportional to m + 1, m+1,and 1; so that though the odds remain m+1 ton +1 for A rather than B, yet it is now m+ 1 to n+ 2 for A against either B or the new event. Thus, suppose a game at which one party or the other must win, and suppose that out of 20 games A has won 13 and B 7: and this is all we know of the game or of the players, Then, it is 134+1 to 7+1, or 14 to 8, or 7 to 4, that A shall win the 21st game. But suppose that it is possible to have a drawn game; then there is some chance that the 2lst may be a drawn game, though but a small one, as might be inferred from such a thing never happening in 20 trials. The 21st game may be either A’s or B’s, or drawn: of which the chances are as 1341, 7+], and 1; or as 14, 8, and 1. Consequently, though in the preceding case it was 14 out of 22 in favour of A’s
ON INVERSE PROBABILITIES. G7
winning, it is now 14 out of 23, and 1 chance out of 23 remains for the next game being drawn.
When a number of different events have happened, A, B, C, &c., write down each number increased by 1, and the results will express the several relative proba- bilities, on the supposition that no events can happen except those which have happened. But if new events may happen, write down 1 for the relative probability of such an occurrence at the next trial. Thus, if out of a box, and in 100 drawings, there have appeared 49 white balls, 377 red, and 14 black ; then if it be known that nothing but white, red, or black can appear, con- sider the chances of these to be as 50, 38, and 15; that is, °°; is the chance of drawing a white ball at the 101st trial. But if another sort of ball may appear, then the chances of the four cases being as 50, 38, 15, and 1, it follows that =°°. is the chance of a ie ball at the 101st trial.
In judging of future events by those which have passed, we must be extremely cautious always to pre- serve the same method of considering the event pro- posed. If, for instance, in 100 trials, A has appeared 49 times, B 37 times, and C 14 times, we know that there is one chance out of 104 that the 101st drawing shall give neither A, B, nor C, but something else. What the new character may possibly be is left un- known ; it may be another letter, or it may be a num- ber, 2 picture, or a blank. Are we to understand that all these are equally probable? Common sense tells us the contrary ; experience makes us feel it much more likely that the letter D should appear at the 101st trial, than any stated number or picture. But we have now changed the question, and, dropping the distinction between A, B, and C, have considered them merely as letters. Having drawn a letter 100 times running, we are to infer (page 64.) that it is 101 to 1 in favour of our drawing a letter at the 101st trial; or that 1° is the chance of this. But it is already 103 to 1 that the next drawing shall be, not merely a letter, but one
F 2
68 ESSAY ON PROBABILITIES.
of the letters A, B, and C: that is, to all appearance, we have this strange result ; — the chances of drawing one of the three, A, B or C, are greater than those of drawing one of the set, A or B, or C or D, &c. &c. up to Z. This paradox will afford me a good opportunity of again inculcating the maxim, that the probability of an event is the presumption drawn from certain obvious principles, as to what the state of our minds ought to be with regard to belief in the happening of that event, as influenced by our knowledge of previous events. Consequently, if John know that A, B, and C have been drawn 49, 37, and 14 times, and nothing more, he has reasonable ground, with his knowledge, for assenting to the proposition that the 101st trial shall give either A, B, or C, as to a proposition which has 1°93 of probability, or 103 to 1 in its favour. But if Thomas only know that 100 draw- ings have all given letters, then he, with his know- ledge, has no ground of inference with respect to A, B, and C, in particular, but may reasonably assent to the proposition, “ the 101st drawing will also give a letter,” as having a probability of 4°1, or 101 to 1 in its favour. But the paradox in question requires that John should make believe he knows no more than Thomas, and then be surprised that a discordance should arise from his using that knowledge in recon- sidering a result, which he has suppressed in the me- thod of attaining it.
It very rarely happens that we meet with a case in which we can so distinctly specify the antecedent cir- cumstances which infiuence our assent or dissent, as to enable us to apply direct calculation to the determin- ation of the rational probability of an event to come. And in most of the practicable cases, the largeness of the numbers employed would check our progress, if it were not for the approximative methods of the higher mathematics, and the tables at the end of this work, I now proceed to the method of using those tables.
USE OF TABLES. 69
CHAPTER IV.
USE OF THE TABLES AT THE END OF THIS WORK.
I nave endeavoured to accumulate in this chapter a considerable part of the uninteresting details of com- putation which accompany the solution of complicated problems. It is at the readers’ pleasure to omit the whole of it, referring to it afterwards in cases where its assistance may be necessary.
In Table I. we see — (I.) a column headed t, con- taining the series -00, °O1 ..... "O09, TU casaw oe every hundredth of a unit from 0 to 2 —(II.) a column headed H, deduced in the manner pointed out in page 17,—(iII.) columns headed A and A®%, which are only the differences of the numbers in column H (marked- A), and the differences of those differences, (marked A’). The following is an extract from the table: —
t H A A2 ‘47 | 49374 52} 900 46] 8 61 48 | 50274 98 | 891 85 | we. P “£9 “SESGGi SSF ccc covccs F veces ‘
The columns A and A? must be made up to 7 places of decimals * by means of ciphers: thus, 90046 means -0090046 ; and 861 means -0000861. The formation of A and A is then as follows: —
-5027498 “5116683 0090046 4937452 5027498 0089185 Subt. 0090046 0089185 0000861
* A little practice will show how to dispense with the decimal points altogether till the end of the process.
ge 3
70 ESSAY ON PROBABILITIES.
Since it is very seldom necessary to use more than five places of the table marked t, the sixth and seventh places, and those which arise from them in the differences, are separated from the rest by a blank space. The sixth and seventh places are allowed to remain, on account of the use which will hereafter be made of the differences derived from them.
In Table IJ. we find to five places only a column marked K, and another marked A, containing the dif- ferences of the former column. This is a modification of the former table, the reason of which will hereafter appear. In the meanwhile, however, observe that we can directly find the value of H and K_ by these tables only, when t is ‘00 or ‘01, &c.; that is, when t is a given decimal of two places. But supposing it required to find H when t lies between two of the values in the table ; suppose, for instance, we ask what is H when
= ‘47694? The method is as follows:
Question. What is the value of H (Table I.) when t = °4'7694, correct to five places of decimals ?
Rute. / EXeEMr.iricaTION. Take out of Table I. the Opposite to +47 we find value of H answering to the 49375 *
two first decimal places and the whole number preceding them, if there be one. Retain only five places of decimals. Take the figures of the first | Three figures remaining, 694,
difference (as far as the blank 900 x 694=62460U. space), and multiply them by Cut away three = Aeuree, the remaining figures in the , (685.2
value of t, and cut away as many places from the result as there were remaining figures.
Add the figures in the last 49375 result to the right hand of the 625 first, and the sum is the an- swer required, - 50000
When t= °47694, H = °50000.
* Whenever decimal figures are rejected, if the first rejected be five or upwards, the last retained is increased by a ‘unit.
USE OF TABLES. ye |
As another example, suppose the value of t to be
1°51209.
When t=1°51, H=.96728 24
When t=1°51209, H=*96752
A=114
209
1026 228
23,826
We shall now take the inverse problem, and sup- posing H to be intermediate between two values in the
table, require the value of t. *93972. Rute. Find in the table the value
of H next below the given | n
value ; note the corresponding value of t, and subtract the nearest value of H from the given value.
Annex three ciphers to the difference just found, and di- vide by the figures of the dif- ference in the table which come before the blank space, rejecting fractions, and taking the nearest whole number.
The quotient cannot have more than three places: if three, annex to the value of t already found; if less than three, place ciphers at the be- ginning to make up the defi- ciency, and annex.
For instance, let H =
EX£EMPLIFICATION. H *93972,
ext below athe aie} 93807t= 1°32
165 Tabular diff. 195.
195)165000(846 1560
900 780 1200 1170
30
t=1°32 846
72 USE OF TABLES AT END OF WORK.
Let H = *97169; required the value of t?
H= *97169 Nearest below =°97162 ceccocseesee t= 1°55
101 )7000(69 t=1°55069 Answer. 606
940 909
ST arama
31 +
The Table II. is used in exactly the same way, except towards the end, from t = 3°40 upwards; in which case the cipher at the end of t must be neglected, and only one decimal place taken out of the table in the value of t. For instance, to find the value of t answering to K= "98222,
K = *98222 Next below °98176 evccceccoses t=3°5
806 )46000( 150 t==-3°5150 306 1540 1530
100
In the first table, there is another result which will frequently be wanted, and which I shall call H’% It arises from adding half the second difference to the first difference *, if the value of t be in the tables, and making five decimal places. But if the value of t be not in the tables, then H’ must be formed for the values of t immediately above and below ; and by means — of the first and the difference of the two, H’ must be found in exactly the same manner as H is found in the first of the preceding rules, page 70., remembering to subtract at the last step instead of adding, if the second H’ thus previously determined be less than the first.
* Meaning the whole differences ; not the parts which precede the blank space, as in the preceding rules. |
—~I Q9
USE OF TAELES.
Examete 1. When t is 1°56, what.is H’?
A= 9745 t= 151 H’ = :09896 : Exampie 2... When t — 1:23412, what is H’? t= 1°23 H’ = :24852 : t=1:24 H’=-24245 Diff. 607 412 "24852 607 250 2884 Subt. -24602=H’ when t=1°23412 2472 250,084
_ We have already seen that when two events, A and B, one of which must happen at every trial, have severally happened m times and n times in m+ n trials, it is m+ 1 to n+ 1 that A shall happen at the next trial. But m + 1 to n + 1 is very nearly m to n, when m and n are considerable num- bers: for instance, 248 to 117 is very nearly 247 to 116. That is, when a great many trials have been made, the numbers of times which A and B have hap- pened express very nearly the odds (relative proba- bilities) for A against B; or, inverted, for B aguinst A. Let us convert the problem, and supposing that we know beforehand the chances of A and B, are we to suppose that in agreat many trials A and B will happen in proportion to their respective probabilities P Common sense tells us that such will always be nearly the case, but that the odds are great against an exact result, Suppose 3000 drawings to be made from a lottery con- taining two As and one B. We must then, it seems clear, draw A twice as often as B, in the long run. Our reason convinces us thus. Let one of the As be distinguished from the other by an accent, so that we have A, A’, and B. If the urn be well shaken before each drawing, it is impossible to believe that, in the
ESSAY ON PROBABILITIES.
74
whole result of 3000 trials, we shall have drawn the three in very unequal numbers ; so that, destroying the distinction between A and A’, we feel secure of drawing A twice as often as B; and it is obviously two to one
in favour of A at each trial.
The following phrases
- seem to common sense to mean the same thing.
It is two to one that A shall happen, and not B.
It is an even chance for head or tail.
It is more than a hundred to one, that a ship at sea will not be lost.
In the long run, A will happen twice as often as B.
In a large number of tosses, the heads and tails will occur in nearly equal numbers,
Of all the ships which sail, the number which is not lost exceeds that which is lost more
than a hundred times.
I now proceed to some problems, which will exhibit the method of applying the tables, and will illustrate and confirm the preceding notions.
Prosiem I. The odds for A against B being a to 3, to find the chance that in n times a + 5 trials, A shall happen exactly n xa times, and B n x b times.
Rue. Divide the H’ belonging to ¢ = 0 (page 72) by the square root of the following: 8 times the product of n, a, and b, divided by a + b.
Suppose, for instance, a die is thrown 6000 times ; what is the chance that exactly 1000 of the throws shall give an ace? Here it is 1 to 5 that an ace shall be thrown in any one trial, and 6000 is 1000 times 1+ 5. Hence a = 1,b = 5, n = 1000: 8 times the product of n, a, and b is 40,000, the sixth part of which is 6667 (sufficiently near), and the square root of this is 81°65. Again, when ¢ = 0, we have in Table I.
A=112833, | A2=11; whence H’ is 1*12844
and 112814 divided by 81:65 gives ‘014 very nearly, This is very near the real probability that 6000 throws with a die shall give exactly 1000 aces: for such an event there are only 14 chances out of a thousand ; and
USE OF TABLES. 75
it is 1000 — 14 to 14, or about 704 to 1, against the event. This result is rather above what we should have expected ; we might have imagined it to be more than 71 to 1 against 6000 throws giving exactly 1000 aces.
As another example, let us find the probability that, out of 200 tosses with a halfpenny, there shall be ex- actly 100 heads and 100 tails Herea=1,b=1, nm = 100, and 8nab* is 800, which divided by a + 3, or 2, gives 400, the square root of which is20. And H’, when ¢=0 (or 1'12844) divided by 20, gives :056. It is therefore about 944 to 56, or 17 to 1, against the proposed event; and (page 42.) we must repeat 200 throws 12 times to have an even chance of the equality of heads and tails happening once.
Generally speaking, the rules in this chapter are very accurate only when the number of trials is considerable. Suppose only 12 tosses; required the chances of 6 heads and. 6 tails. Here. a = 1, b= 1, n = 6; 8nab = 48, which divided by 2 gives 24, whose square root is 4°9 very nearly. And 1°12844 divided by 4°9 gives °23, or 77 to 23, that is 3.8, to 1 against the event. That is (page 47), this rule is not very inaccu- rate, even when the number of trials is as low as 12.
We shall call the event whose chance is sought in the preceding problem, the probable mean; understanding by that term the event which is more likely to happen than any other. Thus, when 12 halfpence are thrown up, 6 heads and 6 tails is the probable mean, being the event which is more likely than any other, though not in itself more likely than not. When 6000 throws are made with a die, the probable mean is 1000 aces, 1000 deuces, &c.
ProsiEM II. The odds for A against B being a to }, required the chance that in n times a + 0 trials, the As shall fall short of the probable mean by a given
* Juxtaposition of numbers, in algebra, stands for their product.
"6 ESSAY ON PROBABILITIES.
number /: J being small, compared with the whole number of As in the probable mean.
Rute. Divide twice / by the square root obtained in the last example ; and find the value of H’ from Table I., taking the preceding quotient for t. Divide H’, so found by the square root just used, and the quotient is the answer required.
N.B. This rule also applies when the number of As
is to exceed the probable mean by 1. _ Examrte I. In 6000 throws with a die, what is the chance that the aces shall fall short of (or exceed) 1000 by exactly 50°? Herea = 1, 6b = 5, n= 1000, and the square root is 81°65, as before. And twice /, or 100, divided by 81°65, gives 1°22: to which the value of H’ is 25162 + 4 of 611, with five decimal places, or ‘25468. This last, divided by 81°65, gives "0031 ; so that it is about 997 to 3, or 332 to 1, against the proposed event: and the 6000 throws must be re- peated 232 times to give an even chance of succeeding once.
ExamptE II. What is the chance that in 200 tosses, there shall be exactly 95 heads? Herea=1,b)= 1, nm = 100, / = 5, and the square root, as before, is 20. And twice J, or 10, divided by 20, gives °50, which being t, the value of H’ is ‘87882, which divided by 20 gives ‘044 very nearly. It is, then, 956 to 44, or about 22 to 1, against the proposed event.
Exampxe III. In 12 tosses, what is the chance of exactly 7 heads? Here a=1, b=1, n=6,/=1, the square root, as before, is 4°9, and 2 divided by 49 is ‘41 nearly ; which being t, H!' is 95384, which divided by 4°9 gives *194. It is therefore 806 to 194, or 4.°, to 1, against the proposed event. In page 47. it is 3304 to 792 against this event, or 4} nearly. Hence the incorrectness of our rule is very small.
Prosiem III. The odds for A against B being a to b, required the chance that in n times a + b throws, the number of As shall not differ from the probable mean by more than /, |
USE OF TABLES. TZ
Rute. Divide one more than twice / by the square root already mentioned, and the quotient being made t, the value of H in Table I. is the probability re- quired.
Exampte I. In 6000 throws with a die, what is the chance that the number of aces shall not differ from 1000 by more than 50; that is, shall lie between 950 and 1050, both inclusive. Here a=1, b= 5, nm =1000, /= 50, and the square root as before is 81°65. Divide 27+ 1, or 101, by 81°65, which gives 1:237, which being t, H is 91977. Hence it is 920 to 80 in favour of the proposed event, or about 114 to 1.
Exampte II. In 200 tosses, what is the chance that the number of heads shall lie between 97 and 103, both incisive? Here a=: 1, b= 1,7 — 100, }= 3: age the square root, as before, is 20. And 2/-+ 1, or 7, divided by 20, gives *35, which being t, H is °379. Hence it is 621 to 379, or about 31 to 19, against the proposed event.
Exampte III. In 12 tosses, what is the chance of the heads being either 5, 6, or 7 in number? Here a=1, b=1, n= 6, i=1, and the square root, as before, is 4°9. And 2/ + 1, or 3, divided by 4°9, gives 61, which being t, H is 6117. Hence it is about 612 to 388, or 153 to 97, in favour of the proposed event, In page 47 the chance of this event is
792 +92944792 : 2508 T 4096 4096
or *612
ProstemIV. The odds for A against B being a to, and n times a + 6 trials being to be made, for what number is there a given probability H that the As shall not differ from the probable mean by more than that number ?
Rue. Find in Table I. the value of t answering to that of H (page 71), multiply it by the square root alréady described, subtract 1, and divide by 2: the
78 ESSAY ON PROBABILITIES.
quotient, or its nearest whole number, is the answer required.
Exampite. In 6000 throws with a die, within what limits is it two to one that the aces shall be con- tained? The square root is 81:65, and H is 2 or ‘66667, to which the value of t is ‘68409, found as follows (page 71) : —
H = *66667 Nearest below ‘66378 t=°68
Tab. Diff. 706) 289000(409 t= °68409 say °6841 2824 81°65
6600 34205 6354 41046 6841 246 54728
55°856765 ] e
2)54+86
27°43 =1.
Answer. It is a little more than 2 to 1, that the aces shall lie between 1000 — 28 and 1000 + 28, and a little less than 2 to 1 that they shall lie between 1000 — 27 and 1000 + 27.
But the most convenient way of solving this problem is by first finding for what degree of departure from the probable mean there is an even chance. In this case, since H =°5 (page’70), t is = °4'76936, which the method in page 41, will show to be very nearly 24. It will be worth while to re-state the whole process.
The odds for A against B being a to b, and the pro- posed number of trials being n times a + 6, required the limits of departure from the probable mean na, within which it is an even chance that the number of As shall be contained.
Ruiz. Multiply together 8,n,a, and b, and divide by
USE OF TABLES. 79
a + b: extraet the square root of the quotient, and multiply it by 31: subtract 65, and divide by 130: the nearest whole number is the answer required. Thus in the preceding instance, where the square root is 81°65, multiply this by 31, which gives 2531-15 ; and 65 less is 2466°15, which divided by 130 gives 18°97. Hence it is very little less than an even chance that the aces in 6000 throws shall be between 1000 + 19 and 1000 — 19, or 1019 and 981.
Having found the limits of departure for which there is an even chance, we can now use Table II. as follows. The values of t in Table II. are the proportions of various departures (each increased by °5) to that depar- ture which has an even chance, as just ascertained, and also increased by 5: the values of K are the proba- bilities of the departures answering to those of t. Having then ascertained 18°97 to be the departure for which there is an even chance, suppose I ask what is that limit
of departure within which it is two to one that the aces shall be contained. Two to one gives 2 for the chance, or ‘66667 : I look into Table II., and find that when K is 66667, t is 1°43433, found as follows: — K= ‘66667 t=1°43 Next below 66521
Tab. Diff. 337) 146000(433 t=1°43433 1348
1120 1011
1090 1011 79
This is the proportion which the departure in ques- tion, increased by °5, bears to 18°97 increased by °5 or 19°47. Multiply 1°43433 by 19°47, giving 27:93 ; from which subtract °5, giving 27°43 for the limit of departure, the same as in page 78.
80 ESSAY ON PROBABILITIES.
Suppose the question to be that of page 77, namely, what is the probability that the number of aces in 6000 throws shall lie within 50, one way or the other, of the probable mean 1000? Now, 18°97 + °5 is 19°47, and 50 + °5 is 50°5, and 50°5 divided by 19°47 gives 2°594, which being t (in Table II.), K is -91981, ex- tremely near to the result in page 77.
There are, therefore, two distinct methods of treating these problems, connected with the two tables: and this is a great advantage, since it is a very strong presump- tion of a correct answer, when the results of the tables agree. The problems III. and IV. being of great import- ance, I shall now recapitulate their details, with the ad- dition of some new phraseology. Let the instance be 6000 throws of a die, and the event A the arrival of an ace. and B the arrival of some other face. The most probable number of aces is 1000, though the arrival of that exact number is not probable in the common sense of the word. There will then most likely be a de- parture from the number 1000 in the number of aces thrown ; of which departure we are now entitled to say, that it is very improbable it should be considerable. Let the term neutral departure mean that degree of de- parture for which it is just an even chance that the actual event shall be contained within its limits: in the present instance it is 18°07. We may explain the fraction as follows: suppose a person to receive 1001. for every unit by which the number of aces falls short of or exceeds 1000. Then, supposing him to try this stake a great many times, he will in the long run re- ceive less than 1897/. at a trial, as often as he receives more. But his receipts will oftener exceed than fall short of 18002. ; while they will oftener fall short of than exceed 1900/. Roughly speaking, there is here the same probability that the aces shall not lie between 1000—19 and 1000+ 19 (both inclusive), and that they shall lie between these numbers.
In all these problems there is a square root to be found, which we call the square root, as there is no other
8
The odds are a to b, for A against B, and n (a + 5) trials are contemplated. Though we have only instanced whole values of n, yet it may be a fraction: thus, if the odds are 3 to 2, and 96 trials are contemplated, n (3 + 2) must be 96, or n must be 191. In this case, the pro- bable mean is that A shall happen 573, and B 382; by which it must be understood, that a person who should repeat 96 throws a great many times, receiving 1/. for every A, would, in the long run, gain on the average 573. per trial of 96 throws.
The square root in question, represented algebrai-
cally, is
USE OF TABLES.
eee
8nab a+b
or the square root of the product of 8, n, a, and 6 di- vided by a+ 8. I now subjoin the two principal pro- -blems, with the two rules in parallel columns. Prosuem. What is the chance~that the number of As in n(a+6) trials shall lie between na +/ and na—Jl, both inclusive? or what is the chance that the departure from the probable mean shall not exceed / ?
By Taste I. By Taste II.
Find the square root, and divide one more than twice / by it; call the result t, and find H answering to t in the table. (Use the rule in p, 70. if necessary.) This H is the probability required.
Prosiem. What is that
Find the square root, and multiply it by 31; then divide by 180. To 7 add ‘5, and divide by the preceding quotient; call the result t, and find the value of K answering to t: this is the probability required.
N. B. The neutral departure is *5 less than the quotient first found.
degree of departure within
which it is p to qg that the number of As in (a + 6)
trials shall lie ?
G
ESSAY ON
By Taste I.
Divide p by p +4, and call- ing the result H, find the cor- responding value of t in the table. Multiply it by the Square root, ‘subtract 1 and divide by 2; the quotient being called /, it is then p to q that the As in 2 (a+b) trials shall be contained between na—l and na+l, both inelu- sive.
PROBABILITIES.
By Tazsvez Il.
Find the square root ; mul- tiply by 31, and divide by 130. Divide p by p+q, and calling the quotient K, find the corresponding value of t in the table; multiply t by the preceding quotient, subtract °5 from the product, and J being the remainder, it is then p to q that the Asin n (a+b) trials shall be contained between na —l and na + 1, both inclusive.
I shall conclude this problem with an example of each
ease, worked by both methods, without explanation. There is a lottery containing 3 white and 2 black balls : what is the chance that in 50,000 drawings the num- ber of white balls shall be between 30,000 + 100 and 30,000- 100°
a =3 b=2, n=10,000, I= 100 809'84 8 x 3x 2x 10,000 a 5 = 96,000 | 130)9605-04(73°885 56,000 = 309-84 73°885)100°5(1°3602=t 309-84) 201(-64873 =t K = 64109 H = 64109
This question shows how nearly a great many trials may be expected to agree with the probable mean: in 50,000 trials, it is nearly two to one against the number of white balls differing from 30,000 by more than a hundred.
In 100,000 tosses, between what limits is it 99 to 1 that the heads shall be contained ?
a@=1, b=1, n=50,000, p=99, g=1 100)99(:$9=H t= 1°8215
447 °21 31
130) 13863°51(106°64 8x1lxIx 50,000 100)99(:99 = K
2
= 200,000
USE OF TABLES. 83
200,000 = 447°21 t=3'821 1°8215 sms 447-21 ihahen 814°6 407 *47
1 5 2)813°6 406°97 =z 406 8 =l
Answer. Between 50,000 — 407 and 50,000 + 407.
Now for the inverse method attached to the preceding problem. If I. be totally unacquainted with the na- ture of the events A and B, except only that one or other, and not both, must happen every time, it is then clear that, as the matter stands, it is to me 1 to 1 forA against B, with avery great chance that, if I were better informed, I should form a different opinion. At the same time (page 10), I choose 1 to 1 as my rule of action, because, though coming events may not justify my pre= diction, I know of nothing to warrant my assuming that the odds are in favour of A, rather than in favour of B. A trial takes place, and A happens; it becomes im- mediately most safe to assume that the odds for A against B are 2 to 1, but still that safety is not very decided. But if 1000 trials be made, and if A have happened 520 and B 480 times, I can then confidently say, that the odds for A against B are very nearly, if not exactly 520 + 1 to 480+ 1, which is nearly 520 to 480. The notion then formed has a strong presumption that it is nearly correct.
Prosriem. In a+ 6 trials A has happened a times and B 6 times: from which, if @ and 6 be considerable numbers, it is safe to infer that it is a to 6 nearly for A against B. What is the presumption that the odds for A against B really lie between a—k to b + k and at+ktob—k?
G2
84
Rute. (Taste I.)
Divide twice the product of a and b by their sum, and ex- tract the square root of the quotient, by which divide &. Then the last quotient being t, the H of the table is the
ESSAY ON PROBABILITIES.
Rute. (Tasrz II.)
Having found the square root, and divided & by it, as opposite, from seven times the quotient, take the hundredth part of the quotient, and take three tenths of the remainder.
Make the result t, and K in the table is the probability re- quired,
Suppose that in a thousand trials, A has happened exactly 600 times, and B 400 times ; what is the pre- sumption that the odds for A against B lie between 570 to 430 and 630 to 370? |
a=600, b= 400, k=
probability required.
2 x 600 x 400 = 480,000 1°369 ” 480,000 _ woo 9°583 014 / 480 =21°91 9569 ai} S6O ct : 21°91 28°707 t= 2: 871
Answer. About 95 to 5, or 19 to 1 in favour of the odds being between the limits specified.
In the preceding problem, A and B have happened a and 6 times; whence the most likely of all individual cases is, that the odds for A against B area tod; or, in other words, the result which has the strongest presumption in its favour is, that
Probability of A was : a+b.
Probability of B wrasse a+b
Now we have found, in the preceding problem, the presumption that
Probability of A lies between prea and —— ees a+b a+b
USE OF TABLES; 85
or, which is the same thing, that
b+k Probability of B lies between x a
For since it is our hypothesis, that either A or B must happen at every trial, whatever presumption there is that the chance of A is x, there is the same presumption that the chance of Bis l1—a&
But we might ask the following questions: A and B having happened a and 6 times in a + 6 trials, what are the values of the following presumptions P
k 1. That the probability of A lies between a4 and ey a+b a+b or its equivalent, that the probability of B lies between b—hk b octane a+b a+b. a k 2. That the probability of A lies between —— and —— oS a+b a+b . : b b—k or that the probability of B lies between and —— - a+b a+b
To solve these by the help of the following rule, remember that, if @ be greater than b, it is more likely that the chance of A falls short of a+(a+b) than ex- ceeds it: and if @ be less than 0b, then it is more likely that the chance of A exceeds a+(a+5) than falls short of it.
Rute. First find the result of the preceding pro- blem, and find from Table I. the H’ (p. 72) belong- ing to the value of ¢t. Subtract this from the H” derived from 0 in the table (which is 1°12844) ; mul- tiply by the difference between b and a, and divide by the product of the square root used in the preceding problem and three times the whole number of trials: call the result V. To one half of the result of the preceding problem add V ; and from it subtract V: and eall these results 3H + V and SH—V.
Then, if B have happened most times, }H + V is the
a 3
86 ESSAY ON PROBABILITIES. a presumption that the chance of A lies between ay and a+
k — or that the odds for A against B lie between ato b
and a+k to b—k. But in this case, 1LH—V is the Prsanep Bion that the chance of A lies between a—k
aoe and ae , or that the odds for A against B lie between a—k to b+ and a to b.
But if A have happened most times, make H+ V and 4H — V change places in the preceding paragraph, every thing else remaining the same.
Exampie. In a thousand trials, A has happened 600 times, and B 400 times. What is the pre- sumption, 1. that the odds for A against B lie between 600 to 400 and 630 to 370; 2. that the odds for A against B lie between 570 to 430 and 600 to 400?
From the preceding problem t=1°369, say=1°37 ; the square root is 21°91, and H is -9471.
t=1°37 A 17036 1°12844 b—a=200 $A2 239 17268 3 3000 x 21°91 = 65730 H’ = +17268 *95576 200 65780)191°152(-0029 = V 4736 =4$H 4765 =$H+V 4707 =4H-—V
Hence (since A has happened most times), it is *4707 to °5293, or about 47 to 53, that the odds for A against B lie between 600 to 400 and 630 to 370. And it is "4765 to °5235, or about 48 to 52, that the odds for A against B lie between 570 to 430 and 600 to 400.
The problems which the preceding part of this chapter has enabled us to solve, are the determination of the chance which exists (under known circumstances} for the happening of an event a number of times which
USE OF TABLES. 87
lies between certain limits, and its converse. The latter problem contains a consideration of some diffi- culty, namely, the probability of a probability, or, as we have called it, the presumption of a probability. To make this idea more clear, remember that any state of probability may be immediately made the expression of the result of a set of circumstances, which being in- troduced into the question, the difficulty disappears. Thus, suppose a large number of urns containing various proportions of black and white balls. Let there be 100 urns, and let one of them only contain equal numbers of black and white balls. If then I lay my hand upon one of these urns with the intention of drawing, it is, before the drawing, 99 to 1 against my having placed my hand upon an urn from which, in the long run, equal numbers of both sorts of balls will be produced : the presumption black and white balls have an even chance is only -},,; the pre- ene that the probability of a white ball is 4, is sy speaking of compound probabilities, writers have employed six words synonymously—probability, chance, presumption, possibility, facility, and expectation. Re- jecting only the word possibility, as indicating a thing of which there cannot be different degrees, the five remaining terms have their advantages, each one pointing out a peculiar and useful view of the main idea. Thus the word presumption refers distinctly to an act of the mind, or a state of the mind, while in the word probability we feel disposed rather to think of the external arrangements on the knowledge of which the strength of our presumption ought to depend, than of the presumption itself. When, therefore, having observed an event, we want to know how strongly we are to suppose that the observed event was preceded -by a given arrangement of circumstances, the term pre- sumption of probability is very appropriate. The word facility applies particularly to the notion which we form when we see one event happen more often than a 4
88 ESSAY ON PROBABILITIES.
another, namely, that it is easier to produce the first than the second. In our problems, however, the facility is not that arising from art, but from previous (it may be accidental) distribution of means. The word expectation will be applied throughout this work to that state of things for the production of which there is an even chance. If (p.79), 6000 throws be made with a die, it is an even chance that the number of aces lies between 981 and 1019: the odds are against any smaller amount of departure on both sides of the probable mean, and against any greater amount ; this is then our expectation of the number of aces.
When one of two possible events happens oftener than the other, it being understood that one, and only one, can happen each time, we are led to suppose that the excess of one event is the consequence of some arrange. ment which would, had we known it, have made us count that event more probable than the other. If A or B must happen, and if in a thousand trials the As outnumber the Bs very much, we feel perfectly cer- tain that such must have been the case. The theory of probabilities confirms this impression, as will appear by the solution of the following
Propitem. In a+b trials, the number of As was a, and that of Bs was 6. If a exceed b considerably *, required the presumption that there was at the outset a greater probability of drawing A than of drawing B, in any one single trial P ;
Ruue. Divide the difference of a and 6b by the square root of twice their sum, and let the result be t.. Find (page 72) the H! corresponding to t: multiply the result by the sum of a and b, and divide by the product of 8, ¢, and the square root of the product of aandb. The result subtracted from unity gives the answer required. Suppose, for instance, that out of 50 trials A occurs 32 times, and B18 times. Then,
* In order that the result may be very correct, a must exceed 6 so much
. that the excess of a above b, multiplied by itself, may considerably exceed the sum of a and 4,
USE OF TABLES. 89
50x2=100, Y100=10 32-18 10 H’=:15891 15891 x 50=7:9455 V 32 x 18=24, 8x 24% 1°40=268°8 7°9 divided by 268°8 is 58; nearly: 1— 8, is 384
=140=
Hence it is about 261 to 8 that A was more probable than B. AppiTionau Rute. When a and? arenearly equal, find t, as in the last rule; find H (not H’) corresponds ing to t, add 1, and divide by 2: the result is the pro~ bability required. ,
The additional rule belongs to the more important case of the two, namely, that in which A has not hap- pened so much oftener than B as to justify an imme- diate conclusion that it was the more probable event of the two. Suppose, for instance, that A has occurred 10,100 times out of 20,000 trials, and B 9,900 times: then t = 200 divided by 200, or 1; to which H is *843, and this increased by 1, and the result divided by 2, gives ‘922. It is, therefore, about 114 to 1 that A was the more probable.
The preceding solution can be applied to various species of observations ; of which we shall see more hereafter. The following may be considered as closely connected with it. If we make two different sets of trials, in circumstances which we suppose to be the same, it will generally happen that the As will not bear the same proportion to the Bs in both sets. If, for instance, we find 1000 As arrive in 2000 trials, the odds are very much against the arrival of exactly 5000 As in anew set of 10,000 trials, though the expectation is that something near that number of As will arrive. Suppose'that the first and second sets of trials give—I1st, 50 As, 30 Bs; 2nd, 112 As, 61 Bs.
In the second set the As bear a larger proportion to the whole than in the first: and our present question is what presumption thence arises that there is some dif- ference of circumstances between the two sets, which
e10) ESSAY ON PROBABILITIES.
gives A a greater facility in the second than in the first, or a greater probability of being drawn at any one trial ? Or, if in a first set of a + 6 trials, A happen a times and B happen 6 times; andif in a second set of a’ + b! trials, A happen a times, and B happen Db’ times ; and if a' be a larger proportion of a’ + 6’ than a is of a + b; required the presumption that there was a greater chance of drawing A at a single trial in the second set than in the first P
Rue. Divide the cube of the sum of a and b by twice their product: do the same with a’ and 6’: mul- tiply the two results together, and add them together : divide the product by the sum, and extract the square root of the quotient.
Divide a’ by a’ + 0’, anda by a + b, and subtract the less result from the greater. Multiply the difference by the square root previously found, and let the product be @ Then the H corresponding to ¢, increased by 1, and divided by 2, is the presumption required.
In the example a = 50, b = 30, a’ = 112, b’ = 61.
80 x 80 x 80= 512000 2x 50x 30=3000, 212000 ~ y794 3000
173 x 173X173 = 5177717 2x 112 x 61 = 13664 an = 3789 170°7 x 878°9 = 64678'23, 170°7 + 378°9 = 549°6 64678'23 = 117°7, 4/1177 = 10°85 5496 112 — 50 — -6474\— 6250 = 0294 "0224 x 10°85 = *24904 = t, H = “2689
4(H + 1) = ‘635, the probability required ; and it is therefore about 16 to 9 in favour of the excess of As at the second set of trials not being accidental fluctua- tion, but arising from some new circumstance or dif- ‘ferent arrangement of the old ones.
If, in 1000 trials, A should happen 520 times, and B 480 times, there is strong presumption that in any fu- ture number of trials the whole number will be divided among As and Bs nearly in the proportion of 520 to
USE OF TABLES, G1
480. But this is not the same set of circumstances as that of the problem in page 77. We are there sup- posed to know exactly in what proportion As and Bs are contained in an urn; and with this positive knowledge we can ascertain the probability of drawing any given number of As in a given number of trials, In the present instance we do not know the contents of the urn, but only the result of a certain number of drawings, from which we can draw presumptions, as in page 53. about the whole contents. The determination of chances relative to a new set of trials depends upon two risks in the latter case, and upon one only in the former. The latter problem is therefore more complicated in its prin- ciples though not so in its results.
Let us suppose two different persons, John and Thomas, thus situated with respect to the contents of an urn; John knows that there are as many As as Bs; Thomas has observed a hundred successive drawings, of which (so let it have happened) fifty have given A, and as many have given B. That which John knows is rendered not improbable to Thomas by the result of the trials, while the same result would have been thought not unlikely beforehand by John. But there is this difference between their degrees of knowledge, that John has the certainty of a fact (the equality of As and Bs), of which Thomas can only say that the fact, or some- thing near it, is extremely probable. No one could argue with John against any particular venture in such a lottery upon the ground of the possibility of the As much exceeding the Bs ; while with Thomas it might be urged as possible, though not probable, that the former might exceed the iatter a hundred-fold. Again, suppose John and Thomas, having equal fortunes, are disposed to venture as far as produce woulda warrant, upon the results of a hundred (to Thomas a second hundred) trials. It is obvious to common sense that Thomas must not venture so much as John; for he runs a larger risk, seeing that he assumes as an average result what possibly may have been a rare occurrence.
92 ESSAY ON PROBABILITIES.
The following is the rule pointed out by the theory of probabilities: —The expectation of fluctuation should be greater to a person who proposes to try g new in- stances, upon the assumption that p preceding instances have fairly represented the Jong run, than it should be to another person, who knows in what proportions the As and Bs really exist ; and greater in the proportion of the square root of p augmented by gq to the square root of p. Thus, if in the preceding case, John and Thomas propose to embark in a matter which depends on 300 more trials,the proportion of the squareroot of 100 + 300 to that of 100, being that of 2 to 1, it follows that, whatever reason John may have to guard against the possibility of 300 drawings giving w more than 150 As, Thomas has as much reason to guard against 2~ more than the same number.
Prospiem (to be compared with that in page 77.). When a + 6 trials have happened to give a As and 6 Bs, required the chance that in m times a + b new throws, the number of As shall not differ from na by more than J.
Ruue. Divide one more than twice 7 by a square root to be immediately mentioned, and the quotient being made ¢, the value of H in Table I. is the probability re- quired.
The square root in the; The square root in the former problem was that of the| present problem, is that of
product of 8, n, a, and b|the product of 8,2, n+1, a, divided by a+. and 6 divided by a+ 6.
The additional rule in page 81. may also be applied verbatim, the square root now meaning the second square root above given ; and the inverse rule (p. 82) may be applied in exactly the same way.
Exampxe. In 600 drawings A occurred 100 times, and B 500 times ; what presumption thence arises that in 6000 more drawings A would occur somewhere be- tween 1000 — 50, and 1000 + 50, or 950 and 1050 inclusive? (See page 77 for the corresponding pro. blem.)
ON THE RISKS OF LOSS OR GAIN. 93 #=10, a=100, b=500; n (n+1) ab+(a+b)=73333 Pras =n °S375 at 270°8 H = °4 very nearly, or about two to three against the proposed event.
Having thus shown the use of the tables at the end of this work, in the solution of complicated questions, I now proceed to the application of the theory to questions involving loss and gain.
/73333=270°8; 21+1=101;
CHAPTER V.