CHAPTER I. ON THE NOTION OF PROBABILITY AND ITS MEASURE- MENT ; ON THE PROVINCE OF MATHEMATICS WITH REGARD TO IT, AND REPLY TO OBJECTIONS.
WHEN 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 cases 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 2 ESSAY ON PROBABILITIES. accidents of life, and contains the principles on which it 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 INTRODUCTORY EXPLANATIONS.S 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 will 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 less, 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 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 Roman em- Two straight Fire will Of 1,000,000 of pire was over- lines cannot burn. children born,some turned by north- 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 10001., 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 answer, unques- tionably, the first, for the second will certainly not hap- 5 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 a, or b, or c, or x, or y, or z, 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 b. He will draw either a, or b, or c. ...................................................... .................................................... He will draw either a or b or or y. He will draw either a or b or or y or z. 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 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 orb must be drawn. other will die aged fifty. Of three children born, one or a, or b, or c must be drawn other will die aged fifty. &c. &c. &c. &c. &c. &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. 7
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 'a The refusal of this substitution is what soldiers call the key of A's position: be 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 he 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 belongs 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
R 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 b, 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 261., and that the 26 letters have been bought. If I buy 10 ESSAY ON PROBABILITIES. up all the vested interests at less than 11. a piece, I am certain to gain ; if at more, I am certain to lose. 11. a piece is what I ought to give for each, if I buy all: it is the universal practice to consider that 11. 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 11. ? 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 INTRODUCTORY EXPLANATIONS. 11 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 1001., 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 26, and in the second by g1,'. Similarly the chance of 1 out of 10 given letters in the first lottery is b, and in the se- cond, 1'0'. 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 o 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 101. 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 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 following amounts of 11. in 1, 2, 3, &c. years:— Yrs. Am. Yrs Am. Yrs. Am. Yrs. Am. 0 1 7 128 14 16364 21 2097152 1 2 8 256 15 32768 22 4194304 2 4 9 512 16 65536 23 8388608 3 8 10 1024 17 131072 24 16777216 4 16 11 2048 18 262144 25 33554432 5 32 12 4096 19 524288 26 67108864 6 64 13 18192, 20 1048576 27 134217728 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 together 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 11. in 11 years yield 20481., and if this 20481. be put out for 12 years more, then, since 11. in 12 years yields 40961., 2048 times as much will yield 2048 x 40961.; 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 11. shall increase to 101. 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 of a table of logarithms as follows : Am. Yrs. Am. Yrs. Am 1 Yrs. Am. Yrs. 1000300043 5232 371867 9997399987 10001 400004 1001 300087 5233 37187.5 9998399991 10002 400008 1002 300130.5239 371883 99991399996 10003 400013 1003 300173 5235 371892 100001400000 10004 400017 &c. &c. &c. &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 so on. We can thus manage to put down every number, within certain INTRODUCTORY EXPLANATIONS. 15 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; that is, the continuation of the pro- cess following. Let [10], for instance, represent the product of all the numbers, from 1 up to 10, both in- clusive, or let [10]stand for1x2xSx4x5x6x7a8x9x10=3628800 [1] is 1 ; [2] is 2 ; [3] is 6 ; [4] is 24 ; [5] is 120 ; [6] is 720 ; [7] is 5040 ; [8] is 40,320 ; and so on. This labour becomes absolutely unbearable when the numbers become larger; thus, [30] 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. RULE.—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. EXAMPLE.—What is [30] or 1 x 2 x 3 x x 29 X30? log. 30 1.4771213 1.4771213 4342945 7981799
Subtract 1.0428268 2)2.2753012 Add. 30 1.1376506 Second step. Multiply 51.284804 First step. 1.137651 Result of second step.
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 360th (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 — 265252859812191058636308480000000 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 7x8x9x10x11x12x13x14x15 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 : —
A N x 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 N P shall be connected with its abscissa A N, by the following law. Measure A N in inches and parts of inches; and multiply the result by itself; and the product by 4342945. Find the number to which this product is the common logarithm, and divide 1.1284 by the result. The quotient is the fraction of au inch in N P, and in the table we find, not N P, but the area AN PB 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 0 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, or 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 capable 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 powers 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 sufficient 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 c2
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 to a 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 education : 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. Norio repenté fait 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..
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 22 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 r(volution francaise, les etats de certaines provinces etaient assembles, on y joua t un jeu terrible, et tel que. l'endroit o1 it se trnait, daus la ci-levant province de Brttagne, s'appellait l'enfer. — Did. dcs Jeua (Enctc. Meth.) 1792. INTRODUCTORY EXPLANATION. 23
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. deistic 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 chalices." c 4 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 . The theory of probabilities, absolutely requires, in its fun- so far as considerations of ab- damental principles, the rejec- solute necessity are concerned, tion of the notion that pure neither denies nor asserts, in chance can produce any two whole or in part, any thing events alike; that is, it pre- whatsoever respecting the mo- sumes causation and order of ral or intellectual character of some kind or other, that is, the providence which it re- providence of some kind or 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.1' It appears 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. t 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. If the 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 t 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, of 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. 27
I. All the eleven planets yet discovered move in one direction round the sun. I 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 to 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, h 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 might, 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 :—
the way of using it? — The necessary preliminary to
application is, the result of the measurement in a case I. What is the measure of probability? II. What is
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 uestion ; 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. II.
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 and that of drawing a black ball by -a =. 'We shall say that these probabilities are iq- and ,',-4. Nothing is more common than to substitute a measure