ni 



PROBABILITY. 



PROBABILITY. 





my, bat everything which disposes the min.l, I. 

 .. to fclopt or reject, including eren the effect of previous know- 

 The value of evidence, that In, the extent it should go toward* 



inducing brlicf, is really the subject ..f in>]iiiry in a branch of exact 

 dence known by the name of the thrort, of prohabilitia. Bat huw 

 <mn the value of evidenoe be made a subject of measurement I why 

 can this be done in the caw of an amount of credibility more than in 

 that of an amount* of benevolence, courage, or talent! We should 

 assumlly think any one miut be in a curious delusion who should 

 suppose M"l* to hare ascertained, from the data given by Homer, 

 that the warlike skill of Achilles was exactly SJ'2 times that of Tin T- 

 sites, and that Sbakspcre had done the former a foul wrung, for that 

 he had made it only 237 times and a fraction. But what is the 

 difference , in the nature of the inquiry we seek to institute, between 

 attempt* to measure prowess and probability? On the mode of 

 answering this question it ilc|vwl/< whether we are to make our subject 

 merely, u 1. kind of artilici.il method of judging the chances 



of a game of liaxard ; or a rational and exact mode of doing, when 

 data are sufficient, that which we daily attempt to do, as well as wo 

 can, with our inaccurate appreciations of the circumstances of common 

 life ; and a science to be used, as are others of a mathematical nature, 

 fur accustoming ourselves to estimate or guess with something like 

 accuracy, by habitual acquaintance with cases in which absolute accu- 

 racy is attainable. 



When we consider all the circumstances which affect belief or opinion, 

 both those which are external and those which depend on the mind 

 which is exposed to them, it may well bewilder the imagination of a 

 pern not accustomed to the idea, when he hears of an attempt to 

 reckon credibility in numbers, and to deduce what are called exact 

 conclusions from hypotheses as to the force of assertions. To remove 

 from the threshold of the subject the incredulity which must exist, 

 and ought to exist in the first instance, let us suppose the other 

 branches of science presented to a student not in their simple begin- 

 ning*, but by a description of their ultimate physical objects. To put 

 this beginner in a state parallel to that of readers in general with 

 respect to the subject of probabilities, he must be of mature age, with 

 very little knowledge of number, none of any other branch of mathe- 

 matics, and no conception of the construction or use of any physical 

 instrument, nor of the object and procedure of any one experi- 

 ment. He might then be addressed as follows : " You arc, without 

 moving from this earth, to track the motions of all the heavenly bodies, 

 to be able to ascertain where they were or will be at any moment of 

 time past or future, to measure their sizes, to weigh their contents, 

 and to find the species and amount of insensible forces which, by some 

 unknown means, they exercise on one another. You are to detect the 

 existence of a subtle fluid which can neither be seen, heard, nor felt, 

 to measure vibrations of which there are millions in a minute, and to 

 tracu the course of effects which travel hundreds of thousands of 

 miles in a second. You are to weigh against each other atoms of 

 matter of which it cannot be shown that millions put together would 

 be visible to the eye." The person so addressed would not be less 

 bewildered nor more disposed to treat the proposed results as fictions, 

 than be who hears for the first time of a numerical theory of probabili- 

 ties. But let us now reverse the method, and suppose the learner allowed 

 to begin at the beginning. He first finds that, step by step, his rude 

 notions of number ore organised into a method of computation which 

 enables him easily to perform more than he could have imagined the 

 most subtle brain to have devised. From notions of the simplest kind 

 connected with space, properties of figure become almost intuitive, of 

 which he could at one time not have comprehended the description, 

 far less the demonstration. By reasoning on the simplest properties 

 of matter, such as can be proved to his senses, he finds no difficulty in 

 tracing remote and complicated combinations of effects from the 

 plainest causes, by which he learns to invert this process, and to 

 reduce observed combinations to their simplest elements. But if, 

 during this long and very gradual process, he were to keep continually 

 before his mind those great results the knowledge of which he had been 

 promised, looking to arrive at the fulfilment of the promise by some 

 sudden acquisition of power, his whole course would be one of dis- 

 appointment. He would be peeping forward a few pages in his Euclid, 

 in the hope of seeing himself almost arrived at the means of calcula- 

 ting in eclipse or explaining the theory of colours, and would find that 

 he was to learn how to make a square equal to a given figure instead. 



Now the application of the preceding description to our present 

 subject is as follows : The beginner in exact science has usually no 

 definite notions as to the end which he is to arrive at ; nor do the terms 



It to rlhrr for the argument that wo endeavour, however imperfectly, to 

 make numerical standards of UK In estimating ever; one of tlirac qualities, 

 which we pot down at huard before the matter of Una note suggests : 

 We are apt to compare the benevolence of two persona by the pecuniary rcsulta 

 of their liberality ; confunndlng the two qualities, but perhaps tho very con. 

 ftuion arUea partly from tin numerical measure which In thereby supposed 

 attainable. We form a vagw notion of the comparative courage of two soldiers, 

 by comparlni the different amount, of danger whloh we think they have faced ; 

 a mwt unfair and fallacious teat, but sti.l indicative of the tendency to meaaurc. 

 Finally, we compare talent in what, (peaking of the way it ii now done, we 

 can hardly call a more rational manner, by trlala of what competitor* can do 

 under the tame circumstances ind In the same time. 



algebra, geometry, mechanics, Ac., suggest any associations beyond a 

 vague notion that they are parts of a learned system. But it is impos- 



Mblc that tlir U'ginncr in the subject of this article should be without 

 an* explicit anil probably 'an exaggerate <f what he is to 



attain. Tin i c is no unknown Qreek or Arabic term the meaning of 

 which must be slowly learned by the study of the science of which it 

 is the name; the word probability, so well known in the common 

 affairs of life, stares him in the face at tho head of every page, and 

 reminds him to be dissatisfied with the extent of power gained, up to 

 the point at which be has arrived. Unless then he can make up his 

 mind to descend, as a student would do who, having in his head the 

 theory of gravitation and the laws of light, should lay by these grand 

 ideas, and set himself to trace the consequences of the simple 

 that two straight lines cannot enclose a space he must be warned that 

 he will be likely to quit the subject in disgust. We now proceed to 

 the fundamental points of the theory. 



That opinion may be formed with more or less strength, particularly 

 when the subject-matters are of different species, is well known to 

 every one from his own experience. The most decided republican in 

 Kngknd, for instance, is not 'so sure of the wisdom of the Long Par- 

 liament as he is that all its members are now dead ; and no royalist, 

 however well persuaded of his tenets, thinks the Restoration was of as 

 much consequence to this country as sun, wind, and rain. It matters 

 nothing that the different degrees of assurance refer to very different 

 matters, and are obtained in very different ways ; that they are separate 

 amounts of the same kind of feeling is universally felt and admitted. 

 To moke something like a gauge for these degrees of belief is not 

 difficult ; to apply it is a harder task, seeing that the cases which present 

 circumstances of sufficiently definite character are seldom met with. 



Suppose a box to contain 3 white and 4 black balls ; it is easily 

 admitted that it is more likely that a black ball should bo drawn than 

 a white one, on the supposition that tho drawer does not see the bolls. 

 Or rather we should eay it is easily admitted that every well regulated 

 mind ought to think a black ball more likely than a white on. 

 that if any one should imagine the contrary, he bos formed an opinion 

 from prejudice, fancy, or want of proper consideration. Just as we 

 should say that if all the balls were black, a black ball would certainly 

 be drawn, so when a majority of the bolls is black, and each one I nil is 

 as likely to be drawn as any other, there are more ways of drawing 

 black than white, and we look upon the former as more obtainable, 

 and more likely to be obtained, than the latter. Common exp. 

 makes us consider the black as more likely than white, when tho 

 number of black balls is much greater than that of white Kills ; as, if 

 there were only 3 white balls, and a million of black ones. Here, as in 

 other questions of magnitude, we can see a difference when the 

 difference is great, which we must perhaps learn to see when it is 

 small : it is plain enough that the black is more likely than the white 

 when there ore a million of black balls to one white ; but not ao easily 

 grasped that the black is more likely than the white when there are 

 five hundred thousand and one block balls to five hundred thousand 

 white. 



The next step to be mode is the assertion that when there are 3 

 white and 4 black bolls, tho probability of drawing white is to that of 

 drawing block in the proportion of 3 to 4 ; that is, if we could by a 

 voluntary act make our impressions about the probability of future 

 events of that strength which our reason tells us they ought to have, 

 we should choose to expect a black ball more strongly than a white one 

 in the proportion of 4 to 3. The principle on which we do this is the 

 main point of the theory, the only objectionable port, if there be one : 

 for all the rest is mathematical deduction. 



The principle is as follows : When any number of events, A, B, c, 

 &c., are such that one and only one con happen at a time, and when 

 a, b, c, to., are the numbers of ways in which they con severally happen, 

 the probabilities of the several events ore in the proportions of the 

 numbers a, b,c, 4c. Returning to the preceding simple instance, ue 

 have on obvious negative reason for supposing that the probabilities 

 should be as 4 to 3, since there is no imaginable ground for assuming, 

 while the excess of block bolls is the sole cause of the superior pro- 

 bability of drawing one of them, that this excess of probability should 

 be in any other proportion than that of tho excess of black balls. If 

 we grant the following, namely, that the probability of having one or 

 other out of two of the different results which a trial may give, is, or 

 ought to be, the turn* of the probabilities of the tv ly, we 



shall be obliged to admit positive reason for the preceding principle, as 

 follows : Suppose a box to contain 10 balls, marked 1, '2, &c. up to 10, 

 and no others. A ball is to be drawn, and the drawer has in his mind 

 on amount of hope, fear, or simple admission of possibility, as the case 

 may be, as to the happening of each number. If the drawing of No. 1 

 be to gain him a prize, there is a certain amount of hope ; if it be to 

 procure him a loss, of fear; if neither one nor the other, of feeling that 

 So, I may be that which is drawn. Now let cither 1 or 2 bring the 

 t loss; is the feeling of hope or fear doubled in strength ? or 

 rather, ouy/it it to be doubled? Ho who admits this, admits the 



Disjunctive mention It a logical aggregation, a Humiliation, of clauea. 

 When we aay " He IB cither fool or mad/' we say he belongs to the genus which 

 contains the specie* foul and mud. To describe him in a word, we want a name 

 for fool + mad, Wo never say "it is either man or brute:" w say " it Is 

 animal." 



