PROBABILITY. 



PROBABILITY. 



779 



UK. urn lie. betwodi that ot 418-10 to 182 + 10 and 418-1 10 to 

 : Out ot 1000 to 471 and 1000 to 402? Here o = 418, 

 4=18S. i=10; the square root above mentioned has 1-2021 {or its 

 logarithm, and 10 divided l.y this square root give* -63 nearly. And 

 A being -6S, B U also -68 rery nearly, which is the probability required ; 

 or it u 63 to 37, or nearly 2 to 1, that for every 1000 white balls in the 

 urn. there are between 408 and 471 black ones. 



i.m 4. When r happens much oftener than Q, we feel a very 

 high degree ot probability (moral certainty) that the same thing would 

 happen In the long run, or would continue how long soever we might 

 go on trying, unleaa there should arrive some change of circumstances. 

 But when P does not happen much oftener than q, we do not feel the 

 same degree of asninncc ; for though q might really happen oftener 

 than r in the long run, yet the casual fluctuations of evcnU might 

 make the contrary appear in any one set of trials. Suppose then that 

 in a + 6 trial*, r has happened a times, and q has happened 6 times, a 

 and 4 being nearly equal : what presumption can thence be derived 

 that the excees will be on the same side in the long rnu on which it is 

 in the a + A trials ? 



Hi I.E. 1 >i\ i.ie the difference of a and 6 by the square root of twice 

 mi : 1ft the result be A, and find the corresponding B; to this 

 add 1, .-mil iiivi.tr l>y '2 : the result is the probability required. 



--Buflbn, in a particular experiment, hod to throw a coin 

 (my a halfpenny) 4140 times; the result was 2048 heads and 2092 

 tail* : what U the probability that some excess of tail over head would 

 hare continued for ever ; that in, that the coin he used had in its con- 

 struction some mechanical tendency to fall tail rather than head ? 

 Hire a = 2048, 4 2092 ; the difference divided by the square root of 

 In- MUM in "48, which being A, B is '50, and 1'50 divided by 2 is 

 75, the probability required. This is j, so that it is 3 to 1 that some 

 excess or other of toils would be found in the long run. Observe how- 

 .n this does not mean an excess to the amount of 44 in every 

 41 40 trials ; hut only tome excess, small or great : observe also that tho 

 character of the mint and its processes are supposed wholly unknown, 

 su that no conclusion con be formed about the character of the coin 

 except from the observed event. 



The experiment of Buffon, and its particular object, will be found in 

 the ' Essay on Probabilities and Life Contingencies,' in the ' Cabinet 

 Cyclo]<axha ; ' in which work will also be found a larger collection of 

 such problems as the preceding, with fuller explanation and more 

 extensive tables. These problems have been here introduced that the 

 reader may hare on opportunity of comparing and verifying the general 

 sameness of results which follows from the two (j-wrhaps) apparently 

 differing notions from which the idea of the measure of probability 

 may be derivi-1. 



Previously to laying down a few general rules, we shall notice the 

 to which persons who hare not thought on the subject are most 

 : this is a confusion between the sort of confidence which is to 

 be given to a result of the theory of probabilities, and that which is 

 claimed by actual demonstration. Many persons ore not aware that 

 out of mathematics the greater number of conclusions are probable 

 results : many of them, it is true, so highly probable that their chance 

 of f.ilxehood does not amount to that of drawing a block ball from 

 among a million of white ones ; but still not absolutely demonstrated. 

 These highly probable results (so probable that the word probable in 

 its common sense is weak as applied to them) form the ordinary know- 

 ledge of common life, and being practical certainties, ore considered 

 and mentioned as certainties, the imperceptibly small chance that they 

 may not be true being disregarded. Hence it happens that when a 

 romlt of this theory is announced, with its proper chance annexed, 

 and though the probability of its truth is so high that it may rank 

 with the moral certainties of ordinary life, there is a morbid disposition 

 to dwell rather upon the one way in which the proposition may fail, 

 than upon the million of equally possible ways in which it may be 

 tm.. Thus if it be said that it is ten millions to one that r will 

 ha]i]>en, and not q, therefore it is morally certain that p will happen 

 it is objected, But how dc we know that the very next event will not 

 be precisely the one of ten millions which is q and not r ? The answer 

 is, we do not kmnc it in the absolute sense of tho word ; if we thus 

 knew it, it would be a certainty that P would happen, and whether p 

 would happen or not would not be a question for the theory of proba- 

 bilities at all : but we do know it in the common sense of the word, 

 since there are hundreds of conclusions which all men call knowledge, 

 which ore not so probable that they can be reasonably shown to have 

 ten millions to one in their favour. 



Another way in which the confusion we have mentioned shows 

 lUclf is, in the habit of reasoning against the probable truths just 

 alluded to, by arguments which could only be valid against an assertion 

 that these truths were absolutely demonstrated. In compliance with 

 the forms of language, those who advance such truths treat them as 

 (moral) certainties : the opponent overthrows their (mathematical) 

 certainty, and the fallacy lies in his supposing that he has thereby 

 shown them not to possess that sort of truth which was claimed for 

 ti.MM. rr example, a medical man gives his opinion that a crime 

 iltcd without any apparent motive in an indication of insanity : a 

 ipOT ridicule* this opinion, and asks, Are there n<> motives then 

 which cannot be discovered f Now,itb\ .< meant apparent 



on the surface, or with slight examination, and if by indication was 



meant an absolute indication, certainly inexplicable except by the sup- 

 position of insanity, the answer U complete, and the opinion shown to 

 be untenable. But suppose the energies of many acute persons and 

 the resources of a whole nation to fail in making the motive of a crime 

 apparent, and that this is what U meant by there being no apparent 

 motive ; suppose moreover that by an indication of insanity is meant 

 circumstance which renders insanity so moderately probable that the 

 hypothesis deserves to be weighed : the answer then is wholly irrele- 

 vant The opinion, expanded into an argument, is a crime committed 

 absolutely without motive or object shows insanity ; a motive, if it 

 exist, may almost certainly be discovered by proper exertions ; con- 

 sequently the appearance of no motive, after all exertions to discover 

 one have been tried, makes it most likely * that the n inn- was an act of 

 insanity ; it is in fact as likely tliat the crime was an act of insanity, 

 as it was unlikely that the exertions to discover a motive shout" 

 failed, if tin re had been a motive. 



The application of this theory to pure logic is contained in the con- 

 sideration of testimony, argument, and their combination. The 

 following rules t will be found demonstrated in De Morgan's ' Formal 

 Lope,' ch. x. 



inoiiy, or authority which means only testimony of great 

 force, as the word is commonly used speaks to the truth or falsehood 

 of the assertion. But argument speaks, not to the truth or fake- 

 hood of the conclusion, but to the validity of the mode of establish- 

 ing it. An argument may be invalid, either by false premises or bod 

 logic : and yet the conclusion may betnie. Consequently, the proba- 

 bility of an argument being valid is a totally different thing from that 

 of the conclusion being true. 



The effect of testimony, as also of argument, must depend upon tho 

 initial state of the receiver's mind with reference to the conclusion 

 asserted. But though the receiver himself may stand in a very 

 different position from the witnesses who bring him testimony, yet in 

 the mathematical formula; he Kinks but as one of those witnesses : if 

 his initial state be that it is 990 to 1 against the conclusion, the results 

 of the theory are the same as if he, being unbiassed, found an addi- 

 tional witness to deny the assertion for whose correctness it is in In 

 mind, a priori, 999 to 1. A person without any bias on his mind begins 

 with an eren chance for the assertion. And a witness of credibility p, 

 who asserts, ranks as a witness of credibili ty 1 jt, who denies. Tho 

 following rules are given in mathematical form, for consistency with 

 the rest of the article: the non-mathematical reader will lind them 

 reduced to arithmetical rules in De Morgan's ' Syllabus of a Proposed 

 System of Logic,' 1860. 



Let a number of witnesses, of credibilities p., p', ic., affirm : let 

 others, of credibilities c, ',..., deny. By the credibility of a v> . 

 we mean the belief we have of an assertion of his being correct, ! 

 we turn our attention to the particular character of that ass.rt i.m. 

 Then the odds in favour of the assertion are jxx/u' x .... x (1 v)x 



(!-') to (1 M) x (1 /') x xrx/ Let the | ro- 



bability of the assertion thence obtained (the first of the two divided 

 by their sum) be called M : and remember that the receiver himself is 

 one of those witnesses, as before explained. 



Let a number of arguments of which the several probabilities of 

 validity (that is, of their proving then- conclusions true) be a, a', .... 

 The probability that the conclusion is proved that is, that one or more 

 are valid is 1 (1 o) (1 a') .... If arguments be produced for tho 

 contradictory conclusion, of validities b, It', &c., then 1 (1 4) 

 (1 b') .... is the probability that the conclusion is disproved. \Ve 

 speak of separate coses : the combination of arguments for and .1 

 will bo instantly seen. Let these results be A and u. \Ve may treat 

 the conclusion, as now asserted to an unbiassed mind by a single, 

 witness of credibility M, supported by one argument of validity A, 

 and opposed by one argument of validity B. The only diilnvu.v is 

 that if there be a plurality of witnesses or of arguments, we have to 

 calculate M, A, B ; which, when there is one only of each kind, are 

 supposed given. 



When testimony and argument are combined, the odds in favour of 

 the .conclusion are M (1 B), to (1 ii) (1-A). The following cases 

 may be distinguished : 



1. No evidence; arguments on both sides. Here M = 4, and the 

 o<MJare 1 Bto I-A. When A and B are very near to unity, or 



* When thin paragraph was written, there wu a run against the hypothesis 

 ot insanity in criminal cases : fince that time there boa been a lon^rr .m.i 

 stronger run In favour of it. Had thi> article been now first written, in all 

 probability the illustration would have been drawn from the CXCCM opposite to 

 that chosen for comment in the text. 



t The account of this matter given In the ' Penny Cyclopaedia ' is not correct, 

 though something nearer the mark than anything which preceded. The logicians 

 had constantly confounded the probability of validity of on argument with the 

 probability of truth of its conclusion ; and the article just referred to followed 

 them In their principle, though not in their details. 



J This word i connected with gambling associations ; and so is the word 

 chance. We substitute probability for the second word, which Is used nmbi- 

 guouMy [CiiANcr] : but we cannot conveniently dispense with odds; the 

 following anecdote, however, may go some way to purify the polluted word. It 

 is told by the distinguished experimenter, Professor r.u .id.. \ , in a lecture ' On 

 Mriiul 'I r. lining' (I2mo, 1854), "When I was young,! nc-iiv.il <<m one 

 well able to aid a learner In his endeavours towards self-improvement, a curious 

 lemon in the mode of estimating the amount of belief one might be induced to 



