♦ KNOWLEDGE • 



[June 23, 1883, 



^ir i«att)fniattral Column. 



THB LAWS OF PROBABILITY (Conduced). 



IT 8houlJ be noticed, in the first ]>lacc, that in dcalin,? with 

 hjr)M>theses, wo must consiiier their aiiteco Jent probability ns well 

 aa the cviJence derived from observed facts. We Bomotimcs hear 

 the contrary course spoken of as the perfection of philosophic 

 reasoning. " I will not suffer myself," some will say, " to consider 

 whether this or that hypothesis is likely or not ; I only desire to 

 know whether observation and experiment appear to favour it or the 

 reverse." This i.i an exaggeration of a principle which is exooUcnt 

 in it«elf. Doubtless observation and experiincnt should bo the test 

 by which every hypothesis is examined j and in the long-run, ob- 

 Eer\-ation and experiment may serve to over-balance any antecedent 

 probability against or in favour of an hypothesis. But during the 

 progress of research, and while ns yet the evidence derive! from 

 obsen-ntion or experiment is not decisive, it is essential to tho 

 formation of the most probable opinion that antecedent proba- 

 bilities shoald be duly weighed. For example, let us suppose 

 that a person submits to my notice three predictions con- 

 cerning tho weather, founded on tho planetary positions, and 

 fal611od in a very striking manner by the event. Let ns 

 suppose that the predictions are couched in tho clearest possible 

 terms, and are such that tho odds must be regarded as heavy 

 against their accidental fnlfilment — say, 9 to 1 tor instance. Then 

 the chance that all these predictions ivill bo fnltilled, if the 



hypothesis of planetary influence is erroneous, is jTJjJn > °^ 

 the same as that of drawing one particular ball out of a bag con- 

 taining a thousand. And tho fact that the predictions have actually 

 been fulfilled seems, therefore, to afford very striking evidence in 

 favour of the hypothesis that the planetary positions do actually 

 influence the weather. If I overlooked tho antecedent probabilities 

 of the hypotheses (1, that the weather is so influenced; and 2, that 

 it is not), I should calculate tho probability of the planetary hypo- 

 1000 

 100 1' 



But this would bo to forget the main porlion of the evidence, that, 

 namely, on which our opinion of the antecedent probability against 

 the planetary hypothesis is based. I do not enter into consideration 

 of this evidence, bnt undoubtedly it is of such weight that the 

 most surprising scries of fulfilled predictions (even though accom- 

 panied by clear evidence that not a single prediction had been 

 unfulfilled) would scarcely justify ns in ev(« giving the hypothesis 

 consideration.* 



It is necessary also to consider most carefully whether we really 

 have before us all the possible hj-potheses by which observed facts 

 may bo explained, and whether the various hypotheses are mutually 

 exclusive, for otherwise the general theory above cnunciatid is not 

 applicable. As an instance, let us consider the views of Dr. Stark 

 re.?pecting the influence of marriage on the death-i-ate. From the 

 olJ«er^■ed mortality among married and unmarried men (the fori-icr 

 class including widowers). Dr. Stark was led to infer that marriage 

 is favourable to longevity. (The statistics are given in my essay 

 on the subject in " Light Science "). In arriving at this conclusion, 

 he took only into account tho particular hypothesis that the 

 observed inferior value of nnmarried lives is due to bachelorhood. 

 The hypothesis that marriage is favourable to longevity is not 

 unlikely. in itself, though, certainly, the very marked difference con- 

 sidered by Dr. Stark seems more than tho liypothesis can account 

 for. But there is another hypothesis of which Dr. Stark 

 took no account whatever": tho superior value of married "lives" 

 might Ix! due to a principle of selection, whereby on tho average of 

 a great naml>erof persons, the healthier and stronger would be 

 found in larL-'-r proportion among tho ranks of the married than of 

 the nnmarried. The neglect of this hypothesis vitiates Dr. Stark's 

 reasoning altogether. Tho consideration of this hypothosis, to 

 the neglect of tho other, would be .-ilso incorrect. Both hypotheses 

 must tx! considered together, with duo reference to tho antecedent 

 probability in favour of each. But as they are not mutually ex- 

 clniive, we cannot apply to them our general rule. Wo can only 

 infer that cither one or the other is the true hypothosis, or else that 

 »X)th hypotheses are partially true ; and this does not lea%-e us three 

 cases only to consider, lierauec the causes dealt with by tho hypo- 

 theses may co-exLst in different proportions. Tho statistics dealt 



• Such an assertion aa this seems incomprehensible to tho 

 paradox mongers (as Do Morgan used to call those who propound 

 Buch theories aa wo aro considering). " Strike, bnt hear," they 

 •ay, and they wonder that none will hear, and that few even think 

 it worth while to strike. 



with by Dr. Stark are, therefore, not capable of loading us to any 

 dehnite conclusion. 



But it is not to bo inferred that in this case, and in other similar 

 cases, the statistics are valueless. The fact that they point to no 

 dctinito conclusion sliows only that they aro insuthcient. Other 

 statistical information may so supplomeut snoh imporfoot know- 

 ledge as to lead to tho deliuite conclusions wo require. And tho 

 consideration of imperfect statistics usually shows us in what 

 direction we aro to look for supplementary facts. For oximple, to 

 return to Dr. Stark's statistics. Wo have seen that thoy aro in- 

 sutficient in themselves. They point, however, to two hypotheses, 

 and we have only to detcrm'ino in what degree the two causes 

 dealt with by these hypotheses actnally operate, to form a sound 

 opinion on tho whole subject. I am not concornod, bo it noted, 

 with tho practical application of such inquiries, only with tho 

 bearing which they would havo on tho special qiiestion considered 

 by Dr. Stark. Now, if we study tho statistii-s relating to tho 

 number of marriages in times of prosperity or adversity, or, again, 

 in healthy and unhealthy seasons, wo should gain some slight in- 

 sight into tho i)robability of tho action of a principle resembling 

 selection. Wo might obtain statistics, again, as to tho trades of 

 those who marry, so as to compare tho number of married and 

 single in the different trades. Other statistical inquiries (which 

 might bo very diflicnlt to carry out in practice, however), such ns 

 inquiries relating to tho insurance of married and single, to tho 

 health and strength of each person throughout his life, and so on, 

 would throw further light on the subject of inquiry. Under the 

 actual circumstances, wo can only form probable inferences on these 

 subjects, but the conclusion wo arrive at is much more likely to be 

 sound when such inferences are consid"red, than when no attention 

 whatever is paid to one of tho possible causes of a result so in- 

 teresting as the superior longevity of tho married. 



Another circumstance to be very carefully considered is the 

 actual number of cases with which our statistics may deal. It is 

 impossible to aj-rivo at correct conclusions unless this number is 

 eufiiciently large. Quitelet has related how a person assured liim 

 that a certain street in Paris was tho healthiest region, not only 

 in Paris, but in Franco. When Quelclct asked how this extra- 

 ordinary conclusion had been arrived at, liis-informant remarked 

 that Quetelet's own method of statistical research liad been 

 applied : " I liave found that more than two hundred persons live 

 "■ that street, and tl;cre havo been only three funerals in three 

 the doath-rato so low aa 

 suppose, then," answered 

 Quetelet, "that if there had been no funerals at all, you would 

 havo concluded that by removing to that street you would live for 



One point remains to bo touched upon. I have spoken of the 

 fact that in any very great number of trials, series of events may 

 bo erpccled to occur which would appear very surprising if they 

 happened in bnt a few trials. It would bo a mistake, however, 

 to infer that the occurrence of a strnngo series of events is to 

 bo regarded as on that occount less worthy of notice. Boswell 

 related that when he mentioned to Dr. Johnson that tho first 

 of tho new hackney-coaches which ho had seen had been actually 

 No. 1, tho Doctor ridiculed tho notion that there was anything 

 remarkable in the coincidence. " You were ns likely," ho rea- 

 soned, "to see that coach as any other." Now, althongh Dr. 

 Johnson was certainly right in this instance, yet it would bo a 

 mi-stako to conclude that such coincidences in general should bo 

 overiooked.» When a coin is tossed, it is as likoly that twenty 

 heads will bo tossed in succession ns that any other specified 

 arrangement will appear; yet undoubtedly if tho event had hap- 

 pened wo should bo justified in suspecting that a two-headed com 

 had been used. If a die is cast twenty times, and ace appears eight 

 times, wo might reason that the observed succession of throws wiis 

 antecedently as likely as any other particular succession, and this 

 reasoning would bo just; but, unquestionably, if such an undue 

 number of acos appeared, wo should think it not unlikely that tho 

 die was cogged. This opinion would be strengi honed if aco was a 



• QuMelot relates a curious story illnslriting this ; " During the 

 first Spanish war, n corps of tho French army, which was employed 

 in tho siege of a town, dreaded tho rccuiTenco of Friday, as a day 

 fatal to it, because tho enemy killed or wounded then more of their 

 men tlinn on any other day of tho week ; a strong prejudice was, in 

 consequence, established against Friday. Now after tho siege, it 

 appeared that tho artillery had been changed every day, and that 

 those who did duty on Friday wore superior marksmen to tho 

 others. Superstitious ideas," adds Quitolot, "have often less 

 foundation; from tho impossibility of discovering the true cause 

 under the circumstances in which wo aro placed, wo aro apt to 

 attribute thom to objects totally extraneous, not being willing to 

 I consider effects as existing without a cause." 



years. Now, in what part of Franco is 

 five per thousand per annum ? " "I 



