3.22 INDUCTION. 



According to his views, indeed, the calculation of chances should be 

 much more universally applicable to things of vv'hich we are com- 

 pletely ignorant, than to things of which we have partial knowledge. 

 Where we have some experience of the occurrence of each of the con- 

 flicting possibilities, it may often be difficult, according to the prescrip- 

 tions of the theory, to reduce those possibilities to a definite number of 

 cases, all equally probable ; but when the case is out of the reach of 

 all experience, so that we have no difficulty in being " equally unde- 

 cided" respecting the possibilities, there is nothing to make us halt or 

 waver in applying the theory. If the question be whether the inhabit- 

 ants of Saturn have red hair, we need only know the number of the 

 prismatic colors, and of th-eir more marked compounds, and we can at 

 once assign the fraction corresponding to the probability ! It is evi- 

 dent that probability, in any sense in which it can operate upon our 

 belief or conduct, has nothing to do with such chimerical evaluations, 

 and that entire suspension of judgment, where we have no evidence, is 

 the only course befitting a rational being. To entitle us to affirm any- . 

 thing positive abovit uncertain facts, whether it be that one supposition 

 is more probable than another, or only that it is equally probable, we 

 must have the testimony of experience, that, taking the whole of some 

 class of cases, the one guess will be oftener right, or as often right as 

 the other. The estimation, in short, of chances, like that of certain- 

 ties, is only rational when grounded upon a complete induction by 

 observation or experiment.* 



§ 3. From these principles it is easy to deduce the demonstration of 

 that theorem of the doctrine of probabilities, which is the foundation 

 of its principal application to judicial or other inquiries for ascertain- 

 ing the occurrence of a given event, or the reality of an individual 

 fact. The signs or evidences by which a fact is usually proved, are 



* Confusion is sometimes introduced into this subject by not adverting to the distinction 

 between the chances that a given event will happen, and the chances that a guess, not yet 

 made, respecting its occurrence, will be right. Supposing that I have rio more reason to. 

 expect one event than another, it is (frotn experience of human actions) an equal chance 

 whether I guess A or B ; but it is not, therefore, an equal chance whether A or B takes 

 place. 



The fallacy has been stated thus. Suppose tliat either A or B must happen: and let 

 the chance that A will happen be x: as certainty is represented by 1, the chance that B 

 will happen is 1 —x. Now, the chance that the event I guess will come to pass, is made 

 up of two chances: the chance that I shall guess A and that A will happen, plus the 

 chance that I shall guess B and that B wil) happen. The chance that 1 shall guess A 

 being \ ; the chance that 1 shall guess A and that A will happen, is compounded of \ and x : 

 it is therefore 1^. The chance that I shall guess B being also .1, the chance that I shall 

 guess B and that B will happen, is-^ (1 — ^)- But the sum of ihcse two is \ : therefore th^ 

 chance that the event I guess will come to pass, is always an even chance. But since it 

 is an even chance that my guess will be right, it is an even chance which of the two e.vents 

 will occur, whatever may be their comparative frequency in nature. 



The whole of this reasoning is sound up to the last step, but that step is a non se.qvUxa, 

 Before I have guessed, or until I have made my guess known, it is an even chance that I 

 guess right ; but when 1 have guessed, and guessed A, it is no longer an even chance that 

 I have guessed right : otherwise there would be an even chance in favor of the most im- 

 probable event. Let the question be. Is Queen Victoria at this moment alive :■ and let me 

 be required to guess aye or no, without knowing about what, in order that I may be equally 

 likely to guess the one and the other. No one will say it is an even chance which is true ; 

 but It really is an even chance whether my guess will be right. The chance of my giu.'ss- 

 ing in the negative and being right, is \ of a very small chance, nay, perhaps -^-5^15-^, but 

 the chance of my guessing in the affirmative, and being right, is \ of the remaining yl^^-l^^ ; 

 so that the two together are \. When, however, I have guessed, and told my guess, the 

 even chance yvluch of the two I should .guess is converted into a certainty, li 1 have 

 guessed aye, the chance that I am right is xffT^^I : i^ no> i' J^ only rsWBTr- 



