352 



INDUCTION. 



must be admitted, that even when 

 we have no knowledge whatever to 

 guide our expectations, except the 

 knowledge that what happens must 

 be some one of a certain number of 

 possibilities, we may still reasonably 

 judge that one supposition is more 

 probable to us than another supposi- 

 tion ; and if we have any interest at 

 stake, we shall best provide for it by 

 acting conformably to that judgment. 



§ 2. Suppose that we are required 

 to take a ball from a box, of which 

 we only know that it contains balls 

 both black and white, and none of 

 any other colour. We know that the 

 ball we select will be either a black 

 or a white ball ; but we have no 

 ground for expecting black rather 

 than white, or white rather than 

 black. In that case, if we are obliged 

 to make a choice, and to stake some- 

 thing on one or the other supposition, 

 it will, as a question of prudence, be 

 perfectly indifferent which ; and we 

 shall act precisely as we should have 

 acted if we had known beforehand 

 that the box contained an equal 

 number of black and white balls. 

 But though our conduct would be 

 the same, it would not be founded 

 on any surmise that the balls were 

 in fact thus equally divided, for we 

 might, on the contrary, know, by 

 authentic information, that the box 

 contained ninety-nine balls of one 

 colour, and only one of the other ; 

 still, if we are not told which colour 

 has only one, and which has ninety- 

 nine, the drawing of a white and of a 

 black ball will be equally probable to 

 us; we shall have no reason for staking 

 anything on the one event rather than 

 on the other ; the option between the 

 two will be a matter of indifference ; 

 in other words, it will be an even 

 chance. 



But let it now be supposed that 

 instead of two there are three colours 

 — white, black, and red ; and that we 

 are entirely ignorant of the proportion 

 in which they are mingled. We should 

 then have no reason for expecting one 



more than another, and if obliged to 

 bet, should venture our stake on red, 

 white, or black, with equal indiffer- 

 ence. But should we be indifferent 

 whether we betted for or against 

 some one colour, as, for instance, 

 white ? Surely not. Erom the very 

 fact that black and red are each of 

 them separately equally probable to 

 us with white, the two together must 

 be twice as probable. We should in 

 this case expect not- white rather than 

 white, and so much rather, that we 

 would lay two to one upon it. It is 

 true, there might, for aught we knew, 

 be more white balls than black and 

 red together ; and if so, our bet would, 

 if we knew more, be seen to be a dis- 

 advantageous one. But so also, for 

 aught we knew, might there be more 

 red balls than black and white, or 

 more black balls than white and red, 

 and in such case the effect of additional 

 knowledge would be to prove to us 

 that our bet was more advantageous 

 than we had supposed it to be. There 

 is in the existing state of our know- 

 ledge a rational probability of two to 

 one against white ; a probability fit 

 to be made a basis of conduct. No 

 reasonable person would lay an even 

 wager in favour of white against 

 black and red ; though against black 

 alone, or red alone, he might do so 

 without imprudence. 



The common theory, therefore, of 

 the calculation of chances appears to 

 be tenable. Even when we know no- 

 thing except the number of the pos- 

 sible and mutually excluding con- 

 tingencies, and are entirely ignorant 

 of their comparative frequency, we 

 may have grounds, and grounds nu- 

 merically appreciable, for acting on one 

 supposition rather than on another ; 

 and this is the meaning of Proba- 

 bility. 



§ 3. The principle, however, on 

 which the reasoning proceeds is suf- 

 ficiently evident. It is the obvious 

 one, that when the cases which exist 

 are shared among several kinds, it is 

 impossible that each of those kinds 



