7 io THE POPULAR SCIENCE MONTHLY. 



We now feel pretty sure that, if we were to make a large number of 

 bets upon the color of single beans drawn from the bag, we could ap- 

 proximately insure ourselves in the long run by betting each time 

 upon the white, a confidence which would be entirely wanting if, in- 

 stead of sampling the bag by 1,000 drawings, we had done so by only 

 two. Now, as the whole utility of probability is to insure us in the 

 long run, and as that assurance depends, not merely on the value of 

 the chance, but also on the accuracy of the evaluation, it follows that 

 we ought not to have the same feeling of belief in reference to all 

 events of which the chance is even. In short, to express the proper 

 state of our belief, not one number but two are requisite, the first 

 depending on the inferred probability, the second on the amount of 

 knowledge on which that probability is based. 1 It is true that when 

 our knowledge is very precise, when we have made many drawings 

 from the bag, or, as in most of the examples in the books, when the 

 total contents of the bag are absolutely known, the number which ex- 

 presses the uncertainty of the assumed probability and its liability to 

 be changed by further experience may become insignificant, or utterly 

 vanish. But, when our knowledge is very slight, this number may be 

 even more important than the probability itself; and when we have 

 no knowledge at all this completely overwhelms the other, so that 

 there is no sense in saying that the chance of the totally unknown 

 event is even (for what expresses absolutely no fact has absolutely no 

 meaning), and what ought to be said is that the chance is entirely 

 indefinite. We thus perceive that the conceptualistic view, though 

 answering well enough in some cases, is quite inadequate. 



Suppose that the first bean which we drew from our bag were 

 black. That would constitute an argument, no matter how slender, 

 that the bean under the thimble was also black. If the second bean 

 were also to turn out black, that would be a second independent argu- 

 ment reenforcing the first. If the whole of the first twenty beans 

 drawn should prove black, our confidence that the hidden bean was 

 black would justly attain considerable strength. But suppose the 

 twenty-first bean were to be white and that we were to go on draw- 

 ing until we found that we had drawn 1,010 black beans and 990 

 white ones. We should conclude that our first twenty beans being 

 black was simply an extraordinary accident, and that in fact the pro- 

 portion of white beans to black was sensibly equal, and that it was an 

 even chance that the hidden bean was black. Yet according to the 

 rule of balancing reasons, since all the drawings of black beans are 

 so many independent arguments in favor of the one under the thimble 

 being black, and all the white drawings so many against it, an excess 

 of twenty black beans ought to produce the same degree of belief 

 that the hidden bean was black, whatever the total number drawn. 



1 Strictly we should need an infinite series of numbers each depending on the prob- 

 able error of the last. 



