OF THE CALCULATION OF CHANCES. 381 



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 knowledge 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 favor of white against black and red ; 

 though against black alone or red alone he might do so without impru- 

 dence. 



The common theory, therefore, of the calculation of chances, appears to 

 be tenable. Even when we know nothing except the number of the possi- 

 ble and mutually excluding contingencies, and are entirely ignorant of their 

 comparative frequency, we may have grounds, and grounds numerically ap- 

 preciable, for acting on one supposition rather than on another ; and this 

 is the meaning of Probability. 



§ 3. The principle, however, on which the reasoning proceeds, is suifi- 

 ciently 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 

 should be a majority of the whole : on the contrary, there must be a ma- 

 jority against each kind, except one at most ; and if any kind has more 

 than its share in proportion to the total number, the others collectively 

 must have less. Granting this axiom, and assuming that we have no 

 ground for selecting any one kind as more likely than the rest to surpass 

 the average proportion, it follows that we can not rationally presume this 

 of any, which we should do if we were to bet in favor of it, receiving less 

 odds than in the ratio of the number of the other kinds. Even, therefore, 

 in this extreme case of the calculation of probabilities, which does not rest 

 on special experience at all, the logical ground of the process is our knowl- 

 edge — such knowledge as we then have — of the laws governing the fre- 

 quency of occurrence of the different cases; but in this case the knowledge 

 is limited to that which, being universal and axiomatic, does not require 

 reference to specific experience, or to any considerations arising out of the 

 special nature of the problem under discussion. 



Except, however, in such cases as games of chance, where the very pur- 

 pose in view requires ignorance instead of knowledge, I can conceive no 

 case in which we ought to be satisfied with such an estimate of chances as 

 this — an estimate founded on the absolute minimum of knowledge respect- 

 ing the subject. It is plain that, in the case of the colored balls, a very 

 slight ground of surmise that the white balls were really more numerous 

 than either of the other colors, would suffice to vitiate the whole of the cal- 

 culations made in our pi-evious state of indifference. It would place us in 

 that position of more advanced knowledge, in which the probabilities, to us, 

 would be different from what they were before; and in estimating these 

 new probabilities we should have to proceed on a totally different set of 

 data, furnished no longer by mere counting of possible suppositions, but by 

 specific knowledge of facts. Such data it should always be our endeavor 

 to obtain; and in all inquiries, unless on subjects equally beyond the range 

 of our means of knowledge and our practical uses, they may be obtained, 

 if not good, at least better than none at all.* 



* It even appears to me that the calculation of chances, where there are no data grounded 

 either on special experience or on special inference, must, in an immense majority of cases, 

 break down, from sheer impossibility of assigning any principle by which to be guided in set- 

 ting out the list of possibilities. In the case of the colored balls we have no diflSculty in mak- 

 ing the enumeration, because we ourselves determine what the possibilities shall be. But 



