OF THE CALCULATION OF CHANCES. 



353 



should be a majority of the whole : 

 on the contrary, there must be a 

 majority against each kind, except 

 one at most ; and if any kind has 

 more than its share in proportion to 

 the total number, the others collec- 

 tively 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 cannot rationally presume this of 

 any ; which we should do if we were 

 to bet in favour 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 knowledge, 

 such knowledge as we then have, of 

 the laws governing the frequency of oc- 

 currence of the different cases ; but in 

 this case the knowledge is limited to 

 that which, being universal and axio- 

 matic, does not require reference to 

 specific experience, or to any con- 

 siderations arising out of the special 

 nature of the problem under dis- 

 cussion. 



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 respecting the 

 subject. It is plain that, in the case 

 of the coloured balls, a very slight 

 ground of surmise that the white balls 

 were really more numerous than either 

 of the other colours would suffice to 

 vitiate the whole of the calculations 

 made in our previous state of in- 

 difference. It would place us in 

 that position of more advanced know- 

 ledge, 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 

 pounting of possible suppositions, but 



by specific knowledge of facts. Such 

 data it should always be our en- 

 deavour 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 ob- 

 tained, if not good, at least better 

 than none at all.* 



It is obvious, too, that even when the 

 probabilities are derived from observa- 

 tion and experiment, a very slight im- 

 provement in the data, by better obser- 

 vations, or by taking into fuller con- 

 sideration the special circumstances 

 of the case, is of more use than the 

 most elaborate application of the cal- 

 culus to probabilities founded on the 

 data in their previous state of in- 

 feriority. The neglect of this obvious 

 reflection has given rise to misapplica- 

 tions of the calculus of probabilities 

 which have made it the real oppro- 

 brium of mathematics. It is sufficient 

 to refer to the applications made of 

 it to the credibility of witnesses, and 

 to the correctness of the verdicts of 

 juries. In regard to the first, common 

 sense would dictate that it is im- 

 possible to strike a general average of 

 * It even appears to me that the calcula- 

 tion 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 setting out the list 

 of possibilities. In the case of the coloured 

 baUs we have no difficulty in making the 

 enumeration, because we ourselves deter- 

 mine what the possibilities shall be. But 

 suppose a case more analogous to thosa 

 which occur in nature : instead of three 

 colours, let there be in the box all possible 

 colours : we being supposed ignorant of 

 the comparative frequency with which 

 different colours occur in nature, or in the 

 productions of art. How is the list of cases 

 to be made out? Is every distinct shade to 

 count as a colour ? If so, is the test to be a 

 common eye, or an educated eye— a pain- 

 ter's, for instance? On the answer to 

 these questions would depend whether 

 the chances against some particular colour 

 would be estimated at ten, twenty, or per- 

 haps five hundred to one. While if we 

 knew from experience that the particular 

 colour occurs on an average a certain num- 

 ber of times in every hundred or thousand, 

 we should not require to know anything 

 either of the frequency or of the number of 

 the other possibilities. 



