KiDD. — On ProhahiUty. Iv 



considered as divisible into so many equal parts as there are units in the 

 denominator ; and the proposition in question predicates respecting so many of 

 those parts as there are units in the numerator. In the first of the three 

 instances that we have examined, the denominated quantity is the days of 

 June, considered as 30 in number, and all equal to each other j of which 30 the 

 proposition in question predicates respecting two-thirds. In the second 

 instance, the unit of the denominated quantity is two throws of a coin ; and 

 the denominator is the number of the modes in which these two throws may 

 be varied. In the last of our examples the unit is a member of each of the 

 classes A and B ; the denominator, 7, is the smallest number of instances that 

 can exhibit an average specimen j and the numerator, 6, is the number of the 

 instances to which the proposition in question applies. 



I have referred thus explicitly to the quantities which furnish the fraction 

 of probability, for the purpose of its being rendered palpably plain that the 

 basis of the probability is objective. For general purposes the fraction of 

 probability is sufficiently interpreted by our saying, that the denominator of 

 the fraction is the number of equally probable alternatives into which the case 

 is considered as distinguishable, and that the numerator is the number of those 

 alternatives which affirm the proposition in question. 



13. (Hypothetic estimate of Alternatives.) — The supposed simple cases of 

 probability which we have taken as examples for analysis, belong to that class 

 of probabilities in which the alternatives can be distinctly perceived to be 

 equal. But in the actual affiiirs of life this condition is less frequently realized. 

 In such cases, nevertheless, the numerical notation of probability is sometimes 

 employed for the purpose of more convenient discussion, especially in the 

 combination of probabilities. In the formation of such hypothetic estimates 

 the import of the probability, and of the fraction representing it, is essentially 

 similar to the foregoing. Let us, for example, suppose the proposition in 

 question to be, that the author of the Letters of Junius was Sir Philip 

 Francis j and let us suppose that we estimate the probability of this proposi- 

 tion as being adequately represented by the fraction four-fifths. We cannot, 

 it may be assumed, distinctly assign five equal alternatives, as constituting the 

 case ; but supposing that we have found four-fifths in favor of the given 

 proposition to be a fitting representative of the probability, we consider that if 

 the case, as known to us, were distinguished into equal alternatives, then 

 about four-fifths of such alternatives would affirm the given proposition. In 

 passing from a case of numerically definite quantities in the basis of the 

 alternatives, to a case not susceptible of a like distinctness of enumeration, we 

 lose the categorical precision in the data ; but it is still the mutual relation of 

 the same two things that determines the probability of the proposition, viz., 

 the relation of what is asserted in the proposition in question to the data that 

 are known. 



