The conception of probability as a matter of fact, i.e., as the proportion of times in which an occurrence of one kind is accompanied by an occurrence of another kind, is termed by Mr. Venn the materialistic view of the subject. But probability has often been regarded as being simply the degree of belief which ought to attach to a proposition, and this mode of explaining the idea is termed by Venn the conceptualistic view. Most writers have mixed the two conceptions together. They, first, define the probability of an event as the reason we have to believe that it has taken place, which is conceptualistic; but shortly after they state that it is the ratio of the number of cases favorable to the event to the total number of cases favorable or contrary, and all equally possible. Except that this introduces the thoroughly unclear idea of cases equally possible in place of cases equally frequent, this is a tolerable statement of the materialistic view. The pure conceptualistic theory has been best expounded by Mr. De Morgan in his Formal Logic: or, the Calculus of Inference, Necessary and Probable.
The great difference between the two analyses is, that the conceptualists refer probability to an event, while the materialists make it the ratio of frequency of events of a species to those of a genus over that species, thus giving it two terms instead of one. The opposition may be made to appear as follows:
Suppose that we have two rules of inference, such that, of all the questions to the solution of which both can be applied, the first yields correct answers to 81/100, and incorrect answers to the remaining 19/100; while the second yields correct answers to 93/100, and incorrect answers to the remaining 7/100. Suppose, further, that the two rules are entirely independent as to their truth, so that the second answers correctly 93/100 of the questions which the first answers correctly, and also 93/100 of the questions which the first answers incorrectly, and answers incorrectly the remaining 7/100 of the questions which the first answers correctly, and also the remaining 7/100 of the questions which the first answers incorrectly. Then, of all the questions to the solution of which both rules can be applied—
both answer correctly 93/100 of 81/100 or 93/100 x 81/100;
the second answers correctly and the first incorrectly 93/100 of 19/100 or 93/100 x 19/100;
the second answers incorrectly and the first correctly 7/100 of 81/100 or 7/100 x 81/100;
and both answer incorrectly 7/100 of 19/100 or 7/100 x 19/100;
Suppose, now, that, in reference to any question, both give the same answer. Then (the questions being always such as are to be answered by yes or no), those in reference to which their answers agree are the same as those which both answer correctly together with those which both answer falsely, or 93/100 x 81/100 + 7/100 x 19/100 of all. The proportion of those which both answer correctly out of those their answers to which agree is, therefore—
((93 × 81)/(100 × 100))/((93 × 81)/(100 × 100)) + ((7 × 19)/(100 × 100)) or (93 × 81)/((93 × 81) + (7 × 19)).
This is, therefore, the probability that, if both modes of inference yield the same result, that result is correct. We may here conveniently make use of another mode of expression. Probability is the ratio of the favorable cases to all the cases. Instead of expressing our result in terms of this ratio, we may make use of another—the ratio of favorable to unfavorable cases. This last ratio may be called the chance of an event. Then the chance of a true answer by the first mode of inference is 81/19 and by the second is 93/7; and the chance of a correct answer from both, when they agree, is—