Page:Chance, love, and logic - philosophical essays (IA chancelovelogicp00peir 0).pdf/128

 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—