440 PROCEEDINGS OF THE AMERICAN ACADEMY 



As another illustration, suppose that a die, of which four sides are 

 white, is thrown thirty times. If the die is fair, white every time is 

 evidently more probable than any other system of throws, supposing 

 that in specifying the system we state what color is thrown on each spe- 

 cific trial. Yet such a result would prove beyond all reasonable doubt 

 that the die was not fair. This statement will not appear paradoxical, 

 if we consider that a constant series of throws of one color would 7iot 

 be very improbable on the supposition that the die was loaded, while 

 they would be very improbable on the supposition that the die was fair. 

 "We therefore select the most probable series of circumstances, and say 

 that the die is not fair. If an indiscriminate series of white and black 

 throws result, the former being about twice as numerous as the latter, 

 such result, describing what color was thrown on each throw, would be 

 still more improbable than a constant series of whites on the supposition 

 that the die was fair. But the former result would be billions of times 

 less probable than the latter on the supposition that the die was loaded ; 

 so that, if the former result occur, we select the supposition of a fair 

 die as the more probable. 



It will be perceived that the degree of speciality with which a phe- 

 nomenon is described affects very materially its a priori probability, 

 but the full development of the results of this fact is reserved for a pa- 

 per on the applications of the theory of probabilities to natural phenom- 

 ena. I may remark, however, that much confusion has arisen from 

 confounding the different degrees of probability which a proposition will 

 have when expressed in the different forms, 



A is X, 



AisV, 



A is Z, Sec, 

 when all V is X, all Z is V, &c., but V is an exceeding small portion 

 of X, Z of V, &c. In such a case the a priori probabilities of the suc- 

 cessive propositions will diminish with great rapidity. If, in the case of 

 the above-mentioned die, supposed fair, one were to guess that the 2d, 

 5th, 8th, Sec. throws would be black, and all the rest white, he would 

 be 1024 times more likely to be wrong than if he guessed that they 

 would all be white. But if he guessed simply that twenty throws would 

 be white, and ten black, which guess would include the former, he would 

 be myriads of times more likely to be right than if he guessed that they 

 would all be white. 



