CALCULATION OP CHANCES. 319 



numl>er of instances in which they have been found together, but by 

 the excess of that number above the number due to the absolute fre- 

 quency of A. If, for example, A exists always, and therefore coexists 

 with everything, no number of instances of its coexistence with B 

 would prove a connexion ; as in our example of the fixed stars. If A 

 be a fact of such common occurrence that it may be presumed to be 

 present in half of all the cases that occur, and therefore in half the 

 cases in which B occurs, it is only the proportional excess above half, 

 that are to be reckoned as evidence towards proving a connexion 

 between A and B. 



In addition to the question, What is the number of coincidences 

 which, on an average of a great multitude of trials, may bo i^xpected 

 to arise from chance alone ?. there is also another question, namely, Of 

 what extent of deviation from that average is the occuiTence credible, 

 fi-om chance alone, in some number of instances smaller than that 

 which constitutes a fair average ? It is not only to be considered what 

 is the general result of the chances in the long run, but also what are 

 the exti-eme limits of variation from that general result, which may 

 occasionally be expected as the result of some smaller number of 

 instances. 



The consideration of the latter question, and any consideration of 

 the former beyond that already given to it, belong to what mathema- 

 ticians term the doctrine of chances, or, in a phrase of greater preten- 

 sion, tlie Theory of Probabilities. An attempt at a philosophical ajxpre- 

 ciation of that doctrine is, therefore, a necessary portion of our task. 



CHAPTER XVIII. 



OF THE CALCULATION OF CHANCES. 



§ 1. " Probability," says Laplace,* " has reference partly to our 

 ignorance, partly to our knowledge. We know that among three or 

 more events, one, and only one, must happen ; but there is nothing 

 leading us to believe that any one of them will happen rather than the 

 others. In this state of indecision, it is impossible for us to pronounce 

 with certainty on their occurrence. It is, however, probable that any 

 one of these events, selected at pleasure, will not take jjlace ; because 

 we perceive several cases, all equally possible, which exclude its oc- 

 cuiTcnce, and only one which favors it." 



Such is this great mathematician's statement of the logical founda- 

 tion upon which rests, according to him, the theory of chances : and if 

 his um-ivaled command over the means which mathematics supply for 

 calculating the results of given data, necessarily implied an equally 

 surt; judgment of what the data ought to be, I should luu"dly dare give 

 utterance to my conviction, that in this opinion he is entirely wrong ; 

 that his foundation is altogether insuflRcient for the superstnu^ture 

 erected upon it ; and that there is implied, in all rational calculation 

 of the probabilities of events, an essential condition, which is cither 



* Essai Philosophique sur les Probabililis, fifth Palis edition, p. 7. 



