OF THE CALCULATION OF CHANCES. 383 



ner, or at least a majority, and thus render a wrong instead of a right de- 

 cision tnore probable the more the number was increased. 



These are but samples of the errors frequently committed by men who, 

 having made themselves familiar with the difficult formulae which algebra 

 affords for the estimation of chances under suppositions of a complex char- 

 acter, like better to employ those formulas in computing what are the prob- 

 abilities to a person half informed about a case than to look out for means 

 of being better informed. Before applying the doctrine of chances to any 

 scientific purpose, the foundation must be laid for an evaluation of the 

 chances, by possessing ourselves of the utmost attainable amount of posi- 

 tive knowledge. The knowledge required is that of the comparative fre- 

 quency with which the different events in fact occur. For the purposes, 

 therefore, of the present work, it is allowable to suppose that conclusions 

 respecting the probability of a fact of a particular kind rest on our knowl- 

 edge of the proportion between the cases in which facts of that kind occur, 

 and those in which they do not occur; this knowledge being either de- 

 rived from specific experiment, or deduced from our knowledge of the 

 causes in operation which tend to produce, compared with those which 

 tend to prevent, the fact in question. 



Such calculation of chances is grounded on an induction ; and to render 

 the calculation legitimate, the induction must be a valid one. It is not 

 less an induction, though it does not prove that the event occurs in all 

 cases of a given description, but only that out of a given number of such 

 cases it occurs in about so many. The fraction which mathematicians use 

 to designate the probability of an event is the ratio of these two numbers ; 

 the ascertained proportion between the number of cases in which the event 

 occurs and the sum of all the cases, those in which it occurs and in which 

 it does not occur, taken together. In playing at cross and pile, the descrip- 

 tion of cases concerned are throws, and the probability of cross is one-half, 

 because if we throw often enough cross is thrown about once in every two 

 throws. In the cast of a die, the probability of ace is one-sixth ; not sim- 

 ply because there are six possible throws, of which ace is one, and because 

 we do not know any reason why one should turn up rather than another — 

 though I have admitted the validity of this ground in default of a better — 

 but because we do actually know, either by reasoning or by experience, 

 that in a hundred or a million of throws ace is thrown in about one-sixth 

 of that number, or once in six times. 



§ 4. I say, " either by reasoning or by experience," meaning specific ex- 

 perience. But in estimating probabilities, it is not a matter of indifference 

 from which of these two sources we derive our assurance. The probabili- 

 ty of events, as calculated from their mere frequency in past experience, 

 affords a less secure basis for practical guidance than their probability as 

 deduced from an equally accurate knowledge of the frequency of occur- 

 rence of their causes. 



The generalization that an event occurs in ten out of every hundred cases 

 of a given description, is as real an induction as if the generalization were 

 that it occurs in all cases. But when we arrive at the conclusion by mere- 

 ly counting instances in actual experience, and comparing the number of 

 cases in which A has been present with the number in which it has been 

 absent, the evidence is only that of the Method of Agreement, and the con- 

 clusion amounts only to an empirical law. We can make a step beyond 

 this when we can ascend to the causes on which the occurrence of A or its 



