CALCULATION OF CHANCES. 321 



conclusions respecting its certainty ; namely, not our ignorance, but 

 our knowledge : knowledge -obtained by experience, of the proportion 

 between the cases in which the tact occurs, and those in whi(;h it does 

 not occur. Every 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 cajios, it occurs in about so many. The fraction 

 \vhich mathematicians use to designate tlie probability of an event, 

 is the ratio of these two immbers ; the ascertained proportion between 

 the number of case^ 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 description of cases con- 

 cerned are tlirows, and the probability of cross is one half, because it 

 is found that if we throw often enough, cross is thrown about once in 

 every two throws ; and because this induction is made under circum- 

 stances justifying the belief that the propt)rtion will be the same in 

 other cases as in the cases examined. In the cast of a die, the proba- 

 bility of ace is one-sixth ; not, as Laplace would say, because there 

 are six possible throws, of vphich aqe is one, and because we do not 

 know any reason why one should turn up rather than another ; but 

 because wo do know that in a hundred, or a million of throws, ace will' 

 be throAvn about one-sixth of that number, or once in six times. 



Not only is this third condition indispensable, but if we have that, 

 we do not want Laplace's two. It is not necessary that we should 

 knoAV how many possibilities there are, or that we should have no 

 more reason for expecting one of them than another. If a north wind 

 blows one day in every ten, the ptobability of a north wind on any 

 given day will be one-tenth, even though of. the remaining possibilities 

 a west wind should be greatly the most probable. , If we know that 

 half the trees in a particular forest are oaks, though we may be. quite 

 igirorant how many other kinds of trees it contains, the chance that a 

 tree indiscximinately selected will bo an oak is an even chance, or, in 

 mathematical language, one-half. So that the condition which Laplace 

 omittqd is not merely one of the requisites for the possibility of a cal- 

 culation of chances ; it is the only requisite. . , 



In saying that he has omitted this condition, I am far firom meaning 

 to assert, that he does not frequently take it into consideration in par- 

 ticular instances: nor indeed could he fail to do so, since whenever 

 any experience bearing upon the case really exists, he would naturally 

 consult that experience to assure himself of the fulfilment of his second 

 condition, that there be no reason for expecting one event rather tjian 

 another. When experience is to be had, he takes that experience as 

 the measure of the probabiUty : his error is only in imagining that 

 there can be a measurement of probability where there is no expe- 

 rience. The consequence of this en-or has been his adoption of con- ' 

 elusions n(»t indeed contrary to, but unsupported by, experience. He 

 has been led to push the theoiy and its applications beyond the bounds 

 which confine all legitimate inferences of the human mind ; by extend- 

 ing them to subjects on which the absence of any giound fur deter- 

 mining between two suppositions, does not arise from our having equal 

 grounds for presuming both, but from our haring an equal absence of 

 gi-ounds for piesuming either. 

 Ss 



