THE INDUCTIVE OK INVERSE METHOD. 299 



may happen in addition to those which have been ob- 

 served, we must assign unity for the probability of such 

 new event. The proportional probabilities then become 

 i for a new event, m + i for A, n + i for B, and so on, and 



the absolute probability of A is 



&c. c 



It is very interesting to trace out the variations of 

 probability according to these rules under diverse circum- 

 stances. Thus the first time a casual event happens it is 

 i to i, or as likely as not that it will happen again ; if it 

 does happen it is 2 to i that it will happen a third time ; 

 and on successive occasions of the Hke kind the odds 

 become 3, 4, 5, 6, &c., to i. The odds of course will be 

 discriminated from the probabilities which are successively 

 i> i, ^, &c. Thus on the first occasion on which a person 

 sees a shark, and notices that it is accompanied by a little 

 pilot fish, the odds are i to i , or the probability ^, that the 

 next shark will be so accompanied. 



When an event has happened a very great number of 

 times, its happening once again approaches nearly to cer- 

 tainty. Thus if we suppose the sun to have risen demon- 

 stratively one thousand million times, the probability that it 

 will rise again, on the ground of this knowledge merely, is 

 1,000,000,000+1 Bu( . then the babilit t]mt it wi jj 



1,000,000,000+ I + 1 

 continue to rise for as long a period as we know it to have 



risen is only I ' 000 ' 000 ' 000+I , or almost exactly 4. The 



2,000,000,000 + 1 



probability that it will continue so rising a thousand times 

 as long is only about 5-^77. The lesson which we may 

 draw from these figures is quite that which we should 

 adopt on other grounds, namely that experience never 

 affords certain knowledge, and that it is exceedingly im- 

 probable that events will always happen as we observe 



c De Morgan's 'Essay on Probabilities/ Cabinet Cyclopaedia, p. 67. 



