258 THE PRINCIPLES OF SCIENCE. [CHAP. 



happened increased by one, by the whole number of times 

 the event lias happened or failed increased by two. 



If an event has happened m times and failed n times, 

 the probability that it will happen on the next occasion is 



* 



. Thus, if we assume that of the elements dis- 

 77i + n + 2 



covered up to the year 1873, 5 are metallic and 14 non- 

 metallic, then the probability that the next element dis- 

 covered will be metallic is . Again, since of 37 metals 

 which have been sufficiently examined only four, namely, 

 sodium, potassium, lanthanum, and lithium, are of less 

 density than water, the probability that the next metal 

 examined or discovered will be less dense than water is 



* + ' or -& . 

 37 + 2 39 



We may state the results of the method in a more 

 general manner thus, 1 If under given circumstances cer- 

 tain events A, B, C, &c., have happened respectively m, n, 

 p, &c., times, and one or other of these events must 

 happen, then the probabilities of these events are propor- 

 tional to m + I, n + i, p + i, &c., so that the probability 



of A will be -p ^ _, -r- -5- But if new 



events may happen in addition to those which have been 

 observed, we must assign unity for the probability of suet 

 new event. The odds 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 = 2LJ! 



i + m + i + n + i + &c. 



It is interesting to trace out the variations of probability 

 according to these rules. The first time a casual event 

 happens it is 2 to i that it will happen again ; if it does 

 happen it is 3 to i that it will happen a third time ; and 

 on successive occasions of the like kind the odds become 

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

 from the probabilities which are successively f , f, , &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 2 to i, or the probability f, that "the next 

 shark will be so accompanied. 



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



