258 THE PRINCIPLES OF SCIENCE. [CHAP. 



happened increased ~by one, ly the whole number of times 

 the event has happened or failed increased ly two. 



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

 the probability that it will happen on the next occasion is 



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



7/1 + 71 + 2 



covered up to the year 1873, 50 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 

 A+JL or J . 



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 



<> f A wm be ,+. -+ .+;*+', +I+fa- But if new 



events may happen in addition to those which have been 

 observed, we must assign unity for the probability of such 

 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 - -p _/" & 



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 , 4, &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 Be Morgan's Essay on Probabilities, Cabinet Cyclopaedia, p. 67. 



