298 THE PRINCIPLES OF SCIENCE. 



pened increased by one, by the same number increased by 

 one and the number of times it is to happen. 



An event having happened m times without fail, the 



probability that it will happen n more times is - 



ra-f n+i 



Thus the probability that three new planets would obey 

 Bode's law is ff , but it must be allowed that this, as well 

 as the previous result, would be much weakened by the 

 fact that Neptune can barely be said to obey the law. 



3. An event having happened and failed a certain 

 number of times, to find the probability that it will happen 

 the next time, divide the number of times the event has 

 happened increased by one, by the whole number of times 

 the event has happened or failed increased by tu?o. 



Thus, if an event has happened m times and failed n times, 

 the probability that it will happen on the next occasion 



m+ I 



m " 



m-\-n+ 2 



Thus, if we assume that of the elements yet discovered 

 50 are metallic and 14 non-metallic, then the proba- 

 bility that the next element discovered will be metallic 



is 6 ^. 



Again since of 37 metals which have been sufficiently 

 examined only four, namely, sodium, potassium, lan- 

 thanum 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, 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 tom+i, n+ i, p+ i, &c. so that the probability 



of A will be - m+1 But if new events 



+ &c. 



