Mb. ELLIS, ON THE FOUNDATIONS OP THE THEORY OF PROBABILITIES. 5 



be very much weakened. At present, we unhesitatingly exclude the comets on account of 

 their striking peculiarities : in the case supposed we should with equal confidence include them 

 in the induction. But at what precise point of their transition-state are we abruptly, from 

 giving them no weight at all in the induction, to give them as much as the old planets ? 



(15.) It is difficult to acquiesce in a theory which leads to so many conclusions seem- 

 ingly in opposition to the common sense of mankind. 



One of the most singular of them may, perhaps, serve as a key to explain their nature. 

 When any event, whose cause is unknown, occurs, the probability that its a priori pro- 

 bability was greater than ^ is £. Such at least is the received result. But in reality, 

 the a priori probability of a given event has no absolute determinate value independent of 

 the point of view in which it is considered. Every judgment of probability involves an 

 analysis of the event contemplated. We toss a die, and an ace is thrown. Here is a com- 

 plex event. We resolve it into, (l) the tossing of the die ; (2) the coming up of the ace. 

 The first constitutes the ' trial,' on which different possible results might have occurred ; the 

 second is the particular result which actually did occur. They are in fact related as genus 

 and differentia. Beside both, there are many circumstances of the event ; as how the die 

 was tossed, by whom, at what time, rejected as irrelevant. 



This applies in every case of probability. Take the case of a vessel sailing up a river. 

 The vessel has a flag. What was the a priori probability of this .'' Before any answer 

 can by possibility be given to the enquiry, we must know (l) what circumstances the person 

 who makes it rejects as irrelevant. Such as, e. g. the colour of which the vessel is painted, 

 whether it is sailing on a wind, &c. &c. ; (2) what circumstances constitute in his mind the 

 ' trial ;' the experiment which is to lead to the result of flag or no flag ; must the vessel 

 have three masts ? must it be square rigged ? (3) What idea he forms to himself of a flag. 

 Is a pendant a flag ? Must the flag have a particular form and colour ? Is it matter of 

 indifference whether it is at the peak or the main ? Unless all such points were clearly under- 

 stood, the most perfect acquaintance with the nature of the case would not enable us to say 

 what was the a priori probability of the event : for this depends, not only on the event, 

 but also on the mind which contemplates it. 



The assertion therefore that | is the probability that any observed event had on an a priori 

 probability greater than i, or that three out of four observed events had such an a priori pro- 

 bability, seems totally to want precision. A priori probability to what mind ? In relation to 

 what way of looking at them .'' 



(16.) Let us see if this will throw any light on the question. Let A be a large number. 

 And suppose we took h trials and that the probability of a certain event from each (considered 



in a determinate manner) was — ; let us take a second set of h trials for which the same quantity 



.2 m- 1 , 



IS — : and so on to and 1. 



m m 



When the trials have taken place, we shall have approximately, 



/I 2 m-1 \ 



h {- + - + + + 1 



\m m ml 



of the sought events. Of these 



Me%i)^(i^3* -)• 



had a priori a probability greater than ^. Summing these series and dividing the second by 



, „ 3m + 2 

 the first, we get , for the ratio which the latter class of events bears to the total number. 



The limit of this, when m is infinite, or when we take an infinite number of sets of trials is |, 

 which is the received result. 



