CHAP. XVI.] OF THE THEORY OF PROBABILITIES. 249 



tions already stated, and may be regarded as indicating the degree 

 in which it has been found possible to render those definitions 

 available. 



PRINCIPLE 1st. If p be the probability of the occurrence of 

 any event, I p will be the probability of its non-occurrence. 



2nd. The probability of the concurrence of two independent 

 events is the product of the probabilities of those events. 



3rd. The probability of the concurrence of two dependent 

 events is equal to the product of the probability of one of them 

 by the probability that if that event occur, the other will happen 

 also. 



4th. The probability that if an event, E, take place, an event, 

 F, will also take place, is equal to the probability of the concur- 

 rence of the events E and F, divided by the probability of the 

 occurrence of E. 



5th. The probability of the occurrence of one or the other of 

 two events which cannot concur is equal to the sum of their se- 

 parate probabilities. 



6th. If an observed event can only result from some one of n 

 different causes which are d priori equally probable, the proba- 

 bility of any one of the causes is a fraction whose numerator is the 

 probability of the event, on the hypothesis of the existence of that 

 cause, and whose denominator is the sum of the similar proba- 

 bilities relative to all the causes. 



7th. The probability of a future event is the sum of the pro- 

 ducts formed by multiplying the probability of each cause by 

 the probability that if that cause exist, the said future event 

 will take place. 



8. Respecting the extent and the relative sufficiency of these 

 principles, the following observations may be made. 



1st. It is always possible, by the due combination of these 

 principles, to express the probability of a compound event, de- 

 pendent in any manner upon independent simple events whose 

 distinct probabilities are given. A very large proportion of the 

 problems which have been actually solved are of this kind, and, 

 the difficulty attending their solution has not arisen from the in- 

 sufficiency of the indications furnished by the theory of proba- 

 bilities, but from the need of an analysis which should render 



