96 Prof. Boole on the conditions by which the Solutions of 

 From the three first equations and the last we find 



^—p—q — r 



furnishing the conditions 



p-^q^r, p + r = q, q + r^p, p-{-q + r~2. 



There still remains the fourth equation of the system (9), in the 

 first member of which, substituting for \ its value, we find 



p + q—r 



the value of Prob. xy sought (Laws of Thought, p. 280). 



There is a peculiarity in these " determinate '^ solutions to 

 which I desire to advert. It is, that if in any series of observa- 

 tions the events referred to in the data occur with a frequency 

 exactly proportional to their assigned probability, the event 

 whose probability is sought will occur in the same series with a 

 frequency exactly proportionate to its determined probability. 

 For instance, in the problem just solved, if in n observations the 

 events a?, y, and x{l — y) -\-y(l—a:) occur exactly np, nq, and nr 

 times respectively, the event xy will in the course of the same 



T) ~f~ O ~~- 7' . 



observations occur exactly n- — ^ times. This is easily shown 



by substituting throughout the demonstration contained in 

 Prop. 2, 'Ncc for Prob. x, Ny for Prob. y, N applied to the ex- 

 pression of any class denoting the number of individuals con- 

 tained in that class; and generally substituting numbers for 

 probabilities. This change will not afiect the truth of the equa- 

 tions. For instance, if we have 



Na?=a ^xy=b, 

 we shall have 



Na? — Na?^ = « — 6, 

 or 



Na?(l— y) = a— Z», 



and so on. I remark that this is a. peculiarity of the above de- 

 terminate solutions. If the probabilities of two independent 

 events x and y are p and q respectively, the probability of their 

 concurrence is pq ; but we are not permitted to affirm, that if in 

 n observations n occurs np times, and y occurs nq times, their 

 concurrence will be observed exactly npq times. « 



When by the method of this chapter we have found the con- 

 ditions of limitation of the solution of a question in the theory 

 of probabilities, we can at once ascertain from those conditions 



