Questions in the Theory of Probabilities are limited. 93 



General proposition. — The probabilities of any events whose 

 logical expression is known being represented hj p, q, r , . re- 

 spectively^ required the conditions to which those quantities are 

 subject. 



Here also it may be well to commence with a particular case. 

 I will take the problem already discussed in this Journal (Oct. 

 1853, Jan. 1854) by Mr. Cayley and myself. The elements of 

 that problem may be thus expressed, w being the element sought 

 in that discussion. 



Prob. x=Ci Prob. y = C2 Prob. xz=:c^p^ Prob. y^f^c^p^ 



Prob. ^=m; Prob. ^(1 -a?) (1-2/)= 0. . (1) 



Here, according to the notation of the calculus of logic, Prob. xsf 

 denotes the probability of the occurrence of the events x and z 

 together. Prob. 2{l—£c){l—y) denotes the probability of the 

 occurrence of z conjointly with the absence of x and ?/, &c. 



The events whose probabilities are given may all be resolved 

 by logical development into disjunctive combinations of events, 

 which do not admit of further resolution with reference to the 

 same elements of distinction x, y, z. Thus 



xz — xzy -f xzi^ — y) 



x-=.xyz -{■ xy[l-' z) -{■ x[].—y)z -\- {\—x){\. — y){Y— z). 



And hence we have 



Vvoh. xz=^Vvoh. xyz+'?voh, x[\—y)zy , . (2) 



and so on. Now assume 



V Yoh. xyz-=X Prob. ^y(l — 2*) =/Lfc Vxoh. x[\—y)z — v 



Prob. <2?(1— 2/)(l— 2')=p Prob. {\'-'x)yz=.o- 



Prob. (1-%(1-^) = T Prob. (1 - ^) (1-2/) (l-^) = t;. 



These represent all the possible combinations of x, y and z, ex- 

 cept z[\^x){\—y)y which by the data is excluded. 

 The equation (2) gives, by virtue of (1), 



and forming all similar equations furnished by the data, we have 



\ + IM + V + p = C^ 



X + v =Cii?i )') ' ' ' • • (^) 



X + O- =^2i^2 ' 



X + v + o- =^w J 

 to which we may add the necessary condition 



X + //- + v + p + cr-f T + f = 1. ... (4) 



