466 Mr. H. Wilbraham on the Theory of Chances 



failing. Let the chances of these four contingencies be respect- 

 ively 6, \, fx,, (f). To determine these we have the equations 

 6 + \ + fi + (}) = l, d + fjb = a, 6 + \ = b. 



Another equation is given in Professor Boole's assumption that 

 A is as likely to happen if B happen as it is if B fail, viz. 



e a e \ 



— or — =— . 



The same equation is given by the condition that B is as likely 

 to happen if A happen as if it fail. These four equations deter- 

 mine the values of 6, X, //., (j). Again, suppose three simple 

 events A, B, C, the chances of which are a, b, c. There are here 

 eight possible cases, {6) A, B, C all happening, (X,) B and C but 

 not A, (ytt) A and C not B, (v) A and B not C, (p) A not B or C, 

 (cr) B not A or C, (t) C not A or B, (0) all failing. Denoting 

 the chances of these several contingencies by the Greek letters 

 prefixed to them, we have the equations 



d + X + fi-\-v-\-p + a- + T + <}) = l 

 6-i-fi + v + p = a 

 d-^\ + v + a- = b 



6 + \+lJb + T=C. 



Professor Boole's assumption of the independence of the simple 

 events completes the system of equations necessaiy to determine 

 the unknown quantities. It gives tlie equations 



fi V _p ^ _^ ^'^ _^ ^_^_'"'_'^ 



\ T <T <^' p. T p (f)' V O- P 4>' 



which comprise, in fact, four independent equations, from which, 

 together with the first four, the unknown quantities may be de- 

 termined algebraically. 



That Professor Boole's method does in such cases as the two 

 just mentioned tacitly make the assumptions stated, is evident 

 as well a posteriori as « priori. For, in the first case, if we seek 

 to find by the Professoi-'s logical equations the chance of A and 

 B both occurring, we find it to be ab, that of B and not A {\. — d)b, 

 and so on, which necessarily imply the condition I have stated 

 to be assumed. So in the second case, we should find the chance 

 of A, B, C all happening to be aba, that of B and C but not A 

 {\ — a)bc, and so on, which imply the four additional assumed 

 conditions. 



Now let us pass to the cases where certain conditions among 

 the chances of the several events are given. In the first question 

 which I have stated, where there are only two simple events A 

 and B, suppose there to be another given relation among the 



