, \ ', . the Theory of Probabilities. 357 



means of balls, dice, &c., but also are enabled to keep more 

 clearly in view the connexion between first principles and the 

 forms of solution in complex cases. I propose now to illustrate 

 their use by deducing from them what may be called the three 

 fundamental propositions of the science. 



8. Theorem I. — IfV be the probabilitij of a hypothesis H, and 

 p the probability that if II be true another hypothesis h is true; 

 then the probability that H and h are true is Pp. 



For the whole quantity of l^elief which we give to the truth of 

 H is made up of the quantities given to the two combinations 

 H trae and h true, H true and h false. The sum of the two 

 latter quantities is therefore P, and their ratio is the same as it 

 would be if H were discovered to be true, namely, p : 1 —pi. 

 Hence their values are Yp and P(l —p). 



This obviously includes the case in which II and h are entirely 

 independent, and P, p their respective probabilities. 



Theorem II. — Let H,, Hg, . . . Hn be mutually exclusive and 

 exhaustive h/potheses, lohose probabilities {relative to a certain state 

 of information) are V^, Po, • • • P„. And let P; be the probability 

 that another hypothesis h is true if U. be true. If it be afterwards 

 discovered that h is true, the probability of H; becomes 



V.p. 



For before the discovery of the truth of h, the quantities of 

 belief wliich we give to the combinations (H,, h), {11^, h), . . . 

 (H„,/i) are P,;^i, V^p^, . . . P„i>„. After the discovery, all other 

 combinations are excluded ; and the probabilities of these com- 

 binations retain the same ratios as before, but their sum becomes 



wards discovered that they cannot be both true, no other knowledge being 

 gained about either of them. What arc their new i)robabiUties ? 



Before the (hscovery, tlie probabihties of the four (exclusive and exhaust- 

 ive) hvpotheses, A true and B false, A false and B true, both false, both 

 true, were i-espectively «(l-i), />(1— a),(l-«)(l-i), cb. After the dis- 

 covery, the last hypothesis is excluded, and the new probabdities (say x, y, z) 

 of the other three are proportioyial to theii- former values ; but their sum 

 being now =1, we have 



X J/ - __ 1 



ail^)" b(X-a) (l-a)(l-i) 1-aO' 



where x and y are obviously the new probal)ilities of .\ and B. If it were 

 further discovered that A and B could not be hot/i false, we should then 

 have 



!c _ y — ___L___, 



u(\ — h)' b(\—a) u + b—2ab' 

 (Compare Prof. De Morgan's pai)er referred to below, \). 27.) 



