638 PROFESSOR BOOLE ON THE COMBINATION 



the conditions of possible experience being that c, c, p, q, and m, should be posi- 

 tive proper fractions, subject to the relation 



c + c'<l + m (10) 



that root of (9) being taken, which satisfies the conditions 



= cp = c'q = .. 



w < — , w < — , w < 1 

 m m 



, =rC'l-q 



l — w<- *•. 



m 



Now if in (9) we suppose m to vanish, we find 



cpc'q (1 — w) = wc 1 — p d 1 — q 



.'. pq l — w = w \—p 1 — q 



whence w = ,-. pq . ... , (11) 



The condition (10) becomes simply c + d < 1. The remaining conditions are all 

 satisfied by the value of w. 



The formula (11), which in the present investigation appears as a kind of 

 limiting value, applicable only to cases in which the presumption for or against 

 the event z increases most by the combination of the testimonies given, is usual- 

 ly regarded as expressing the general solution. The reasoning by which it is 

 supposed to be established is the following. 



Let p be the general probability that A speaks truth, q the general probability 

 that B speaks truth ; it is required to find the probability, that if they agree in a 

 statement they both speak truth. Now, agreement in the same statement im- 

 plies that they either both speak truth, the probability of which beforehand is 

 pq, or that they both speak falsehood, the probability of which beforehand 

 is (1— p) (1 — ?)• Hence the probability beforehand that they will agree is 

 pq+(l-p) (1-g), and the probability that if they agree, they will agree in speak- 

 ing the truth, is accordingly expressed by the formula (11).* In the case of n, tes- 

 timonies whose separate probabilities are p x p 2 . • ./>„, the corresponding formula is 



P1P 2 • • P* M 2 ) 



PlP2 ■ ■ Pn+(±-Pl) (l-2> 2 ) " • Q-Pn) 



In applying which, it is usual to regard one of the testimonies as the initial testi- 

 mony of the mind itself, f Substantially the same reasoning is applied to deter- 

 mine the probability of correctness of a decision pronounced unanimously by a 

 jury, the probabilities of a correct decision by each member of the jury being given 

 In this reasoning there is no recognition that it is to the same fact that the 

 several testimonies are borne. Take the case of two testimonies, and the problem 



* Cournot Exposition de la Theorie des Chances, p. 411. De Morgan, Formal Logic, p. 191. 

 f Formal Logic, p. 195. 



