BISHOP TEKHOT ON PROBABILITIES. 373 



may have been np white and nq—np black, where n is any whole number from 

 one to infinity. In like manner, the second may have consisted of nr white 

 and ns-nr black, where n is any number from one to infinity. Any one as- 

 sumed state of the first introduction may have co-existed with any assumed state 

 of the second ; and thus assuming that the first contained p white and q-p black, 

 we have the infinite series of probabilities, 



p + r p + 2 r p + nr 



q+s—p—r , q+2s—p—2r ' q+ns—p—nr' 



Again, assuming that the first contained 2p white, and 2q-2p black, we have 



2p + r 2p + 2r 2p + nr 



2q + s — 2p — r , 2q+2s — 2p — 2r ' 2 q+ns — 2p — nr' > 



and so on ad infinitum,. 



This infinite series of infinite series I cannot sum. If they can be summed, 



then their sum divided by the infinite of the second order n 2 , is the probability 



required. 



In no case, except when -=-, so far as I see, can the sum of their sums, or 



the whole probability, be determinately expressed. When -=-, the fractions being 



in their lowest terms, p=r and q=s. The two pieces of information are then 

 identical; the same information is given by both observers; and the information, 

 unaffected by the repetition, is absolutely received by the third party : and this 

 is the result, if, in the foregoing series, we substitute p for r and q for s. 



(8.) If we revert to the expression (3) given in the Encyclopaedia Metropoli- 

 tan, where the separate probabilities are a and e, and their conjoint force is 

 stated to be a + e — a f, it would follow that the effect produced by two observers 

 making the same statement as to the probability of an event should be twice the 

 asserted probability minus its square. Now, in the case of a repetition of the 

 same probability by two observers, it must, I think, be allowed that my result is 

 conformable to that of which we are all conscious. If, for example, the North- 

 ampton and the Carlisle Tables both give ^ as the probability that a man of 



thirty will live to the age of fifty, and are both implicitly believed, we believe that 

 there is an even chance of his living to fifty, and not, as would follow from the 

 expression given in the Encyclopaedia, that the chances are three to one in his 

 favour. 



(9.) It perhaps deserves to be noticed, that when a second series of observa- 

 tions or experiments is added to one previously admitted, the probability is not 

 increased by the mere preponderance of favourable over unfavourable cases in 

 the second series. To increase the probability, the ratio of favourable to unfavour- 

 able cases must be greater in the second series than in the first. For the first 



received probability is -, and the composite is ^— -. (6.) 



vol. xxi. part in. 5 i 



