BISHOP TERROT ON PROBABILITIES. 373 
may have been mp white and x q—np black, where m is any whole number from 
one to infinity. In like manner, the second may have consisted of m7 white 
and ms—n7 black, where ” 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 g—p black, 
we have the infinite series of probabilities, 
ptr fears _ ptnr 
Qts—p—r’?qt2s—p—zr--"*** qt+ns—p—nr- 
Again, assuming that the first contained 2 white, and 2 g—2p black, we have 
2p+r 2pt+2r 2p+nr 
2q+s—2p—r’ 2qg+2s—2p—Q@r- °°" 2qtns—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 ’, is the probability 
required. 
Tr 4 
In no case, except when fat so far as I see, can the sum of their sums, or 
the whole probability, be determinately expressed. When ar fractions being 
in their lowest terms, p=7r and g=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 Encyclopedia Metropoli- 
tana, where the separate probabilities are @ and ¢, and their conjoint force is 
stated to be a+ e—ae, 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 manus 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 5 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 Encyclopzedia, 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 a and the composite is == (6.) 
VOL. XXI. PART III. 5I 
