370 BISHOP TERROT ON PROBABILITIES. 



2 3 fi 



The opposite probabilities are now ^ and y, and their product is ^, the probability 

 against the conclusion whose probability we are now seeking. Consequently, 

 1-—=^ is the probability for our conclusion, namely, that he did not write the 

 book. But by the former calculation, the probability of the same conclusion was 



12 



found to be -~: and, as these incompatible results follow from the same principle 



and method, the principle and method must be erroneous, 



(3.) The only mathematical attempt at the solution of this problem which I 

 have met with, is at section 15 of the Article Probability, in the Encyclopedia 

 Metropolitana. It is given there as follows :— 



" It is an even chance that A is B, and the same that B is C ; and, therefore, 

 1 to 3 on these grounds alone, that A is C. But other considerations of them- 

 selves give an even chance that A is C. What is the resulting degree of evidence 

 (or the probability) that A is C ?" There is a previous solution which I omit, and 

 then the passage proceeds as follows :— " Let us now treat the preceding question 

 as having two contingencies, the compound argument 1 to 3 for, and the inde- 

 pendent evidence an even chance. We have, therefore, four possible cases. 



Puob. A is C. 



" Argument and Evidence both true, . . -7 x - = - 



3 13 



Argument false, Evidence true, . . . 4 * 2 = 8 



Argument true, Evidence false, . . . - x - = - 

 Argument and Evidence both false, . . =0 



"The sum of these is g as before (for the resulting probability that A is C). 



The above generalized is as follows: — Let a and (I -a) be the probabilities for 

 and against the argument (the conclusion from the argument) ; and e and (1 — e) 

 be the probabilities from any other source. Then the chance that both are wrong 

 is (1-a) . (1-e), and of the contradictory, namely, that (A is C) follows from the 

 one or the other, is 1 — (1— a) . (1 — e)=a + e— a e." 



This is the formula adopted by Whately ; and it is open to the same objec- 

 tion, namely, that by applying it we can arrive at two contradictory conclusions. 

 But, further than this, what is the meaning of A rgument true, Evidence true f 

 The argument and the evidence are here treated as two independent events hav- 

 ing respectively the probabilities of j and ^ ; and their coincidence is represented 



by g. But nothing corresponding to this goes forward in the mind. The argument 



merely affords the information, that for every reason for believing that A is C, there 

 are three equivalent reasons for believing that A is not C. This information we 



