370 BISHOP TERROT ON PROBABILITIES. 
6 
35? 
against the conclusion whose probability we are now seeking. Consequently, 
6 29. 
iy 35 18 
book. But by the former calculation, the probability of the same conclusion was 
The opposite probabilities are now A and > and their product is =, the probability 
1 the probability for our conclusion, namely, that he did not write the 
found to be =F 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 Ais 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. 
Pros. A is C, 
G 1 i Uaeliga 
« Aroument and Evidence both true, ao aie a 
: ose lie 
Argument false, Evidence true, ASRS Ss 
5 ere lava ne 
Argument true, Evidence false, ies 
= (0, 
Argument and Evidence both false, 
‘The sum of these is : as before (for the resulting probability that A is C). 
The above generalized is as follows:—Let a and (1—a) be the probabilities for 
and against the argument (the conclusion from the argument); and € 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—€)=a+¢€—a €.” 
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 Argument true, Evidence true? 
The argument and the evidence are here treated as two independent events hay- 
ing respectively the probabilities of 5 and ES and their coincidence is represented 
1 : 5 - ‘ 
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 
