318] 
ON A QUESTION IN THE THEORY OF PROBABILITIES. 
81 
part of the original data, and that the}- ought to be of such a kind that they can 
be established by experience in the same way as the other data are. For instance, 
if experience, as embodied in a sufficiently long series of statistical records, establish that 
Prob. A — a, Prob. B = ft, 
the very same experience may, by establishing also that 
Prob. AB = aft, 
whence in conjunction with the former it follows that 
Prob. AB' = a/3', Prob. A'B = a!A, Prob. A'B' = aft', 
enable us to pronounce that A and B are in the long run, as to happening or not 
happening, in the position of mutually independent events. 
3rdly. I think it may be shown to demonstration, from the nature of the result, 
that the solution you have obtained does not apply simply and generally to the problem 
under the single modification of the assumption that A and B are independent. The 
completed data under this assumption are 
Prob. A = a, Prob. B = /3, Prob. AB = aft, 
Prob. AE = ap, Prob. BE = ftq. 
You may deduce all these from your Table of Probabilities of ‘ compound events ’ given 
in your paper. Now you may easily satisfy yourself that the sole necessary and 
sufficient conditions for the consistency of these data are the following: 
(1) 
(2) 
(3) 
<*p’ + P ( 1 > a &> ' 
ap + ftq' S aft, 
( a^ 
<1, 
/3 
P 
<7 ; 
5 0. 
(M). 
But your solution requires the following conditions to be satisfied, viz., 
q-ap> 0, p- ftq S 0, 
together with the system (3). Now (1) and (2) are expressible in the form 
ft(q- ap) + aft'p! 5 0, 
a(p - ftq) + fta7/ 5 0 ; 
from which you will see that your conditions are narrower than those which the data 
are really subject to. If your conditions are satisfied, the data will be consistent; but 
the converse of this proposition does not hold. 
C. Y. 
II