Ill 
of Edinburgh, Session 1878-79. 
tively. The probability that if the cause A 1 present itself, an event 
E will accompany it (whether as a consequence of the cause A ± or 
not) is p v and the probability that if the cause A 2 present itself, 
that event E will accompany it, whether as a consequence of it or 
not, is q. Moreover, the event E cannot appear in the absence of 
both the causes, A x and A 2 . Eequired the probability of the 
event E. 
The data are — 
x = a, 
y = b, xz = ap, yz = bq } 
and 
o 
II 
!\! 
1 
rH 
"iT 
I 
and ^ is required. 
How (1 - 
x)(\ - y)z — z - xz - yz +xyz, 
z = xz +yz - xyz 
by the last datum ; 
z < xz + yz - x - yz + 1 
a-) 
< xz + yz - y - xz + 1 
(2.) 
< xz + yz . 
(3.) 
z < 1 - a + ap . 
(1-) 
<1 - b +bq- 
(2-) 
< ap + bq. 
(3.) 
Also 
(1 - x)yz = yz - xyz , 
= yz + z — xz — yz 
by the last datum ; 
z — xz + ( 1 - x)yz ; 
z > ap- 
Also 
z>Vq- 
This problem was discussed in the “Philosophical Magazine,” 
by Boole, Wilbraham, and Cayley. Cayley’s solution is different, 
applying to a modification of the problem. Boole goes further, and 
finds the most probable value of the probability. Wilbraham con- 
siders only mathematical probability, and maintains, quite rightly, 
that we cannot proceed further than above without making assump- 
tions. He says that the disadvantage of Boole’s method is, that it 
does not show whether a problem is determinate. This desideratum 
is supplied by the method of indeterminate coefficients to which I 
have referred above. 
VOL. x. 
Q 
