622 PROFESSOR BOOLE ON THE COMBINATION 
whence 
age BBR SE en BS LY oe ee 
In like manner we find 
a Dy Cy Po = - me oe he ‘ Sy. : (6) 
Again, since we have from the above 
Gg Oy Cae Vg giey a eg Bi 
l—a,¢,° y+1 l-a,c, «+1 
we have 
a, (L=¢,) ay (1c) _ ay 
(l—a, ¢,) d—a, ¢,) ~ (+1) +1) 
moreover, 
v+1) (y+1)z 
Te ig CPs cy py= 24) Yt) yy Le 
Multiplying the two last equations together we find 
a6, Vaso, OM by cee 
(1—a, ¢,) (1-4, ¢) (7-6, Py — My Cy Pa) = X - : bitte 2 (7) 
Now, to Ee 
Substituting in this expression the values found for its several terms in (5), (6), 
and (7), we have 
a,(1—c¢,) a, (1—<¢,) a(1—¢y) 
= Gio ; ! Gy Cy Po + (%#— ay ¢, Py — Mp CP) 
1—a,¢, 7 9"? (1=a, ¢,) (1—a, ¢,) yA Oa 
1l—a, ¢, 
U 
1% Pit 
This is the value of Prob. wyz. That of Prob. wyz will be found by simply changing 
in the above expression p,, p., and r, into 1—p,, 1—p,, and 1—r respectively. These 
expressions admit of some reductions, and give 
_% (1—<¢,) a, 1—«,) §1—a, 1-a, 
ee (l—a,¢) d—a,¢,) (1-e, te Tae; ahh } ; ‘ 5 
—_ a, (l—¢,) a, (1—e,) {7 9. 1-a, . ES. } 
Prob. vyz= (=a elas.) laa ¢, (l=p,) tesa: ¢, (1—p,) +1—r ne) 
whence we find for the & posteriori value of Prob. 2, 
1—a, 1-a, 
Prob. vyz 1—e, C) Py + Sr Cy Po +r 
Prob. ay 1—a, Xa, 
1l—¢, a ae 1—c, ¢,+1 
Equating this to we have 
Whence 
oF as SO aC a 
