REVERSIONS. 
151 
(Art. 104.) 
Pm, n—1 ■ 
4re+n—1 ___ (n+n—l K , Kn—\ 
In 
An-1 
In, 
An-f-n—I , Im Pm—1, n 
An-1 An—l Pm—1,1 
when there are more lives .than one it may be similarly shown that 
P (m—l,mi—1, m 2 —1), n 
V(jn>m v m 2 , &c.), n—i 1 — “ • 
jr (wi— ly rr%i—1, mg—1), I 
178. To determine the probability that one, in particular, of two 
given lives, A and B, aged m and m 1} shall die before the other. 
This event happening in any year, as the ?ith, must take place from 
one or other of these two events, either by A dying in that year and B 
surviving it, or by both dying in the ?zth year, A having died first. 
That there will die 
in the nth 
year 
after it 
the probability is 
A 
B 
(.pm, n—1 Pm, n) Pm x , » 
A and B, 
A having 
died first 
jneither 
3 (Pm, n-\~~Pm,n) (Pm x ,n— 1 Pm x , n) Art. 174. 
their sum 
\(Pm, n—l ““Pm,n) (.Pm x , Pm x ,n) 
is the probability of A dying before B in the Tith year, which we write 
thus:— 
Q(m, mO, n , and 2<J( m> nt\), n j 
o o 
is the total probability of A dying before B in any year during the pos 
sible term of their joint existence. 
179. To find 2<7 (m , m) , „, when 2q (m+1 , mi+1)> „ is given. 
(0 o 
If A and B at the ages m, m x , were certain of jointly surviving one 
year, the probability of A dying before B would then be 2q (m+1 , mi +i),«l 
0) 
but the probability of A and B jointly surviving one year is p (m , mi) ,» 
therefore jp (M , Wl)t i 2g (m+1>ifni+1)> , is the probability of A dying 
0) 
before B after the first year, and the probability of his dying before 
B in the first year is ^(1— p m>1 )(l +p mv i) '> the sum of the probabili 
ties of A dying before B in the first year, and A dying before B after 
the first year, is the total probability required. 
^7cm, m,), n— i(l — Pm, l) (1 "h Pm x , 1) ~^~P(m,mi), l X mj+l), n • 
0) (0 
When the age of the older of the two lives is the oldest age in the Table,