152
LIFE ANNUITIES.
mj+i),» Xpim.mo, i=0, and the remaining part of the expression
is easily computed.
180. By means of this formula a table may be formed of the pro-
hahility of one life failing before another at any ages; for, if we com
mence with finding the probability of the event happening at the oldest
ages they jointly complete, we can, by means of the result, find the
probability for lives each one year younger; and this new probability
again enables us to find the probability on lives each one year younger
than these last; continuing in the same manner, the probabilities can
be found for all the ages these two lives can jointly complete.
181. The probability of the failure of the joint existence of the two
lives in the next t years is 1 — and as this event must happen
either by A dying before B, or by B dying before A,
2 17(m, m,), n “b 2 q («¡j, m), n — 1 P (in, mi), t >
n (1 ) n (0
by transposing 2 q lm m) ,= 1 —p (m . mi) , t - 2 q (m< m )i „ .
n C) n (>)
When t is greater than the difference between the age of the older
life and the extreme age in the table, it becomes
2 q(m v m), n'— 1 2 Q(m, m x ), n •
C) (0
182. The probability of the life of A failing in the next t years is
1 — p m , i, and this event must take place by A dying either first, or
second of the two lives A and B ;
2 <2(m,»n,),?»T"2 q(m, mi),n— 1 Pm,ti
n oo n (.)
by transposition, 2 q (m , mi) .„= 1 — p m , t - 2
n (0 (')
the probability of A dying second in the next t years.
W hen t is greater than the difference between the age of A and the
extreme age in the table it becomes
^ m x ), n'— 1 2 q^ m> m{),n •
o . o ..
183. The probability of both the lives failing in the next l years is
(1 —p m , t ) (I—p mx ,t) , which probability is the sum of the chances of
A dying after B in the term, and of B dying after A in the term.
2 7(m, mi), n~f" 2 q(m x ,m),n'— (1 Pm, t) (1 Pm\,t)>
1 C 2 ) q ( 2 )
^ “ (1 Pm,t) (1 "~P mi ,t) ~ A •
n («) n (0
184. To find the present value of an annuity on the life of A aged
rn, after the failure of the joint existence of two other lives, B and C,
aged m x and m 2 , provided that event take place by the death of B.'
The present value of the nth year’s payment of the annuity is
2 9( m ii m%),n X p m>n T ,
"1 (0
