112 
LIFE ANNUITIES. 
(by logarithms) log—— X 850 x .81309151 = 2.8193518 by last example 
OOo5 j i — q htiK.AaAo 
log /«= 3.7754648 
-log l l9 — 4.2033563 1 
2.7981729 £628.308= 
¿£628 6 2 
107. Whatever may be the number of lives, if the receipt of the money 
depend on all of them surviving a given period, the present value of the 
sum must be multiplied by the continued product of the fractions which 
express the chance of each surviving separately. 
108. As certainty is expressed by unity (Prob. Art. 6), the pro 
bability of a life dying before the end of a given time is found by 
subtracting from unity the probability of the life surviving that time, it 
being evident that one or other of the events must happen. 
The same rule is obtained by dividing the number of deaths that take 
place in the given time by the number living at the present age.’ 
109. If there be two or more lives, the probability of their joint 
existence failing in n years is 
7—, Ac. 
P (m, tiij , »ig > Ac.), n ■— 
110. The probability of any number of lives all dying in a given term 
is obtained by finding the product of the 'chances of each separate indi 
vidual dying in that term. 
If we call the respective ages of the lives m, m u m iy &c., then 
(1 -p m . n ) (1— p»,.«) (l-p m2 ,n) s &c.= 
&c. (Art. 107), is the chance that the lives aged m, m u &c., will all 
die in n years. 
111. Since it is certain that the lives will either all fail, or that one or 
more will survive the term, the probability that at the end of the term 
they will not all have died, that is, that one of them at least will 
be in existence, is 
1 ~ (1 - Pm.n) (1 - Pm l>n ) (1 - p mi .n) &C.J