150 
LIFE ANNUITIES. 
If A or C die first, it is equally probable that in the remainder of the 
term B will die either second or last of the three. 
If A or C die last, it is equally probable that B will die first or second 
of the three. 
We have shown that if A or C die first of the three it will be equally 
probable that B will die second or last, and if A or C die last it is 
equally probable that B will die first or second; and we have also proved 
that the probability of B failing first is the same as of his failing last; 
whence it is evident that for every way in which the life of B can fail 
second there is another way, equally probable, in which it can fail first, 
and a third way, equally probable, in which it can fail last. 
And since it is certain that B must die first, second, or third, the sum 
of these three probabilities is unity, and as each event is equally pro 
bable, the fraction is the probability of any one of them happening. 
176. The probability of the three lives failing in any particular order, 
as C, A, B, is L; for the probability of the life of C failing first is -¡j-, as 
just shown, and the probability of A dying before B is ^; therefore 
^ x i = ■§• is the probability of the particular order of survivorship 
taking place. 
177. To find the probability of a life failing in any particular year. 
-i The probability of any life or lives failing in any particular year will 
be expressed by using the letter q, in the same manner as we have used 
the letter p to denote the chance of living. 
Let the probability required be that a life aged m will die in the nth 
year from the present time. The number now aged m, who, by the 
Tables, survive n — 1 years, or, which is the same thing, enter 
upon their (m + n)th year, is l m+n -i, and the number who complete 
their (m-j-n)th year is Z m+n .; the difference between these is the number 
who die in the ?ith year, which, divided by the number living at the age 
of m years, gives the present chance of an individual aged m dying in 
the nth year from this time. 
q m ,n = —■— = p m , n -1 (Art. 104.) 
If there be any number of lives, and we call x the probability of the 
last v survivors jointly entering upon nth year, y the probability of their 
jointly surviving it, and z the probability of their failing in that year, 
then, since it is certain that they must either die before the nth year, the 
probability of which is (1 —<?), in the nth year, or after it 
l~~x + y + z=:l, 
by transposition, z == x — y; hence the following general rule. The 
probability of the joint existence of the last v survivors of any number of 
lives failing in the nth year is equal to the excess of the probability of 
their jointly entering upon it above the probability of their jointly 
surviving it.