ON PROBABILITY. 35 
annuity which is called the premium, the value of this premium with present 
Payment 
i 27 (1 a.) — $8 a, Sa, 
JUS IY = 2h =Tiss 
fod] l1+a, 1+ a, 
m+ I 3 . . opie 
is The value of any sum s to be paid if either of two individuals, aged m and 
; m' years respectively, are alive after n years is 
‘0 De taken 7° fe (1 1 — 2s 
gh PLL = (1 = pun) (L = pr) } 
= 37" Pun +S Di, = 8D, Hp 
and the value of an annuity to be paid as long as either of two individuals, 
aged m and 7 years, are alive 
+ oubabili = SE Pua + EP a = EP X Pol a} 
- = 80, + $a, — $a, 
understanding by the symbol a, the value of an annuity of £1. on the 
Joint lives of two persons aged m and m/ years. 
If life is considered valuable in proportion to its duration the expectation 
which any individual has of life will be measured by the sum of the probabi- 
lities of his dying after each given age, that is by an annuity without interest, 
ihe thegr of Hence we have these expressions : 
ration of vw: Expectation of life = 2 Pua 
Y payments : 
versal Value of annuity sg, =s3 Pry 
*+ +... ..insurance of a given sum s in one payment 
7 of a given =sr(l +a.) — sa, 
oh »++e.... premium of insurance of a given sum s 
sums fo be 
the value of Eg i 
i fo receive T+. 
pu ++ +... . annuity during two joint lives 
value 0 
s value of an = 82 Pun X Pty 
«+. .....annuity to be paid as long as either of two individuals, aged m 
irance com- and m/ years respectively, are alive 
Aenever that =o a, ne $ a, — 8 [7% we 
ridual, aged . ‘ = —Or) . 1 
a Thus if m — 20, r= —, @,=20 1428, according to Table III. for 
he vale of 1:03 
males. 
Value of the single premium required for a male aged 20 to secure the 
payment of 1 at the end of the year in which the life shall fail 
- 21'1428 
= ——— — 20°1428 = + 38419. 
1:03 
Value of the annual premium required for a male aged 20 to secure the 
payment of 1 at the end of the year in which the life shall fail 
1 20°1428 
Ee — = (1817. 
uf 1:03 21-1428 
que 1M 
. insi- 53. In calculating the values of annuities the labour is much diminished 
ff pa 0 by observing that the probability of an individual aged m years living at 
Dn 2