858 
DE. EAEE ON THE CONSTEUCTION OE LEFE-TABLES. 
(20.) The Life-Table serves to determine the value of Life Annuities, the value of 
policies, and the premiums of insurance. 
This is effected by introducing a new unit, such as £1, 1 franc, 1 dollar, or any other 
monetary unit. Thus if £1 is payable at each death, the series vdll show the number 
of pounds falling due in each year of age ; so if £1 is payable by each person on attain- 
ing the age x, and each subsequent year of age, the series 1., shows the number of pounds 
payable every year by the 4 persons ; and will be the number of pounds payable in 
N £1 
the whole course of life after the age x : thus — ) — = the aveeage AJ^ioryi of an annuity 
of £1 payable on each life at and after the age x. The money-unit may be introduced 
Y 
into the other columns ; and ^ • £1 would show the aveeage amount payable under an 
annuity of £1 on each of lives. The present value of these future payments can 
always be determined by assuming a given rate of interest. The estimates thus obtained 
are also always read subject to the qualification that by hypothesis the Life-Talle is 
based on a law of mortality actually to rule for a definite time in the population to 
which it is applied. The probability of the hypothesis is not here in question. 
Under the same circumstances masses of mankind appear to experience, at the same 
ages, the same rates of mortality. Consequently if for several years d, persons have 
died annually on an average out of persons living at the beginning of the year, other 
things being equal, the probability that the same number will die out of C persons in a 
year to come is greater than any other that can be named, and the fraction expressing 
that probability is y- We know that d^ expressing the numbers djfing in a year, C+j 
must express the numbers surviving as C+i + ^i=4- The chances may be represented 
by 4 balls; 4+i white balls in an urn will represent the chances of Imng, d^ Mack balls 
in the same urn will represent the chances of dying. Now let each of 4 persons pay the 
sum z for a ticket, and each person that draws a white hall be entitled to £1. Before 
the drawing commences the value of each ticket is for 4 (the total chances): 4+i 
(the chances in favour of winning on one ticket) : : 
Put 4=30,007, and 4+^=29,647; then- ^^^V'^^ = 
29,647.£l 
30,007 
£•98802. 
The amount of 
money to be paid on 4+i white balls is £29,647, and £-9802 X 30,007=s .4=£29, 647. 
In like manner it may be shown that if £1 is paid to each person who di’aws a Mack 
d 2 
ball, the value of each ticket is — =^£1 ; for y. 4-£l = ^,,£l 5 and £1 is to be paid on 
each of d^ tickets. 
Should £1 be paid alike to those who draw white balls and to those who cffaw black 
balls, the value of a ticket will be equal to the sum of the two factions expressing the 
several probabilities, namely. 
4+1. £l . 4£l 
— +-,-=Z+y: 
4+1 + <4 
4 
£1=^£1=£1. 
