DE. FAEE ON THE CONSTEUCTION OE LIFE-TABLES. 
859 
As one or other of the two kinds of halls must hy hypothesis he drawn, and £1 is paid 
for each ball, the receipt of the £1 is certain ; certainty is thus in all cases expressed by 
unity. 
If every baU as it was drawn were replaced in the urn, although in 30,007 trials 
white halls were not actually drawn 29,647 times, black balls 360 times, still 
30 ^ 00 ^ 
would express the probability of drawing a white ball, and the value of £1 contingent 
on that event, more accurately than any other fraction that could be named. 
Again, if an urn contained by hypothesis an indefinite number of balls, out of which 
29,647 white balls and 360 black balls were di-awn and then replaced, the probability 
of again di-awing a white ball on trial, and the value of £1 contingent on that 
event, would be expressed more accurately by than by any other fraction that 
could be named ; past experience being by hypothesis the only means we have here of 
judging of the future. 
Thus a Life-Table applicable to the case furnishes the fractions to determine the 
value of any sums of money dependent on the life or death of a given person, or a 
certain number of given persons in a given time. 
The probability of living two years expressed by the fraction 
less than the probability of living one year. 
Making n any number of years and fractional parts of years, the fraction ~ will 
invariably expiess the probability of living n years after the age x. As n approaches 
zero the fraction will approximate to 1, the symbol of certainty ; thus a person is more 
hkely to live a day than a year, a minute than a day. As n increases diminishes 
in value; and when x-\-n expresses a year after the age co in the Life-Table, is by 
hypothesis zero, .-. The chance of living so long is expressed in this case 
by zero, the chance of dying in the time by 1, the symbol of certainty. 
(21.) l^^„ expresses the number of chances in favour of surviving 7 i years, and 4 — 4+« 
the number of chances of dying in the same time, the sum of the two together (4) 
expressing the total number of chances. 
Thus the fraction 
expressing the pro- 
bability of Ihing a given time ranges from 1 to 0, and^^”=l-^, or the chance of 
dying in a given time also ranges from 1 to 0 as varies. When the two fractions are 
equal 
I* 
then 4+„ = 4-?,+„, and 24+„=4 
7 — 
2 * 
To verify the equations, an age x-\-n must be chosen at which 4+m is exactly equal to \l^. 
Thus by the Life-Table of healthy districts 100,000 children born alive are reduced to 
50,-8ol in 58 years, and to 49,895 in 59 years; so the chances are rather in favour of 
* The addition of 1 to the numerator, and of 2 to the denominator, may be neglected, when, as in this 
case, the numbers are large. 
