ON PROBABILITY. 
0 at least 9 
M+. =10 1 147" pag,10 + 7° po, 00 + &e. } — 5) 
: 9 
=.10 { 7° Pw, 0 +r? Pao, 20 -4- &e. } + 2 
57. In calculating annuities the values of the annual payments, except, per- 
haps, at birth, vary so gradually that the result thus obtained will be a suffi- 
ciently accurate approximation, and, probably, within the limits of the errors 
of which the values of p, that is of the table of mortality which is used, are 
susceptible ; the correction, however, in all cases may be considered as con- 
stant for different tables of mortality, and may, therefore, be determined by 
calculating the annuity first accurately and afterwards by the approximate 
method from any table of mortality in which the deaths are given for every 
age, the difference between the two values so obtained will be the correction 
mn wticed by eignived, : Cal : : 
at whieh pce The method of calculating annuities hitherto adopted by Dr. Price and 
other writers, has been, first to interpolate living between those which are 
actually given from ten years to ten years by the observations, to calculate 
{able IT, and probabilities of surviving each number of years from the numbers so inter- 
polated, then to discount these probabilities so obtained, and finally to obtain 
the value of the annuity by adding together all these discounted probabili- 
ties. This labour, though diminished by means of the equation noticed by 
Mr. Barrett, is still unnecessary, and would lead to the same result as that 
given by the series of the last page. The same method is, we think, gene- 
7 payments, rally the simplest which cart be applied to calculating annuities on two or 
If these pay more lives, and, in fact, to the summation of any series of which the law is 
¢ those which too complicated to admit of the ordinary methods. 
58. We have, as yet, said nothing with respect to the method of determin- 
He g, and if ing p and ¢, and this isa question of very considerable difficulty, whether as 
¢ vite of y regards theory or practice. 
Suppose 1000 infants to be carefully registered at birth, and the ages at 
which they die to be noted. If of these 900 are alive at the end of the first 
year the probability of an infant at birth living one year under similar circums- 
900, 9 : 
seessary, how- stances would be nearly = 1000” °* io and if the number of infants registered 
watitv in the 
were infinite, this would cease to be an approximation, the ratio of the number 
alive at the end of the first year to the whole number registered at birth 
would be exactly equal to the probability of an infant under the same cireum- 
stances living 1 year. The problem is, in fact, similar to the one we solved, 
4. page 33, when we supposed a bag to contain a number of balls of different 
colours, and that a certain number of drawings had been made. The different 
dratures, a0C ages at which the individuals can die correspond to the different colours in the 
1 are required former problem. 
blste, VOL IV If we suppose 101 ages at which deaths take place, that is, if we sup- 
, teversionary pose 7 to vary from 0 to 100 in the values of Gm, n> Pm, n> and if dy, dy, &e. d,,, 
f the values are the number of the 1000 infants who have been observed to die in their 
the term first, second, and n” years respectively, we have 
2 ny + | 
ir LT 
ZI snp tbugpit oo. 3-101 =m — 5 
np A080 Pon = dad} ¥ rsa rien 101 — 771 T 
19