138 
LIFE ANNUITIES. 
156. To find the value of an annuity payable so long as two out of 
three lives shall jointly he in existence. 
If the annuity he on three lives, A, B, and C, respectively aged m, 
rrii, m 2 years, the chance of its being received at the end of any par 
ticular year depends on either of the following events : 1°, that the three 
lives, A, B, C, he all in existence ; 2°, that A and B be alive, and C 
dead ; 3°, that A and C be alive and B dead ; 4°, that B and C be both 
alive and A dead : the following table shows the chance of each sepa 
rate event happening in the ?ith year, the sum of which shows the pro 
bability of the wth year’s payment of the annuity being received ; 
That^ 
there’ 
will be 
Alive. 
Dead. 
The probability is 
1 
ABC 
None 
Pim, m i 
2 
AB 
C 
P(m, m, ), n (1 Pm 2 , n) “ P(m, tHj )> n P(m,m l , n> 2 ), n 
3 
A C 
B 
P(m,m a ), n (1 Pmj ,n) P(m, m 2 ), n P(m, inj ,m 2 ),n 
4 
BC 
A 
PC m U m a )>« n) —Pim l ,m 2 'ì,n P (m, nij , n? 2 ), n 
their SUm, Pimtmy ),n 4" ),n ”1" P(m 1 ,m 2 ),n ^P(m, m l ,m 2 ), n } 
multiplied by r", gives the present value of the wth year’s payment of 
the annuity. If n be made equal to 1, 2, 3,4, &c., and the correspond 
ing values of the expression be found, they will show the present value 
of the payments of the annuity in the 1st, 2nd, 3rd, 4th, &c., years, and 
the sum of these values for every year in which, by the tables, the 
annuity can be received, will be the present value of the required 
annuity : this sum (Art.^-43} will be 
m l 4~ ^m, m 2 4“ , m 2 ” 2 a m> 5 m a • 
157. When the value of any expression is found for the successive 
values of n when made equal to 1,2, 3, 4, &c., years, the sum of these 
values continued for the whole term of existence may be denoted by 
prefixing the symbol 2 to the expression; when the sum is to be found 
only for a limited number of years, as t, it may be denoted by the 
character 2q. 
Example. What is the present value of £80 per annum, to cease on 
the failure of the joint existence of the last two survivors of three lives 
aged 23, 25, and 30 ? (Northampton 3 per cent.)