C 492 1 
nuitant’s receiving the whole drd, fecond, third, &c» 
payment, or at lead: the halves of them (being, in 
faCt, the probabilities of his living the whole of thofe 
years, or at lead the halves of them), are called the 
expectations of life, in thofe years. 
Mr. De Moivre has proved, on another occafion, 
that, if there be an expectation, to take place upon 
the happening of two independent events, the proba- 
bility of the compound event, will be the product of 
the two tingle probabilities : Now, if there be an ex- 
pectation, depending on one man’s fur viving another, 
it mud necedarily be compounded of the probabi- 
lities of the one’s living, and of the other’s dying. 
Now let the perfon, upon whofe furviving the ex- 
pectation depends, be of fuch an age, that he may 
podibly live 36 years j and let the perfon, who is to 
be furvived, be of that age, that he may podibly live 
43 years : Then dnce the furvivor’s expectation of 
life will, in the drd, fecond, third, &c. year, be 
+ 7T> if + 7?> U + 7V, &c. and dnce the pro- 
bability of the other’s dying, in any one year, will be 
73 ; therefore, if the former be feverally multiplied 
by the latter, the products aridng, viz. + 77 X 
7: X T ' T , + will exhibit the 
probabilities of the fur vivor drip’s taking place, in the 
drd, fecond, third, &c. years. 
For the above expectations contain the probabilities 
of the furvivor’s living the whole, or (at lead) the 
halves of the drd, fecond, third, &c. years ; and the 
condant faCtor, is *be probability of the other’s 
dying, fome time within one of thofe years ; which 
may, therefore, be either in the drd, or in the fecond, 
half year, of either of them 5 thro* both of which 
periods, 
