A L G E 
probabilities is 
a 1 ’ -\-na h 'b-\-n. 
-a n ~ l 
and 
B R 
n —i 
to n~~t- J-i terms 
A * 31.7 
n 2 n 3 , &c. will be the probabilities or 
Ex. What is the probability of throwing an ace, twice 
in three trials, with a lingle die ? ' • 
In this cafe n— 3, t—2, a— 1, b=z$ ; and the probabi- 
.. ... i-)~ 3 , 5 2 
lity required is ■ ■ - ■■ ——- = —. 
1 6.6.6 216 27 
Scholium. Much might be faid on a fubject fo exten- 
five as the doctrine of chances; tlie learner will however 
find the principal grounds of calculation in the above 
rules; and, if he wifli for farther information, he mu ft 
confult De Moivre’s work 011 this fubjedt. It may not be 
improper to caution him againlt applying principles which 
on the firft view' may appear felf-evident, as there is no 
fqbjedt in which he will be fo likely to miftake as in the 
calculation of probabilities. A fingle inftance will fliew 
the danger of forming a hafty judgment even in the.mod 
Ample cafes. The probability of throwing an ,ace with 
one die is -, and, fince there is an equal probability of 
6 . 
throwing an ace in the fecond trial, it might be fuppofed 
n n n 
his living 1, 2, 3, &c. years ; hence the prefent value of 
f 
an annuity of il. to be paid during his life is 
■ 1 
the feries 
n> 3 
n— 
Sec. continued to n terms. 
,x n —2..V 2 n —3.x' 
nr 
The firm of 
to n terms is 
found to be — 
of the feries 
! .x — nx 2 -\~x nJr ' , . 1 
-= =~ —— ; let x—~. 
and the fum 
x.i- 
n —1 
4 - See. to n terms. 
B_i.r_a_j.__. 
-; the prefent value of the annuity. 
with 
n.r —1 ( 
Cor. i. This expreflion for the fum is the fame 
1 1 
r -— 1- 
r 
X 
nr~n 
or . 
n.r —1 
n.r —1 i 
r- 
n 
let P b e 
-1 
that the probability of throwing an ace in two trials is the P refent value of an annuity of il. to continue certain 
6 1 
This is not a juft conclufion ; for, it would follow', by the 
fame mode of reafoning,. that in f;x trials a per'fon could 
not fail to throw an ace. The error, which is not eafily 
feen, arifes from a tacit fuppofition that there mull necef- 
farily be a fecond trial, which is not the cafe if an ace be 
thrown in the firft. 
for n years, then Pz 
and the expreflion be- 
r „ 
1- P 
n 
conies 
On Life Annuities. 
To find the prefent value of an annuity of il. to be 
continued during the life of an individual of a given age, 
allowing compound intereft for the money. Let r be the 
amount of il. in one year ; A the number of perfons, in 
the tables of the given age; B, C, D, &c. the number 
left at the end of 1, 2, 3, &c. years; then — is the 
D 
&c. its value 
value of the life for one year, —, —, 
A A 
for 2, 3, Sec. years ; and this feries muft be continued 
to the end of the tables'. Now the prefent value of il. 
to be paid at the end of one year is - ; but it is only to 
be paid on condition that the annuitant is alive at the 
end of the year, of w hich event the probability is —; 
A 
therefore the prefent value of the conditional annuity 
. B 
is —; in the fame manner, the prefent value of the fe¬ 
cond year’s annuity is the prefent value of the third 
year’s annuity is Sec. therefore the whole value re- 
Ar 3 
. , . I BCD 
quired is—x —■-\—- -|-j- &c. to the end of the tables. 
A r r 2 r3 
De Moivre fuppofes that out of eighty-fix perfons born, 
one dies every year, till they are all extindt. This fup¬ 
pofition is fufficiently exadt, if our calculations be made 
for any age above ten, as will appear from an infpedtion 
of the tables; and on this fuppofition the fum of the 
r . x B~~~C D ~ 
feries + i -f Sec. may be found. 
A -r r 2 r 3 J 
Let 11 be the number of years which any individual 
wants of eighty-iix ; then will n be the number of per- 
i'ons living, of that age, out of which one dies every year; 
Vol. i. No. 20. 
Cor. 2. The prefent value of the annuity to continue 
for ever, from the death of the propofed individual, 
r P 
is — . For the whole prefent value of the annuity 
n.r —1 
to continue for ever, is —-—; and if from this, its 
r —1 
value for the life of the individual be taken, the re- 
rP 
mainder — . is the prefent value of the annuity to 
n.r —1 
continue for ever, from the time of his death. 
To find the prefent value of an annuity of il. to be 
paid as long as two fpecified individuals are both living. 
Find, by the foregoing rules, the probability that they 
will both be alive at the end of i, 2, 3, Sec. years, 
to the end of the tables; call thefe probabilities a, 
b, c, Sec. and r the amount of il. in one year; then 
a b c 
- + — + — 4-&c. is the prefent value of the annuity 
required. 
To find the prefent value of an annuity of il. to be 
paid as long as either of two fpecified individuals is living. 
Find, as above, the probability that they will not both be 
extinift; in 1, 2, 3, &c. years, to the end of the tables, and 
call thefe probabilities A, B, C, Sec. then the prefent 
ABC 
value of the annuity is -!— --4 — - 4 - &c. 
* 7" 7^ 7 3 
Cor. If the annuity be M\. the prefent value is 
M times as great as in the former cafe, or 
A B C 
A/X-+ — -\-Sec. 
v 7' 7 * X J 
Thefe are the mathematical principles on which the 
values of annuities for lives are calculated, and tire rea¬ 
foning may e’alily be applied to every propofed cafe. But. 
in pradftice, thefe calculations, as they require the combi¬ 
nation of every year of each life with the correfponding 
years of every other life concerned in the queftion, will be 
found extremely laborious, and other methods muft be 
reforted to when expedition is required. 
4M 
PA RT 
