3 si6 A L G E 
be drawn out, the probability that thefe will be «and£ 
From the Bills of Mortality in different places, tables 
have beenconftructed which (hew how many perfons, upon 
an average, out of a certain number born, are left at the 
end of each year, to the extremity of life. From fucli 
tables, the probability of life, under any propofed cir- 
cumftances, is known. 
Ex. To find the probability that an individual of a gi¬ 
ven age will live one year. Let A be the number, "in the 
tables, of the given age, B the number left at the end of 
Q 
the year; then—is the probability that the individual 
will live one year; and 
A—B 
the probability that he will 
die in that time. In Dr. Halley’s Tables, out of 586 of 
the age of 22, 579 arrive at the age of 23; hence the pro¬ 
bability that an individual aged 22 will live one year is 
or — nearly ; and —, or — nearly, is the proba- 
5S6 84 586 84 
bility that lie will die in that time. 
If two events be independent of each other, and the 
probability that one will happen be — , and the probability 
that the other will happen be-, the probability that they 
both will happen is 
For each of the m ways in 
which the firft can happen or fail, may be combined with 
each of the n ways in which the other can happen or fail, 
and thus form mn combinations, and there is only one in 
which both happen; therefore the probability that this 
will be the cafe is . 
mn 
Required the probability of throwing an ace, with a 
fingle die, in two trials. The chance of failing the firft 
time is-, and the chance of failing the next is there- 
6 6 
fore the chance of failing twice together is 
36 
and the 
chance of not failing both times is 1-or—. 
36 36 
In how many trials may a perfon undertake, for an even 
wager, to throw an ace with a (ingle die? Leix be the 
number of trials; then, as the chance of failing x times 
together is d 
, and this by the queftion is equal to the 
chance of happening, or 
log. 2, 
hence xX log.-=log. 
2 6 0 
or xX log-5 
2 • 
log. 1—log. 2 
■log.6 — log. j 
log. 2 
and 
, fince log. 1 
log. 5 * 
BRA. 
the probability that Q will die is 
bability that they will both die is 
w—t 
; therefore, the pro- 
-, and the pro¬ 
bability that they will not both die is 1—~——-—- or 
mn 
m-\-n —1 
In the fame manner, if - be the probability that P will 
P 
live t years, and - the probability that 0 will live the fame 
9 
time j the probability that one of them at leaft will be 
p—\ . (j 1 
alive at the end of the time is 1- 
p+q—i 
pq 
pq 
If the probability of an event’s happening in one trial 
be reprefented by — -j-p to find the probability of its hap¬ 
pening once, twice, three times, &c. exactly, in n trials. 
The probability of its happening in any one particular 
trial being — , the"probability of its failing in all the 
a-\-b 
b n —‘ 
others—r trials is - - - ; therefore the probability of 
its happening in one particular trial and failing in the reft is 
cib n — * 1 
■ ; and, fince there are n trials, the probability that 
it will happen in fome one of thefe and fail in the reft is » 
times as great, or ——. The probability of its hap- 
pening in any two particular trials and failing in all the reft 
2 71 I 
is . ■ ■. and there are n. - ways in which it may 
a-\-b\ 2 
happen twice in n trials and fail in all the reft; therefore 
the probability that it will happen twice in n trials ri 
n n JZ± a *b*— a - 
—. In the fame manner, the probability of its 
n —1 n^—7. 
a?b n ~ 
happening exactly three times is 
and 
the probability of its happening exadtly t times is 
2 —j-i 
-a'b n ~ 
log. 5 — log. 6 log. 6- 
Rui.e. To find the probability that two individuals, 
P and Q, whofe ages are known, will live a year. 
Let the probability that P will live a year be — ; and 
7 71 
the probability that Q will live a year ~; then the pro¬ 
bability that they will both be alive at the end of that time 
1 r 1 
is — x-> or-. 
, m n mn 
Ri?le. To find the probability that one of them, at 
lea(t, will be alive at the end of any number of years. 
The probability that P will die in a year is ———, and 
Cor. i. The probability of the event’s failing exadtly 
n —1 n —2 n —t- 4 -i „ 
- --- .... - -—a 
. , , , -3 “ * 
t times in n trials is- 
~‘b { 
a v -\-na n 'b-\-n . -—-a 
2 
the ev< 
n .... to terms 
Cor. 2. The probability of the event’s happening at 
leaft t times in n trials is 
For if it happen every time, or fail only once, twice, 
.... n—t times, it happens t times ; therefore the whole 
probability of its happening at leaft t times is the fum of 
the probabilities of its happening every time, of failing 
only once, twice, .... n— (times; and the fum of thefe 
probabilities 
