CHANCES. 
2 rt & -)- a o 
probability required will be = ^^2 • 
2. Let the probability be required of its 
happening once in three times. Provided 
it has happened once at least in the first two 
trials, the event will have equally succeed- 
ed, whether it happens or fails in the third 
-I- 2 a /> 
trial, and therefore Will represent 
the probability in this case. But it may 
have failed in the first two and happened in 
the third trial, the probability of which is 
6 6a 
adding this to the preceding frac- 
pening the fourth, after having happened 
once in the three preceeding, be =r=s'' and 
a 6j * 
therefore the whole probability willl be 
-f- 3 6 _a’-^-4g’/)-|-6 0^6* 
Tplr — • 
By proceeding in the same manner, it may 
be found that the probability of an event's 
happening twice at least in five trials, will 
, a‘'-(- 4 g ’6 4 . 6 a' 
be = — — _f. 
4 g 6’ 
tion we have 
a’ 3 g^ 6 -j- 3 g 6^ 
2a6xa-l-6-|-6"g 
g 6'’ 
for the probability 
“ + *1 
required. In like manner the proba- 
bility of its happening once at least in 
c.. * • I -,,1. g’ 3 g^ 6 -1- 3 g 6 6 . 
tour trials will be — ^ .4. 
g'* -1- 6 g^ 6 -|- 6 g^ 6' -|- 4 g 6’ , 
the probability of its happening once at 
least in n times will be = — ~ — . In 
“+ ^1’ 
other words, since tlie event must happen 
once at least, unless it fails every time, the 
probability required (by Def. t) will always 
bo expressed by the difference between 
6” 
unity and ===; n- 
a-j-61 
3. Let the probability be required of an 
event’s happening twice at least in three 
trials. In this case it will succeed if it 
happens the first and second, and fails the 
third time, if it happens the first and 
third, and fails the second time, if it hap- 
pens the second and third, and fails the first 
time, or if it happens each time successively. 
The first three probabilities are =r=), and 
g* 
the fourth is 3 . therefore the proba- 
g -4" bl ^ ^ a -j- 6 
_ g'^ -4- 5 g'*6 4- 10 g’ 6^ -}- 10 g" h‘ 
g + &l' " • 
the probability of the event’s happening 
thrice in four, five, six, &c. trials be requir- 
ed, they may, by pursuing the same steps, be 
found = a’^4-5g^6 4.l0g»6 2 
g + 6l‘' ’ g -|- 6 ^ ’ 
g'' 4- 6 6 + 15 g^ 6^ 4- 20 g= 6’ 
- i — t &c. res- 
g-4-6| 
pectively.^ Hence it follows, that if the 
binomial a . be raised to nth power, the 
probability of an event's happening at least 
d times in n trials will be = 
a”-^b^ (m 4- 1 — d) 
’ 1 
2 ' 
bility required will be 
_ a^ + 3an 
If the 
g4“^^ 
event is to happen twice at least in four 
times, the probability of its happening dur- 
ing the first three times has been already 
found. Let it be supposed to have happen- 
ed only once in these times, the probability 
of which, by the preceding problem, is 
Sabh 
^ j then will the probability of its hap- 
a’'4-wa''~' 54-n. 
that i.s, the series in the numerator must be 
eontinued till the index of a becomes equal 
tod. 
Cor. From this solution it appears that the 
6’’-4-w 6” — 'g 4-n.— j~'6' — V to d term.s 
series ^ 
will express the probability of the event’s 
not happening so often as d times in n 
trials. 
Ex. Supposing a person with six dice un- 
dertakes to throw two aces or more in the 
first trial, what is the probability of his suc- 
ceeding ? In this case g, 6, n, and d, being 
respectively equal to 1, 5, 6, and 2, the 
above expression will become = 
1 -f 30 4- 1,6 X 25 4- 20 X 125 4- 15 X 62.o 
, 
12281 
= Hence the odds against his suc- 
ceeding will be as 34375 to 12281, or 
nearly as three to one. 
We have already observed, that the doc- 
trine of chances is particularly applicable 
to the business of life annuities and assu- 
rance. This depends on the chance of life 
in all its stages, which is found by the bills 
of mortality in different places. These bills 
exhibit how many persons upon an average 
out of a certain number born are left at the 
end of each year to the extremity of life 
