CHANCES. 
failing twice in three trials, may be found 
3 lib a 
equal 3- 
3. Let the probability of the event’s hap- 
pening only once in four trials be required. 
In this case it must either happen the first 
and tail in the three succeeding trials ; or 
happen the second and fail in the first, third, 
and fourth trials ; or happen the third, and 
fail in the first, second, and fourth trials ; 
or happen the fourth, and fail in the first, 
second, and third trials. The probability of 
n Zi’ . 
each of these being the required 
probability wfill be j— ; and for the same 
reason the probability of its happening 
three times and failing only once in four 
4 b (ri 
trials will be — 
a b\ 
4. Let the probability be required of its 
happening twice and failing twice in four 
trials : here the event may be determined 
in either of six different ways : 1st, by its 
happening the first and second, and failing 
in tlie third and fourth trials ; 2dly, by its 
happening the first and third, and failing the 
second and fourth trials ; 3dly, by its hap- 
pening the first and fourth, and failing the 
second and third trials ; 4thly, by its hap- 
pening the second and third, and failing the 
first and fourth trials; 5thly, by its happen- 
ing the second and fourth, and failing the 
first and third trials ; or, 6thly, by its hap- 
pening the third and fourth, and failing the 
first and second trials. Each of these pro- 
babilities being expressed by r-ti4, it fol- 
failing in the rest, it is evident that this pi'i- 
bability ought to be repeated as often as d 
things cab be combined in n things, which, 
by the known rules of combination, are = 
■ continued to d terms ; 
hn-d' 
n. n — 
tX-T 
the general rule therefore will be ^ 
multiplied into 
n — 3 
n X 
-.7— X 
- 2 
a 6) 
6a^V 
lows that the sum of them, or ^ 
express the probability required. 
By proceeding in the same manner, the 
probability in any other case may be deter- 
mined. Hut if the number of trials be very 
great these operations will become exceed- 
ingly complicated, and therefore recourse 
must be had to a more general method of 
solution. 
Supposing n to be the whole number of 
trials, and d the number of times in which 
the event is to take place, the probability 
of the event’s happening d times succes- 
sively, and failing the reinainini 
continued to d terms. 
4 
Ex. Supposing a person with six dice un- 
dertakes to throw two aces and no more ; or, 
which is the same thing, that he undert-kes 
with one die to throw an ace twice, and no 
more, in six trials, it is required to deter- 
mine the probability of his succeeding, a 
being in this case = 1, Z> = 5, » = 6, and 
d = 2, the above expressions will become 
t,4 ’ . 5 6'2b X 15 
= gy, multiplied into 6X5— 
— i- vei-y nearly. Hence since there is 
5 ^ 
only one chance for his succeeding, while 
there are four for his failing, the odds 
against him will be as four to one. 
Prob. 2. To determine the probability that 
an event happens a given number of times 
in a given number of trials ; supposing, as 
in the former problem, the probability of 
its happening each time to that of its failing 
to be in the ratio of a to 6. 
Sol. It will be observed that this problem 
materially differs from the preceding, 'in as 
much as the event in that problem was 
restrained so that it should happen neither 
more or less often than a given number of 
times, while in this problem the event is 
will determined equally favourable by its hap- 
* pening either as often or ofteiier than a. 
given number of times, so that in the pre- 
sent case there is no further restriction 
than that it should not fall short of that 
number. 
1. Let the probability be required of an 
event happening once at least in two trials. 
If it happens the first and fails the second 
time, or tails the first and happens the 
second time, or happens both times, the 
event will have equally succeeded. The 
ah 
probability in the first case is the 
ha 
will be 
’ ^ “-a . 
But as there is the' same probability of its 
happening any other d assigned trials and 
nd times, 
ad .b’l — d 
and the 
probability in the second is ^ 
a a 
probability in the third is ^ ; hence the 
