CHANCES. 
that Expectation will be so much less than 
it was on the former supposition as ~ is less 
than unity. Hence we have 1 :y 
b a bd 
for the tnie expectation in this case. 
Cor. By the same method of reasoning it 
will appear that the probability of the hap- 
pening of any number of subsequent events 
is equal to the “ product of the probabilities 
of those events separately considered,” and 
therefore if a always denote the probability 
of its happening, and b the probability of its 
oP 
happening and failing, the fi-action j- will 
express the probability of its happening n 
times successively, and (by Def. 1) the frac- 
tion — — — will express the probability of 
its failing n times successively. 
Rem. It should be observed that in some 
instances the probability of each subsequent 
event necessarily differs from that which 
preceded it, while in others it continues in- 
variably the same through any number of 
trials. In the one case the probabilities are 
expressed, as in the theorem, by fractions, 
whose numerators and denominators con- 
tinually vary ; in the other they are expres- 
sed, as in the corollary, by one and the 
same invariable fraction. But this perhaps 
will be better understood by the following 
examples. 
1. Suppose that out of a heap of counters, 
of which one part of them are white and the 
other red, a person were twice successively to 
take out one of them, and that it were requir- 
ed to determine the probability that these 
should be red counters. If the number of 
the white be 6, and the number of the red 
be 4, it is evident, from what has already 
been shown, that the probability of taking 
out a red one the first time will be : but 
the probability of taking it out the 2nd time 
will bo different ; for since one counter has 
been taken out, there, are now only nine 
remaining; and since, in order to the 2nd 
trial, it is necessary that the counter taken 
out si lould have been a red one, the number 
of th ose red ones must have been reduced 
to 3. Consequently, the chance of drawing 
out a red counter the 2nd time will be |, 
and the probability of drawing it out the 
1st and 2nd time will (by this theorem) be 
4 X 3 __ 
10 )< 9 “ 15' 
2., Suppose next, that with a single die, a 
person undertook to throw an ace twice 
successively: in this case the probability, 
of throwing it the first, does notin the least ' 
alter his chance of throwing it the second 
time, as the number of faces on the die is 
the same on both trials. The probability, 
therefore, in each will be expressed by the 
same fraction, so that the probability, be- 
fore any trial is made, will, by the preced- 
ing corollary, be | X i = On these 
conclusions depend all the computations, 
however complicated and laborious, in tlie 
doctrine of chances. But this, perhaps, 
will be more clearly exemplified in the two 
following problems, which will serve to ex- 
plain the principles on which every other 
investigation is founded in this subject. 
Prob. 1. To determine the probability 
that an event happens a given number of 
times and no more, in a given number of 
trials. 
Sol. 1 . Let the probability be required of 
its happening only once in two trials, and 
let the ratio of its happening to that of its 
failing be as a to b. Then since the event 
can take place only by it happening the 
first, and failing the second time, the pro- 
bability of which is — r X ^ 
ab 
o -j- 6 a -J- & ' 
or by its failing the first and hap- 
pening the second time, the probability of 
which is - I , - 2 , the sum of these two frac- 
9, ah 
tions, or ===,2 
required. 
will be the probability 
2. Let the probability be required of its 
happening only twice in three trials. In this 
case, the event, if it happens, must take place 
in either of three different ways : 1st, by its 
happening the first two, and failing the third 
aa b 
time, the probability ot which is 
2dly, by its failing the first and happening 
the other two times, the probability of 
baa 
which is : 
or, 3dly, by its happen- 
ing the first and third, and failing the second 
• 
time, the probability of which is 
a -|- o|’ 
The sum of these fractions, therefore, or 
3b aa 
^ '"'•H be the required probability. 
By the same method of reasoning the pro- 
bability of its happening only once in three 
trials; or, which is the same thing, ot its 
