CHANCES. 
required to determine before the event what 
chance or probability that person has of 
receiving the 51. and what sum he may ex- 
pect should be paid to him in consideration 
of his resigning his chance to another. 
Solution. Since there is nothing in the 
form of the metal that can incline it to shew 
one face rather than the other, and since it 
must shew one, it will follow, that there is 
an equal chance for the appearance of either 
face, or there is one chance out of two for 
the appearance of the white face, and con- 
sequently the probability ot it may be ex- 
pressed by the fraction - ; if, therefore, 
any other person should be willing to pur- 
chase his chance, he must give for it the 
half of 31. or 21. 10s. This is one of the 
most simple cases: before, however, we 
proceed, it may be proper to give some de- 
finitions introductory to the doctrine. 
Def. 1. The probability of an event is the 
ratio of the chance for its happening to all 
the chances for its happening or failing: 
thus, if out of six chances for its happening 
or failing there were only two chances for its 
happening, the probability infavour of such an 
event would be in the ratio of 2 to 6 ; that is, it 
would be a fourth proportional to 6, 2, and 
1, or 1. For the same reason, as there are. 
four chances for its failing, the probability 
that the event will not happen, will be in 
the ratio of 4 to 6, or in other words, it will 
be a fourth proportional to 6, 4, and 1, or |. 
Hence, if the fractions expressing the pro- 
babilities of an event’s both happening or 
failing be added togetlier, they will always 
be found equal to unity. For let a be the 
number of chances for the event’s happen- 
ing, and b the number of chances for its 
failing, the probability in the first case being 
, and in the second case — |— ^, their 
' (1 0 
Having there- 
a-{-b 
sum will be = = 1. 
a -)-0 
fore determined the probability of any 
event’s either happening or failing, the pro- 
bability of the contrary will always be ob- 
tained by subtracting the fraction expres- 
sing such probability from unity. 
Def. 2. The expectation of an event is 
the present value of any sum or thing which 
depends either on the happening or on the 
failing of such an event. Thus, if the receipt 
of one guinea were to depend on tiie throw- 
ing of any particular face on a die, the ex- 
pectation of the person entitled to receive 
it would be worth 3s. 6d.; for since there 
are six faces on a die, and only one of them 
can be thrown to entitle the person to re- 
ceive his money, the probability that such 
a face will be thrown being i (according to 
Def. 1), it follows that the value of his in- 
terest before the trial is made, or, which is 
the same thing, that his expectation is equal 
to one-sixth of a guinea, or 3s. 6d. AFere 
his receiving the money to depend on his 
throwing either of two faces, his expecta- 
tion would be equal to two sixths of a 
guinea, or 7s. And, in general, supposing 
the present value of the money or thing to 
be received to be A, the probability of the. 
event’s happening to be denoted by w, and 
of its failing by b, the expectation will be 
Aa , . A b 
either expressed by ^ o*' oy ^ 
according as it depends either on the event s 
happening, or on its failing. 
Def. 3. Events are independent, w'hen 
the happening of any one ot them does 
neither increase nor lessen the probability 
of the rest. Thus, if a person undertook 
with a single die to throw an ace at two 
successivKi trials, it is obvious (however his 
expectation may be affected) tliat the pro- 
bability of liis throwing an ace in the one 
is neither increased nor lessened by the re- 
sult of the other trial. 
Tlieor. The probability that two subse- 
quent events will both happen, is equal to 
the product of the probabilities of the hap- 
pening of those events considered sepa- 
rately. 
Suppose the chances for the happening 
and failing of the first event to be denoted 
by b, and'those for its happening only to be 
denoted by a. Suppose, in like manner, 
the chances for the second event’s happen- 
ing and failing to be denoted by d, and those, 
for its happening only by c ; then will the 
probability of the happening of eacli of 
those events, separately considered, be (ac- 
cording to Def. 1 ) ^ and ~ respectively. 
Since it is necessary that the first event 
should happen before any tiling can be de- 
termined in regard to the second, it is evi- 
dent that the expectation on the latter must 
be lessened in proportion to the improbabi- 
lity of the former. Were it certain that the 
first event would happen, in other w'ords, 
were a = 6, o»' | = ‘I*® expeetatioii on 
c 
the second event would be = But if a 
is less than b, and the expectation on the 
second event is restrained to the contin- 
gency of its having happened the first time. 
