[ 377 ] 
Suppofe there be three fuch events, and whichever 
of them happens I am to receive N, and that the pro- 
bability of the i ft, 2d, and 3d are refpeCtively ^ 
1 , 1 . Then (by the definition of probability) the va- 
lue of my expectation from the ift will be a , from 
the 2d b , and from the 3d c. Wherefore the value 
of my expectations from all three will b e a-\~ b c. 
But the fum of my expectations from all three is in 
this cafe an expectation of receiving N upon the hap- 
pening of one or other of them. Wherefore (by de- 
finition 3) the probability of one or other of them is 
or A 4 - A _U -L. The fum of the proba- 
N N 1 N 1 N A 
bilities of each of them. 
Corollary. If it be certain that one or other 
of the three events muft happen, then a -j- b -f- c 
— N. For in this cafe all the expectations to- 
gether amounting to a certain expectation of re- 
ceiving N, their values together muft be equal 
to N. And from hence it is plain that the proba- 
bility of an event added to the probability of its fai- 
lure (or of its contrary) is the ratio of equality. For 
thefe are two inconftftent events, one of which ne- 
ceftarily happens. Wherefore if the probability of 
P N — P 
an event is — that of it's failure will be x t-» 
P R O P. 2. 
If a perfon has an expectation depending on the 
happening of an event, the probability of tire event 
is to the probability of its failure as his lofs if it fails to 
his gain if it happens. 
Suppofe a perfon has an expectation oi receiving 
N, depending on an event the probability oi which 
is 
