[ 378 ] 
p 
i* N • Then (by definition 5) the value of his ex- 
pedition is P, and therefore if the event fail, he lofes 
that which in value is P ; and if it happens he re- 
ceives N, hut his expedition ceafes. His gain there- 
fore is N — P. Likewife fince the probability of the 
P 
event is — , that of its failure (by corollary prop. 1) 
. N p ^ p XT p 
1S ~ N ~' ^ Llt N * S t0 ”n~ aS ^ * S t0 ^ u e * 
the probability of the event is to the probability of it’s 
failure, as his lofs if it fails to his gain if it happens. 
PROP. 3. 
The probability that two fubfequent events will 
both happen is a ratio compounded of the probabi- 
lity of the 1 ft, and the probability of the 2d on fup- 
pofttion the ift happens. 
Suppofe that, if both events happen, I am to receive 
p 
N, that the probability both will happen is ^ , that 
the 1 ft will is — (and confequently that the ift will 
N a w 
not is — — ) and that the 2d will happen upon fup- 
pofttion the 1 ft does is jq-. Then (by definition 5) P 
will be the value of my expedation, which will be- 
come b if the ift happens. Confequently if the ift 
happens, my gain by it is b — P, and if it fails my lofs 
is P. Wherefore, by the foregoing propofition, — is to 
i. e. a is to N — a as P is to b — P. Where- 
fore (componendo inverfe) a is to N as P is to b. 
But the ratio of P to N is compounded of the ratio 
of P to b , and that of b to N. Wherefore the 
c fame 
